Oscillation Theory for Functional Differential Equations

Theorem 1.1.1. Assume that (1.1.10) and (1.1.11) hold. Let t 0 ~ t 0 and C(t_ 1 , t 0 ),Rm) be given. Then the initial value problem (1.1.9) and (1.1.13) has exactly one solution in the interval [t 0 , oo ).

In oscillation theory we only discuss those solutions y of (1.1.9) which exist on some ray [T11 ,oo) and satisfy sup {ly(t)l: t ~ T} > 0 for every T ~ T11 •

1.2. Oscillation and Nonoscillation We first give the definition of the oscillation of solutions for the scalar differential equation, for example (x(t)

+ p(t)x(t- r))' + Q(t)x(t- u) = 0,

t ~to.

(1.2.1)

A nontrivial solution x of Eq. (1.2.1) is said to be oscillatory if x has arbitrarily large zeros. Otherwise, x is said to be nonoscillatory. That is, x is nonoscillatory if there exists a t 1 > t 0 such that x(t) '=I 0 for t ~ t 1 . In other words, a nonoscillatory solution must be eventually positive or eventually negative. When we say every solution of Eq. (1.2.1) is oscillatory, we mean that for every initial point t 1 ~ t 0 and every initial function

5

Preliminaries

to prove that Eq. (1.2.1) has a nonoscillatory solution, it is sufficient to prove that there exists a point t 2 2:: t 1 2:: t 0 such that x is a solution on [t 2, oo) and x(t) > 0 for all t 2:: t2 or x(t) < 0 for all t 2:: t2. For equations with complex deviating arguments we do not have existence and uniqueness theorems for global solutions. In fact, the formulation of the initial value problem for such equations is not always clear. In these cases we say all solutions oscillate in case every continuous function which satisfies the equation on some infinite interval [r, oo) has arbitrarily large zeros and we do not care about the formulation of the initial value problems. For the system of first order equations with deviating arguments of the form (1.1.9) the solution (x 1 (t), ... ,xm(t))T is said to be oscillatory (strongly oscillatory) if at least one (each one) of its components is oscillatory. We shall now present some examples to describe the oscillation and nonoscillation problems in this book.

Example 1.2.1. The delay equation y I (t)

+y

(

7r = 0 t - -)

2

(1.2.1)

has an oscillatory solution y(t) = sint which is caused by the delay.

Example 1.2.2. Consider the system x'(t) = 2x(t)- y(t) y'(t)

= x(t) + y(t).

(1.2.2)

From the analysis of the characteristic equation we know that every solution (x(t),y(t)) of (1.2.2) oscillates. But the delay system

x'(t) = 2x(t)- y(ty'(t) = x(t-

~en 4)

~en 4) + y(t)

(1.2.3)

has a nonoscillatory solution x(t) = exp((3/2)t), y(t) = exp((3/2)t). This nonoscillation is caused by delay. Therefore we must study the effect of deviating arguments on the oscillation of systems.

Chapter 1

6

Example 1.2.3. In Chapter 2, we shall show that the equation

x'(t)+px(t-1)=0,

t"2_to,

p>O

(1.2.4)

has a nonoscillatory solution if and only if pe

~

(1.2.5)

1.

In Chapter 3, we shall prove that every solution of (x(t)- cx(t- 1))'

+ px(t -1) = 0,

t "2. to

(1.2.6)

is oscillatory if pe>1-c,

cE(0,1).

(1.2.7)

Assume that 1- c < pe ~ 1, c E (0, 1), then the oscillations of (1.2.6) are caused by the neutral term. If pe > 1, then every solution of (1.2.4) oscillates. In Chapter 3, we shall show that if c < 0, pe > 1, then Eq. (1.2.6) has a nonoscillatory solution. This nonoscillation is caused by the neutral term. Therefore the presence of the neutral term can cause or destroy the oscillation of the solutions. In this book we will investigate systematically the effect of deviating arguments and neutral terms on the oscillation of systems in the sense that their presence causes or destroys the oscillation of the solutions. We shall present many of the recent developments in the following problems ( 1) (2) (3) (4) (5)

Criteria for all solutions to be oscillatory. The classification and existence of nonoscillatory solutions. The existence of oscillatory solutions. The distance between adjacent zeros of the oscillatory solutions. Some applications to ecology models.

1.3.

Formulation of Boundary Value Problems for Functional Differential Equations

The main sources of boundary value problems for ordinary differential equations without deviating arguments are boundary value problems for partial differential equations.

7

Preliminaries

For functional differential equations the situation is different. The usual sources of boundary value problems for functional differential equations arise from variational problems for these equations. For example, we consider the problem of an extremum of the functional

V(x(t)) =

i

t,

F(t,x(t),x(t- r),x(t),:i:(t- r))dt

(1.3.1)

to

with fixed or movable boundary values and initial functions. These variational problems lead to the following simple boundary value problem statements. 1. Find the continuous and smooth solutions of the equation

x"(t) = f(t,x(t),x(t- r1(t)), ... ,x(t- Tm(t))), x'(t),x'(t- r1(t)), ... ,x'(t- rm(t)), x"(t- r1(t)), ... , x"(t- Tm(t))

(1.3.2)

fortE (to, tl) having a given initial function
x"(t) = f(t,xt,x'(t))

0~t ~T

(1.3.4)

xo =
x(T) =A where x 1(8) = x(t

+ 8),

8 E [-r, 0].

(1.3.5)

Chapter 1

8

By a solution of the BVP (1.3.4) and (1.3.5) we mean a function x E

C([-r, T], Rn)nC 2 ([0, T], Rn) which satisfies the equation (1.3.4) and the boundary conditions (1.3.5). As in ordinary differential equations the BVP (1.3.4) and (1.3.5) can be transformed into the integral equation

{T

I

x(t)= { Jo G(t,s)f(s,x 8 ,x(s))ds+

A- cp(O)

T

t+cp(O), 0


~t

-r ~

cp(t)

t

~

(1.3.6) 0

where G(t,s) is the Green's function for the BVP

x"(t) = 0,

x(O) = 0,

x(T) = 0

and is given by the formula G(t,s)=.!_{(t-T)s, T t( s - T),

O~.~~t~T 0 ~ t ~ s ~ T.

(1.3.7)

Then the BVP (1.3.4) and (1.3.5} reduces to finding a fixed point for the mapping S defined by

{T

I

(Sx)(t)= { Jo G(t,s)f(s,x.,x(s))ds+

cp(t},

A - cp(O) T

t+cp(O),

O~t=::;T -r ~

t

~

0.

The method which is used to prove that the equation x = Sx has a solution, is usually the application of a suitable fixed point theorem.

1.4.

Fixed Point Theorems

We will introduce some fixed point theorems which are employed throughout the book. A vector space X on which we have defined a norm II· II is called a normed vector space. A subset S of a normed vector space X is said to be bounded if there is a number M such that llxll ~ M for all x E S. A subset S of a normed vector space X is called convex if, for any x, y E S, as+ (1 - a)y E S for all a

E [0, 1].

A sequence {xn} in a normed vector space X is said to converge to the vector x EX if and only if the sequence llxn- xll converges to zero as n-+ oo.

Preliminaries

9

A sequence {xn} in a normed vector space X is a Cauchy sequence in X if for every c > 0 there exists an N = N(c) such that llxn- xmll < c for all n,m ~ N(c). Clearly a convergent sequence is a Cauchy sequence but the converse may not be true. A space X where every Cauchy sequence of elements of X converges to an element of X is called a complete space. A complete normed vector space is said to be a Banach space. Let M be a subset of a Banach space X. A point x E X is said to be a limit point of M if there exists a sequence of vectors in M which converges to x. We say a subset M is closed if M contains all of its limit points. The union of M and its limit points is called the closure of M and will be denoted by M. Let N, M be normed spaces, and X be a subset of N. An operator T: X --t M is continuous at a point x E X if and only if for any c > 0 there is a 6 > 0 such that IITx- Tyll < c for ally EX such that llx- Yll < 6. Tis continuous on X, or simply continuous, if it is continuous at all points of X. The following result is well-known. Theorem 1.4.1. Every continuous mapping of a closed bounded convex set in Rn into itself has a fixed point.

A subset S of a Banach space X is compact, if and only if every infinite sequence of elements of S has a subsequence which converges to an element of S. We say M is relatively compact if every infinite sequence in S contains a subsequence which converges to an element in X. That is, M is relatively compact, if M is compact. A subset Sin C([a, b], R) with norm

11!11

= sup lf(x)l :rE[a,b]

is relatively compact if and only if it is uniformly bounded and equicontinuous on [a, b] (Arzela-Ascoli Theorem). A family S in C([a, b], R) is called uniformly bounded if there exists a positive number M such that

lf(t)l :5 M

for all

t E [a, b],

all

f

E S.

Sis called equicontinuous iffor every c > 0 there exists a 6 = 6(c) > 0 such that lf(tt)- j(t2)l < c for all t~, t2 E [a, b] with lt1 - t2l < 6 and for all f E S.

Chapter 1

10

Theorem 1.4.2. (Schauder's Fixed Point Theorem). Let S be a closed, convex and nonempty subset of a Banach space X. LetT : S --1 S be a continuous mapping such that TS is a relatively compact subset of X. Then T has at least one fixed point in S. That is, there exists an x E S such that Tx = x. Remark 1.4.1. In oscillation theory we usually want to prove that the family of functions is uniformly bounded and cquicontinuous on [to, +oo ). According to Levitan's result, the family S is equicontinuous on [to, oo) if for any given c; > 0, the interval [to, oo) can be decomposed into a finite number of subintervals in such a way that on each subinterval all functions of the family S have oscillations less than c. A topology T on a linear space E is called locally convex if and only if every neighborhood of the element 0 includes a convex neighborhood of 0. A real valued function p( x) defined on a linear space X is called a semi norm on X iff the following conditions are satisfied:

p(x

+ y) ~ p(x) + p(y)

p(ax)

=

jajp(x),

a

is scalar.

From this definition, we can prove that a seminorm p(x) satisfies p(O) = 0, p(x 1 - x 2 ) ~ jp(x 1 ) - p(x 2 )j. In particular, p(x) ~ 0. However, it may happen that p(x) = 0 for x f 0. A family P of seminorms on X is said to be separating iff to each x =/= 0 there corresponds at least one p E P with p( x) =/= 0. For a separating seminorm family P, if p( x) = 0 for every p E P, then x = 0. A locally convex topology T on a linear space is determined by a family of seminorms {Pa- : a E I}, I being the index set. Let E be a locally convex space, x E E, {xn} E E. Then and only if Pa-(Xn- x) --1 0 as n --1 oo, for every a E I.

Xn --1

x in E if

A set S C E is bounded if and only if the set of numbers {Pa(x), xES} is bounded for every a E I. A complete metrizable locally convex space is called a Frechet space.

Theorem 1.4.3. (Schauder-Tychonov Theorem). Let X be a locally convex linear space, let S be a compact convex subset of X, and let T : S - t S be a continuous mapping with T(S) compact. Then T has a fixed point in S.

11

Preliminaries

For example C([t 0 , oo ), R) is a locally convex space consisting of the set of all continuous functions. The topology of C is the topology of uniform convergence on every compact interval of [to, oo ). The seminorm of the space

C([t 0 , oo ), R) is defined by p0

(x)

=

max Jx(t)J,

:r:E[to,<>]

x E C,

a E (to,oo).

Let X be any set. A metric in X is a function d : X x X following properties for all x, y, z EX :

-+

R having the

1. d(x,y) ~ 0 and d(x,y) = 0 if and only if x = y 2. d(y,x) = d(x,y) 3. d(x,z) :S d(x,y) + d(y,z) A metric space is a set X together with a given metric in X. A complete metric space is a metric space X in which every Cauchy sequence converges to a point in X. Let (X, d) be a metric space and let T : X -+ X. If there exists a number r E [0, 1) such that d(Tx,Ty) :S rd(x,y)

for every

x,y EX,

then we say T is a contraction mapping on X.

Theorem 1.4.4. (The Banach Contraction Mapping Principle). A contraction mapping on a complete metric space bas exactly one fixed point. Theorem 1.4.5. Banach space, let maps of Q into X contraction and B

(Krasnoselskii's Fixed Point Theorem).

Let X be a

Q be a bounded closed convex subset of X and let A, B be

such that Ax + By E Q for every pair x, y E Q. If A is a is completely continuous, then the equation Ax +Ex =x

bas a solution in

n.

A nonempty and closed subset ]{ of a Banach space X is called a cone if it possesses the following properties:

Chapter 1

12 (i) if a E R+ and x E K, then ax E K (ii) if x, y E K then x + y E K (iii) if x E K- {0}, then -x fl. K.

We say that a Banach space X is partially ordered if X contains a cone K with a nonempty interior. The ordering :5 in X is then defined as follows: x :::; y

if and only if

y- x E K.

Let M be a subset of a partially ordered Banach space X. Set

M == {x E X : y $ x for every

y E M}.

We say that the point x 0 E X is the infimum of M if xo E M and for every x EM, x 0 :5 x. The supremum of M is defined in a similar way. Theorem 1.4.6. (Knaster's Fixed Point Theorem). Let X be a partially ordered Banach space with ordering :5 . Let M be a subset of X with the following properties: The infimum of M belongs to M and every nonempty subset of M has a supremum which belongs to M. Let T : M ~ M be an increasing mapping, i.e., x :5 y implies Tx :::; Ty. Then T has a fixed point in M. Let X be a Banach space let K be a cone in X, and let

:5 be the order in

X induced by K, i.e, x :::; y if and only if y - x is an element of K. Let D be a subset of K, and T: D ~ K a mapping. We denote by (x, y) the closed order interval between x andy, i.e., (x,y) == {x E Z: x $ z :5y}. We assume that the cone K is normal in X, which implies that order intervals are norm bounded. The cones of nonnegative functions are normal in the space of continuous functions with supremum norm and in the space Lp. Theorem 1.4. 7. Let X be a Banach space, K a normal cone in X, D a subset of K such that if x, y are elements of D, x $ y, then ( x, y) is contained in D, and let T : D ~ K be a continuous decreasing mapping which is compact on any closed order interval contained in D. Suppose that there exists x 0 E D such

Preliminaries

13

that T 2 x 0 is defined (where T 2 xo = T(Tx 0 )) and further more Tx 0 , T 2 x 0 are (order) comparable to xo. Then T has a fixed point in D provided that either:

(I) Txo $ x 0 and T 2 xo $ xo or Txo ;::: xo and T 2 xo ;::: xo, or (II) The complete sequence of iterates {Tnxo}~=O is defined and there exists Yo E D such that Tyo E D and Yo $ Tnxo for all n.

Theorem 1.4.8. Let X be a Banach space, A : X -+ X be a completely continuous mapping such that I - A is one to one, and let n be a bounded set such that 0 E (I- A)(!l). Then the completely continuous mappingS: n-+ X has a fixed point in 0 if for any .X E (0, 1), the equation

x = .XSx + (1 - .X)Ax has no solution

X

on the boundary

an of il.

The following theorem is a very useful version of the Topological Transversality Theorem of Granas. Theorem 1.4.9. Suppose that X is a normed space (possibly incomplete). Let S be a bounded, closed, convex subset of X, containing the origin 0 in its interior D. Let H : [0, 1] x S-+ X be a homotopy of compact transformations such that H(O, as) c sand H(t, X) =F X (0 $ t $ 1, X E as). Then tp = H(1,.) has a fixed point in S. Theorem 1.4.10. (Leray-Schauder Alternative). Let C be a convex subset of a normed linear space E and assume 0 E C. Let F : C -+ C be a completely continuous operator and let

£(F)= {x E C: x = .XFx

for some

0

Then either £(F) is unbounded or F has a fixed point.

< .X < 1}.

2 Oscillations of First Order Delay Differential Equations

2.0. Introduction In this chapter, we will describe some of the recent developments in the oscillation theory of first order delay differential equations. This theory is interesting from the theoretical as well as the practical point of view. It is well known that homogeneous ordinary differential equations (ODEs) of first order do not possess oscillatory solutions. But the presence of deviating arguments can cause the oscillation of solutions. In this chapter, we will see these phenomena and we will show various techniques used in the oscillation and nonoscillation theory of differential equations with deviating arguments. We will present some criteria for oscillation, for the existence of positive solutions and results in the distribution of zeros of oscillatory solutions of DDEs of first order. In the last section we will discuss the oscillation and nonoscillation behavior of some basic ecological delay equations.

14

15

Oscillations of First Order Delay Equations

2.1. Stable Type Equations with a Single Delay 2.1.1.

Oscillation results

We consider the linear delay differential inequalities and equations of the form

x'(t)

where p, T E Set

+ p(t)x(r(t))

::=; 0

(2.1.1)

x'(t) + p(t)x(r(t)) 2:: 0

(2.1.2)

x'(t) + p(t)x(r(t)) = 0

(2.1.3)

C([to,oo),~),

r(t) ::=; t and lim r(t) = oo.

R+ = [O,oo),

1--+oo

m = liminf [' p(s)ds 1--+oo

(2.1.4)

1T(I)

and

M =lim sup [' p(s)ds.

(2.1.5)

1T(I)

1--+oo

The following lemmas will be used to prove the main results in this section. All the inequalities in this section and in the latter parts hold eventually if it is not mentioned specifically. Lemma 2.1.1.

Set

6(t)

= max {T(s);

s E [t 0 ,tl}

(2.1.6)

and m > 0. Then we have lim inf [' p( s )ds = lim inf [' p( s )ds = m. l--+oo

16(1)

t--+oo

1 T(l)

Proof: Clearly, 6(t) ;::: r(t) and so

[' p(s)ds ::=; 16(t)

1 1

T(l)

p(s)ds.

(2.1. 7)

16

Chapter 2

Hence liminf [' p(s)ds::; liminf [' p(s)ds. t-+oo

}6(t)

t-+oo

If (2.1.7) does not hold, then there exist m' tn - oo as n --t oo and

lim n-+oo

l tn

6(tn)

}r(t)

> 0 and a sequence {tn} such that

p(s)ds::; m' < m.

By definition, li( tn) = max ( r( s) : s E (to, tnl), and hence there exists t~ E [to, tn] such that li(tn) = r(t~). Hence

l tn

p(s)ds =

ltn

6(tn)

p(s)ds >

lt~

r(t:,)

p(s)ds.

r(t:,)

It follows that {J:rt:.)p(s)ds}~=l is a bounded sequence having a convergent subseque nce, say

l

t~.

p(s)ds- c::; m',

as

k--. oo,

r(t:..>

which implies that liminf [' p(s)ds::; m', t-+oo

lr(t)

contradic ting (2.1.4). Lemma 2.1.2. then

D

Let x(t) be an eventually positive solution of (2.1.1). Hm > ~ ,

lim x(r(t)) = oo. t-+oo

x(t)

(2.1.8)

H m ::; ~' then liminf x(r(t)) >A t-+oo

x(t)

-

(2.1.9)

17

Oscillations of First Order Delay Equations

where A is the smaller positive root of the equation (2.1.10)

Proof: Let t 1 be a sufficiently large number so that x(r(t)) Hence

> 0 fort !:: t1.

x'(t) < -p(t) x(r(t)) < -p(t). x(t) x(t) -

(2.1.11)

Integrating it from r(t) tot we have that eventually

r

x(r(t)) ;:::: exp( p(s)ds). x(t) lr(t) Then for any e > 0, there exists Te such that x(r(t))

~!::e

m

-e,

t!::Te.

Substituting (2.1.12) into (2.1.11) we have ~N? ~ -(em - e)p(t), hence

. . x(r(t))

limmf t-+oo

X

(t )

(2.1.12)

t !:: Te, and

!:: exp(m expm).

Set Ao = 1, A1 = exp(m>.o), ... ,>.n = exp(mAn-1), .... For a sequence {en} with en > 0 and en --+ 0 as n --+ oo, there exists a. sequence {tn} such that tn --+ oo as n--+ oo and (2.1.13) If m > 1, then n-..oo lim An= oo, and (2.1.8) holds. If m = 1, then lim An= e; and e e n-+oo if m < ~. then An tends to the smaller root of Eq. (2.1.10). 0

Remark 2.1.1. From Theorem 2.1.1 we will see that (2.1.1) has no eventually positive solutions if m > ~ .

Chapter 2

18

Lemma 2.1.3. In addition to the hypotheses of Lemma 2.1.1, assume T is nondecreasing, 0 :5 m :5 ~ , and x(t) is an eventually positive solution of (2.1.1). Set

. 'nf r = 11m1 t.-oo

x(t) t

(2.1.14)

- (( )) . X T

Then 2

A(m) := _1_-_m_-_v __ 1_-_2_m_-_m_

2

< r < 1. -

-

(2.1.15)

Proof: Assume that x(t) > 0 fort > T1 ;::: t 0 , and there exists a sequence {Tn} such that T1 < Tz < T3 < ... and r(t) > Tn fort> Tn+1, n = 1, 2, .... Hence x(r(t)) > 0 fort> T2 • In view of (2.1.1), x'(t) :50 on (T2 , +oo). Clearly, (2.1.15) holds form= 0. If 0 < m :5 ~ , for any c E (0, m), there exists Ne such that

t

p(s)ds>m-c,

for

t>Ne.

(2.1.16)

JT(t)

For a fixed e we will show that for each t > Ne there exists a At such that r(At) < t < At and {At

lt In fact, for a given t,

(2.1.17)

p(s)ds = m- c.

f(A) := ftA p(s)ds is continuous and lim /(A)> m-e> A-+oo

0 = f(t). Hence there exists At > t such that j(>.t) = m- c, i.e., (2.1.17) holds. From (2.1.16) we have

1

At

p(s)ds >m-e=

T(At)

!At

p(s)ds,

f

therefore r(At) < t. Integrating (2.1.1) from t (> max {T4 ,Ne}) to At we have At

x(t)- x(At);:::

1 t

p(y)x(r(y))dy.

We see that r(t) ~ r(y) ~ r(At) < t fort~ y ~At.

(2.1.18)

19

Oscillations of First Order Delay Equations

Integrating (2.1.1) from r(y) tot we have that fort::;; y::;; At

x(r(y))- x(t) 2::

lt

p(u)x(r(u))du

r(y)

2:: x(r(t))

lt (lt

p(u)du

r(y)

= x(r(t))

p(u)du

r(y)

-fy p(u)du) t

> x (r( t)) [(m - c) -

J.Y p( u )dul·

(2.1.19)

From (2.1.18) and (2.1.19) we have

x(t) <:: x(A,) +

J."' p(y)x(r(y))dy

> x(At) + [ ' py{ x(t) + x( r(t)) = x(.At)

[(m- c)- [

p(u)du]} dy

+ x(t)(m- c)

+ x(r(t))

[(m- c)'-[' p(y) [

p( u) du dy

l·

Noting the known formula

f At dy fy p(y)p(u)du = fAt du 1At p(y)p(u)dy, t

t

t

u

or

At ly 1At 1At 1 dy p(y)p(u)du = dy p(y)p(u)du, t

t

t

Y

(2.1.20)

Chapter 2

20

we have

1A, 1" t

dy t p(y)p(u)du

=

1A' 1"t 11A' 1A' 2

=

1 [ [A' ] 2 lt p(s)ds

1[ t =2 t

dy

dy t

1A' }y{A'

p(y)p(u)du + t

dy

p(u)p(y)du

]

p(y)p(u)du 2

=

1

2(m- e?.

Substituting this into (2.1.20) we have

x(t) > x(.At) + x(t)(m- e)+

t (m- e)

2

x(r(t)).

{2.1.21)

Hence

x(t) (m-e? x(r(t)) > 2(1- m +e)

d :=

(2.1.22)

b

and then

Substituting this into (2.1.21) we obtain

x(t) > x(t)(m + d1- e)+

t {m- e) 2 x(r(t)),

and hence

x(t) {m-e) 2 x(r(t)) > 2{1- m- d1 +e)

d :=

2.

In general we have

x(t) {m-e) 2 x(r(t)) > 2(1-m-dn+e) :=dn+I>

n= 1' 2 ' " ' '

It is not difficult to see that if e is small enough, 1 ~ dn Hence lim dn = d exists and satisfies

> dn-l,

n--+oo

-2d2 + 2d{1- m +e)= (m- e) 2 ,

n = 2, 3, ....

Oscillations of First Order Delay Equations

21

i.e., d= 1-m+e± y'1-2(m-e)-(m-e) 2 2

Therefore, for all large t

x(t) x(r(t))

-----> Letting e

-+

1-m+ e- y'l- 2(m- e)- (m- e) 2 ----~----~--~----~~--~2

0, we obtain that

x(t) 1-m-v'l-2m-m2 x(r(t)) ;::: 2 = A(m). D

This shows that (2.1.15) holds. Assume that 0 < M $ 1, eventually positive solution of (2.1.1). Set

Lemma 2.1.4.

r is nondecreasing. Let x(t) be an

liminf x(r(t)) =f.. t-+oo

x( t)

Then (2.1.23)

Proof: For a given e E (O,M), there exists a sequence {tn} such that tn-+ oo as n-+ oo and

1

tn

p(s)ds >M-e,

tn

> T,

n = 1, 2, ....

r(tn)

Set ()~ = 1- y'1- (M-e) . It is easy to see that 0 < e. Hence there exists Pn} such that r(tn) < An < tn and

n = 1,2, ....

()E

<M-e for small

22

Chapter 2 Integrating (2.1.11) from An to tn we obtain

Similarly, we have

x(r(tn))- x(An) ~

!.

>. ..

p(s)x(r(s))ds

r(t,.)

~ x(r(An))

!.

>. ..

p(s)ds

r(t,.)

= x(r(An)) [

['" p(s)ds- ['" p(s)ds]

Jr(t,.)

J>.,.

> x(r(An))[(M- c)- 8,.]. From the above inequalities we get

x(An) > 8,.x(r(tn))

> 8,.[x(An) + x(r(An))(M- c- 8,.)] and then

x(r(An)) 1-8,. x(An) < 8,.(M- c- 8,.) '

n

= 1,2, ...

which implies that

e<

1- 8,. " ) , - 8,. ( M- c- u_.

When c -+ 0,

e<

8,.

-+

cE

(0 M) ,

.

1 - V1 - M , then we obtain

~

-(1+~)2

- (1 - V1 - M)(M- 1 + V1 - M) -

M

.

We are now in a position to state the oscillation criteria for Eq. (2.1.3).

0

23

Oscillation& of First Order Delay Equations

Theorem 2.1.1. Assume m > ~. Then (i) (2.1.1) has no eventually positive solutions, (ii) (2.1.2) has no eventually negative solutions, (iii) every solution of Eq. (2.1.3) is oscillatory.

Proof: It is sufficient to prove (i), (ii) and (iii) follow from (i). Suppose the contrary is true, and let x(t) be an eventually positive solution of (2.1.1). By Lemma 2.1.1, we may assume that T is nondecreasing. In view of Lemma 2.1.2,

x(r(t))

lim inf _(_)_ = oo. t-+oo

On the other hand, from (2.1.22), proves (i).

X

t

z(~~;>)

is bounded above. This contradiction D

Remark 2.1.2. If in addition to the conditions of Theorem 2.1.1, r is nondecreasing, ME (0, 1]. Then the condition m > ~can be replaced by ln b m > -b-,

b =min (e,B(M)),

(2.1.24)

where B(M) is defined by (2.1.23). In fact, under the above assumptions Lemma 2.1.4 holds. Let x(t) be a positive solution of (2.1.1) and w(t) = i :5 B(M). From (2.1.1), we obtain

. By Lemma 2.1.4, liminfw(t) = z((<,)>) z

x'(t) - x(t) ~ p(t)w(t),

t~~

t ~ T.

Integrating it from r(t) tot we obtain lnw(t)

~

[' p(s)w(s)ds = w(e,)

r

lr(t)

lr(t)

and hence liminf lnw(t) :2::: im t-+oo

p(s)ds,

24

Chapter 2

and lnf In b m<-<- £ - b which contradicts (2.1.24). Theorem 2.1.2.

0

Assume 0::;

m::;

! , and r

is nondecreasing. Furthermore

M > 1-A(m)

(2.1.25)

where A(m) is defined by (2.1.15), or M

where

>

1n>.+1 >.

(2.1.26)

>. is the smaller positive root of the equation (2.1.27)

Then the conclusion of Theorem 2.1.1 is true. Proof: The same as in Theorem 2.1.1 it is sufficient to show that under our assumptions (2.1.1) has no eventually positive solutions. Integrating (2.1.1) from r(t) tot we obtain

x(r(t))- x(t)

~

lt

rOO

p(s)x(r(s))ds

~ x(r(t))

(

h~)

p(s)ds.

Then if (2.1.25) holds, by Lemma 2.1.3, we have

M =lim sup ( t-oo

Jr( t)

p(s)ds::; lim sup t-+oo

. .

x(t) = 1- hmmf (( )) t ..... oo X T t

[1-

(x((t)))] X T

t

= 1- r::; 1- A(m),

(2.1.28)

which contradicts with (2.1.15). If (2.1.26) holds, choose m 1 < m sufficiently close tom such that

M =lim sup t--+oo

l

t

r(t)

p(s)ds >

ln >.' + 1 >.'

(2.1.29)

Oscillations of First Order Delay Equations

25

where ).' is a smaller root of the equation >. = em' A. Clearly >. 1 < >. and hence In ~;+ 1 > In ~± 1 • By Lemma 2.1.2, we have

x~~?)

> >. 1 for all large t.

(2.1.30)

From (2.1.29), there exists a t 1 so large that (2.1.30) holds for all t > r(r(ti)), and

1 tt

p( s )ds >

ln ).' + 1 >.' •

(2.1.31)

r(tt)

Without loss of generality denote t 0 = r(t 1 ). We shall show that x(t) > 0 on [to,td will lead to a contradiction. In fact, let t2 E [to,td be a point at which x(t 0 )/x(t 2) =>..'.If such a point does not exist, take t 2 = t 1 • Integrating (2.1.1) over [t 2,t 1 ] and noting that x(r(t)) ~ x(t 0 ), we have

i

tt

1 p(s )ds $ >. 1

(2.1.32)

•

12

On the other hand, dividing (2.1.1) by x(t) and integrating it over [t 1 , t 2] we have

1.

12

to

1

p(s)ds <--

).1

1 -12

to

x 1(s) ds x( s)

ln >..' =>.' .

(2.1.33)

Combining (2.1.32) and (2.1.33) we get

i

tt

p(s)ds $ ln

~~+ 1

(2.1.34)

to

which contradicts (2.1.31 ).

D

Example 2.1.1. Consider the equation

I( )

x t +

0.85 ;n(2a+cost)x(t-f)=O a71'+v2

where a= 1.137. We have p(t ) =

0.85 ( ;n 2a+cost) a71'+v2

(2.1.35)

26

Chapter 2

and

l

t

p(s)ds =

t-t

0 85 · y'2 (a'lt' + a'lt'+ 2

V2

cos(t-

f)).

Hence . 1'nf m = 11m t--+oo

1' ( t-t

p s )ds

= 0.85 a'lt' - V2 . /i\ = 0.367837 < -1 mr + v2 e

and M = lim sup t--+oo

1'

t-f

p( s )ds = 0.85.

It is easy to see that (2.1.25) holds. Therefore every solution of Eq. (2.1.35) is oscillatory.

2.1.2. The existence of positive solution

First we consider a linear delay differential equation of the form

x'(t) + x(t- r(t)) = 0

(2.1.36)

whererEC([t 0 ,oo),J4)and lim(t-r(t))=oo.SetT0 = inf {t-r(t)}. t~to

t--+oo

Definition 2.1.1. A solution xis called a positive solution with respect to the initial point t 0 , if x(t) is a solution of(2.1.36) on [t 0 ,oo) and x(t) > 0, on [To,oo). Theorem 2.1.3. Eq. (2.1.3) has a positive solution with respect to to if and only if there exists a real continuous function Ao(t) on [T0 , +oo) such that Ao(t) > 0 fort;::: t 0 and

r(t) :5 t- A01 (A 0 (t) -ln Ao(t)), where A0 (t) = Proof: i)

t;:::

(2.1.37)

to,

J;. Ao(s)ds, A0 (u) denotes the converse function of A

Nece~sity.

1

0•

Let x 0 (t) be a positive solution of (2.1.36) with respect to

27

Oscillations of First Order Delay Equations t 0 • That is, x(t) > 0 on [T0 ,+oo). Set xo(t)=xo(To)exp(-

~~ >.o(s)ds), t~To.

(2.1.38)

Then >. 0 (t) satisfies the equation

>.o(t) =

exp(1'

>. 0 (s)ds),

t

t-r(t)

Clearly >.o(t) > 0 for t

~

~ t0 •

(2.1.39)

to. From (2.1.39) we have A0 (t)- A0 (t -r(t)) = In>. 0 (t).

Then

t- r(t) = A0 1 (Ao(t) -In >.o(t)) and r(t) = t - A01 (A 0 (t) -In>.o(t)). ii) Sufficiency . . If there exists a function >.o(t) such that (2.1.37) holds. Then

Ao(t -r(t)) ~ A 0 (t) -In>. 0 (t), and

>.o(t)

~ exp(it

>. 0 (s)ds).

t-r(t)

Define

>.1(t) = {

exp(lt

>.o(s)ds),

>.l(to) + >.o(t)- >.o(to), Clearly, >.1(t) ~ >.o(t),

t

t

t-r(t)

~

~to

t E [To, to).

T0 and 0 ~ >. 1(t) ~ >.0 (t),

t

~ t0 •

28

Chapter 2

In general, we define

An-t(s)ds), t ~to An(to) + Ao(t)- Ao(to), t E [To, to).

An(t) = { exp

(JLr(t)

(2.1.40)

Thus

and An(t)

~

0,

t

~to.

lim n-+oo

Then limn-+oo An(t) = A(t),

it

=it

An(s)ds

t-r(t)

t

A(s)ds,

~To

t

t-r(t)

exists and

~to.

Hence .A(t)=expi

1

.A(s)ds,

t2::to.

t-r(t)

Set

~~ A(s)ds),

t2::To.

(2.1.41)

Then x is a positive solution of (2.1.36) with respect to t 0 •

0

x(t)=exp(-

Remark 2.1.3. (2.1.37) becomes

We take A(t) = A > 0 in Theorem 2.1.3. Then condition

r(t) :5

ln A

T

,

t ~to.

(2.1.42)

t 2: to,

(2.1.43)

In particular, set A= e. (2.1.42) becomes 1

r(t) :5 - , e

i.e., (2.1.43) is a sufficient condition for the existence of positive solutions of (2.1.36). Let t 0 = ~ , A(t) = 2t. Then by Theorem 2.1.3, if

r(t) = t- vft 2 -In 2t

(2.1.44)

29

Oscillations of First Order Delay Equations then Eq. (2.1.36) has a positive solution with respect to to = e-t• is such a solution. We note that

t . In fact, x(t) =

e 2e- V(e)2 2 - 1 > ;1 ·

(2.1.45)

r( 2) =

This example shows that (2.1.43) is not necessary for the existence of a positive solution of (2.1.36). We now consider the linear equation of the form

x'(t) + p(t)x(t- r(t)) where p, r E C([to, oo ), .R.r ),

=0

(2.1.46)

r(t) $ t and lim (t- r(t)) = oo. t-+oo

Set T0 = inf {t- r(t)}. t>to

Applying;; similar idea to Theorem 2.1.3 we can prove the following results. Theorem 2.1.4. Eq. (2.1.46) has a positive solution with respect to t 0 if and only if there exists a continuous function >. 0 (t) on (T0 , oo) such that >. 0 ( t) > 0 for t ~ t 0 and

.Ao(t)

~ p(t)expjt

.Ao(s)ds,

t

~

t0.

(2.1.47)

t-r(t)

Remark 2.1.4. If p(t)

> 0, then (2.1.47) can be replaced by

-1(Ao(t)

()

r t $ t- A0

Corollary 2.1.1.

-ln .Ao(t)) p(t) ,

t

~to.

(2.1.48)

H

j

t t-r(t)

p(s)ds$~, t~t 0 •

(2.1.49)

Then Eq. (2.1.46) has a positive solution with respect to t 0 . Proof: We take Ao(t) = ep(t), then (2.1.47) is satisfied. Then the corollary follows from Theorem 2.1.4. D

Chapter 2

30

=

Theorem 2.1.5. Assume that r(t) r > 0, ft': p(t)dt = oo. Then Eq. (2.1.46) has a positive solution with respect to to if and only if there exists a continuous function ..\ 0 (t) on [to - r, oo) such that

(2.1.50)

Proof: Set u

=
t;::: t 0 • Then

( 1

.,.-'(u)

)

t-r=
p(s)ds.

u- 1 (u)-r

Denote

z(u)

= x(a-

1

(u)).

Then (2.1.46) becomes

( 1

.,.-t(u)

z'(u)+z u-

)

p(s)ds

=0.

(2.1.51)

u- 1 (u)-r

By Theorem 2.1.3, (2.1.50) is a necessary and sufficient condition for Eq. (2.1.51) to have a positive solution with respect to 0. From the transform, it is equivalent to that Eq. (2.1.46) has a positive solution with respect to to. 0 Remark 2.1.5. If we choose ..\ 0 (t)

i

= e in (2.1.50), then we obtain 1

t

t-r

p(u) du

::5 - , e

t ;::: to.

(2.1.52)

As we have mentioned, (2.1.52) is a sufficient condition and is not a necessary condition for the existence of a positive solution of (2.1.46). Combining Theorem 2.1.1 and (2.1.52) we obtain the following corollary. Corollary 2.1.2. Let p(t) :::: p > 0, r(t) = r > 0. Then a necessary and sufficient condition for all solutions of (2.1.46) to be oscillatory is that pre > 1.

31

Oscillations of First Order Delay Equations

Remark 2.1.6. The above techniques can be used on the first order advanced

type equations

= x(t + r(t))

(2.1.53)

= p(t)x(t + r(t))

(2.1.54)

x'(t) and

x'(t)

where p, T E C((t 0 ,oo),R+)· For example, (2.1.53) has a positive solution if and only if there exists a continuous function ,\ E C(To, oo ), R) such that

r(t):::; A- l (A(t) + ln ,\(t))- t

(2.1.55)

where A(t) = 1: .\(s)ds. If we let .\(t) = ,\ > 0, then (2.1.55) becomes r(t):::; ~, which is a sufficient condition for the existence of positive solutions of (2.1.53). For Eq. (2.1.54), assume

J

liminf t-oo

l

t

t+r(t)

1

p(s)ds >-. e

(2.1.56)

Then every solution of (2.1.54) oscillates. If r(t) = T > 0, then (2.1.54) has a positive solution if and only if there exists a continuous function ,\(t) such that

(+r

lt

p(u)du:::; A- 1 (A(t)+ln ,\(t)) -t.

(2.1.57)

Corollary 2.1.2 is also true for Eq. (2.1.54).

2.2. The Distribution of Zeros of Oscillatory Solutions In this section we will estimate the distances between adjacent zeros of oscillatory solutions of the first order delay differential equation (2.1.46). Let us first consider the differential inequality

x'(t) where

T

> 0 is a constant.

+ x(t- r):::; 0

(2.2.1)

32

Chapter 2

Lemma 2.2.1. Assume that r > ~ . Let x(t) be a solution of (2.2.1) on [Tx, +oo ). If there exists T > to ~ Tx such that x( t) > 0 for t E [to, T], then

w(T) > 1 + ~ (1 + ln r),

if T ~ t 0

T

where w(t)

+ (n + 2)-r,

(2.2.2)

== x~~)) , n is a nonnegative integer.

Proof: If n = 0, from (2.2.1), x'(t) < 0 for t E (to + r, T]. Therefore w(t) = x~~:;.;> > 1 for T ~ t 0 + 2-r. lfn ~ 1, dividing (2.2.1) by x(t) and integrating it from t -r to t 0 we have ln x(t-r) > [' x(s-r) ds. x(t) - lt-r x(s)

(2.2.3)

Hence w(t);:::expjt

t-r

Set w(to) =

min w(s),

T-t~s~T

w(s)ds~exp(r· t-r~s~t min w(s)).

(2.2.4)

toE (T- r, Tj, and noting that

1

er"'~x+-(l+lnr), r

for

xER,

(2.2.5)

+ lnr).

(2.2.6)

i == 1, 2, ... , n- 1.

(2.2.7)

r>O,

we have

w(T)

~

exp (r · w(eo)]

~

1 w(eo) +- (1 T

Define a sequence {to, 6, ... , en-d by

w(e;)

= {minw(s), s E [ti-I- r,e;-1]},

Clearly ei E [ei-1 -

T,

ei-2]· In view of (2.2.4) and (2.2.5) we have (2.2.8)

33

Oscillations of First Order Delay Equations

From (2.2.6) and (2.2.8) we obtain

2 1 w(r) 2:: w(~o) +- (1 + lnr) 2:: w(6) +- (1 + ln r) T

T

2:: · · • 2::

w(~n-1)

n

+-T (1 + ln

r).

It is easy to see that to+ 2r ::; ~n-1 ::; t and hence w(~n-d Therefore (2.2.2) holds.

> 1 by the first step. D

Theorem 2.2.1. Assume that T > ~. x(t) is a solution of (2.2.1) on [T:r, +oo ), and I is any closed interval in [T:r, +oo) with length nor. Then x(t) cannot be always positive on I, where

2 - 2r - r 2 ] no = max { 4, 3 + [ r( 1 + ln T)

}

,

(2.2.9)

and [·] denotes the greatest integer function. Proof: Assume the contrary, then there is a t 1 2:: T.,, such that x(t) > 0 on [t1, t1 +nor]. Set t* = t1 +noT. Integrating (2.2.1) from t* - T tot* we have

r

t•-r

x(t*- r)- x(t*);::::

x(s)ds.

lt*-2r

Define a line

eas follows: z( t) - x(t* - r) = k( t - t*

+ r)

where k = -x( t* - 2r ). Hence z(t*

+ 2r) =

x(t*- r)

+T

•

x(t*- 2r).

(2.2.10)

From (2.2.1), x'(t*- r) < k, and so x(t) lies above the line e at a small leftneighborhood of the point t = t* - r. If there is a point t on [t* - 2r, t* - r] so that x(t) lies below£, then there exists~ E [t*- 2r,t*- r) such that x'(~);:::: k. Hence -x(~- r)

> x'(~) 2:: k

= -x(t*- 2r),

Chapter 2

34

i.e.,

x(e- r) :5 x(t*- 2r).

(2.2.11)

Noting that no ~ 4 and hence t*- 2r > e- T ~ t*- 3r ~ tl + T. On the other hand, x(t) is decreasing fort~ t 1 + r, so x(e- r) > x(t*- 2r), which contradicts (2.2.11). Therefore x(t) lies above the lineR. on [t*- 2r, t*- r]. We see that

1

t•-r

( X S

)d

S

x(t•- r)

>

+ z(t•- 2r)

T.

2

t•-2r

Using (2.2.10) we have

J

t•-r

t•-2r

x(s)ds >

2x(t•- r) + rx(t•- 2r) r. 2

(2.2.12)

By Lemma 2.2.1 no+ 3 (1 + ln r) } x(t*- r). x(t*- 2r) > { 1 + r-

(2.2.13)

Combining (2.2.12) and (2.2.13) we obtain

l

t•-r

x(s)ds>

2+r(1+

no+ 3 (1+lnr))

;

r·x(t*-r).

(2.2.14)

t•-2r

Integrating (2.2.1) from t•- r tot* and using (2.2.14) we obtain { 1- 2 + r( 1 + ; ( 1 + ln r)) r }x(t*- r)- x(t*) > 0 and hence 3 (1+lnr)) } { 1- 2-r(1+ ""; T >0

which contradicts the definition of n 0 as given in (2.2.9). From Theorem 2.2.1 and Lemma 2.2.1 we have the following corollary.

0

Oscillations of First Order Delay Equations

35

Corollary 2.2.1. Assume that T > ~, x(t) is a solution of (2.2.1) and positive and decreasing on [tl> t 1 + T], then x(t) can not always be positive on [t 1 + T, t1 +(no -1)T]. Then we consider

x'(t) + p(t)x(t- T) = 0.

(2.2.15)

By the method of Theorem 2.1.3 and 2.1.5 the results of (2.2.1) can be extended to Eq. (2.2.15) Theorem 2.2.1. Assume that (i) p E C(t 0 ,oo),(O,oo)), p(t) ¢. 0 on any subinterval of[t 0 ,oo), liminfjt p(s)ds=k>!.

(ii)

t-+oo

(2.2.16)

e

t-r

Let x(t) be a solution of (2.2.15) on [T.,, +oo), T., ;::: t 0 • Let c: > 0 be a small number such that k - c: > ~ . Then there exists t~ ;::: T., such that the distances between adjacent zeros of x(t) is less than (1 + ne)T on [t~, +oo ), where

n~

=max{4,

3+ [ 2 - 2(k-c:)-(k-c:)2]}· (k- c:)(1 + ln(k- c:))

Proof: In view of the assumptions there exists

i

t

p(s)ds>k-c:

for

t~ ;::: t 0

(2.2.17)

such that

t;:::te.

(2.2.18)

t-r

Set u = a(t) = ft: p(s )ds and z( u) = x(u- 1 (u)). As in the proof of Theorem 2.1.4, (2.2.15) reduces to

( 1

cr-t(u)

z'(u)z u-

)

p(s)ds

= 0.

(2.2.19)

cr- 1 (u)-r

We shall show that x(t) has zeros on any interval [t1, t 1 + (1 + n.)T], t 1 ;::: t •. If not, x(t) > 0 on [t1, t1 + (1 + ne)T], then x'(t) < 0 on [t 1 + T, t 1 + (1 + n~)T]. Set

Chapter 2

36

Then z(u) > 0 for u ~

Ut

and z'(u)

1

< 0 for u ~ u2 and

.,.-t(u)

u-

p(s)ds

~

u 2 for

u

~

u

~ Ut.

u3 .

.,.-•(u)-r

Since

1

.,.-•(u)

p(s)ds > k- e:,

for

.,.- 1 (u)-r

we have

z'(u) + z(u- (k- e:)) < 0,

u ~ u3.

for

z(u) is positive and decreasing for u ~ u3. By Corollary 2.2.1, z(u) has zeros on [u3, U3 +(n., -1)(k -e:)]. Hence there exists u E [u3, u3 + (n., -1)(k- e:)] such that z(u) = 0. Set t = u- 1 (U), then x(i) = 0. On the other hand, from (2.2.18) we have

i

t 1 +(n.+t)r

p(s)ds > (n., -1)(k- e:).

•+2r

Hence

i

t, +(n.+l)r

to

p(s)ds >

1t

p(s)ds >

1t'

to

+2r

p(s)ds,

t0

which derives that

This implies that x(t)

> 0, contradicting that x(t) = 0.

Remark 2.2.1. Consider the autonomous equation

x'(t)+px(t-r)=O,

p>O,

r>O,

(2.2.20)

where pr > ~ . Then according to Theorem 2.2.2 the distance between adjacent zeros of solutions of (2.2.20) is less than

2- 2pr- (pr) 2 ]} r max { 5,4 + [ pr(l + ln(pr)) .

(2.2.21)

Oscillation.s of First Order Delay Equations

37

2.3. Unstable Type Equations We now consider the first order linear delay differential equation with unstable type of the form

x'(t) = p(t)x(r(t)) where p, r E C([t 0 ,oo),~), Theorem 2.3.1.

r(t)

~

(2.3.1)

t and limt.....oor(t) = oo.

Assume that 00 {

lto

p(t)dt

= 00.

(2.3.2)

Then Eq. (2.3.1) always has an unbounded positive solution, and every bounded solution of (2.3.1) is oscillatory. Proof: The proof of the first part can be seen from Theorem 3.5.1. Now we will prove that every bounded solution of (2.3.1) is oscillatory. Let x(t) be a bounded positive solution of (2.3.1), then x(t) is nondecreasing and bounded. Hence lim x(t) =I.> 0 exists. From (2.3.1), we obtain t-+oo

x'(t)

~ ~ p(t)

for all large t.

(2.3.3)

(2.3.1) and (2.3.2) imply that lim x(t) = oo, contradicting the boundedness of 0 x(t).

Example 2.3.1. Consider the equation

r.::a; 2 ).

y'(t) = 2ty
(2.3.4)

By Theorem 2.3.1, every bounded solution of (2.3.4) is oscillatory. In fact, y(t) = cos t 2 is such a solution. On the other hand, y( t) = e~t 2 is an unbounded positive solution of (2.3.4) where ~ is the unique positive root of the equation ~ =

3 exp (- 2 1r~).

Chapter 2

38

Assume that

Theorem 2.3.2.

roo p(t)dt <

ito

(2.3.5)

00.

Then Eq. (2.3.1) has a bounded positive solution. Proof: Let BC be a Banach space of bounded and continuous functions x : [t 0 , oo) -4 R with the super norm. Define a subset n of BC as follows: Q

= {x E BC:

0 :5 x(t) :51,

to :5 t < oo}.

Clearly, n is a bounded, closed and convex subset of BC. Define a mapping T: fl -4 BC as follows: (Tx)(t) = {

!

+ J;p(s)x(r(s))ds, (Tx)(T),

t;:: T to5t5T.

(2.3.6)

where Tis chosen sufficiently large so that r(t) ;:: t 0 fort ;:: T and

!.

J; p(s)ds :5 Clearly, for

X

En,

0:5 (Tx)(t) :51, i.e., Tfl

JTxt)(t)- (Tx2)(t)J5

c n.

Let

Xt,X2

En, then

£

p(s)Jxl (r(s))- x2(r(s))Jds

1

52Jixl-x2JI,

for

t;::T

and hence

i.e., T is a contraction on fl. Therefore by Banach fixed point theorem there is a X En so that x(t) = and so x'(t) = p(t)x(r(t)), (2.3.1).

r

21 + lr p(s)x(r(s))ds, t ;:: T. Hence x is a bounded positive solution of

0

39

Oscillations of First Order Delay Equations Corollary 2.3.1.

Every bounded solution of (2.3.1) oscillates if and only if

(2.3.2) holds. Corollary 2.3.2.

Eq. (2.3.1) bas an unbounded positive solution if and only

if (2.3.2) holds. For the existence of oscillatory solutions of (2.3.1) we have the following result. Theorem 2.3.3.

Ifr(t) = t- r 1 (t) in (2.3.1) and (2.3.7)

then Eq. (2.3.1) has an oscillatory solution. We consider (2.3.1) with p(t)

=p > 0,

r(t) = t- r,

r > 0.

By the analysis of the characteristic roots we have the following conclusion. Theorem 2.3.4.

Assume that p > 0,

r > 0, then

(i) if

then every oscillatory solution x(t) of (2.3.1) tends to zero as t-+ oo; (ii) if

then every oscillatory solution of (2.3.1) is bounded on R+, and (2.3.1) bas an oscillatory solution which does not tend to zero as t -+ oo.

(iii) if

then (2.3.1) bas an unbounded oscillatory solution.

Chapter 2

40

The proof of Theorem 2.3.3 and 2.3.4 can be seen from the original paper

[72].

2.4. Equations with Oscillatory Coefficients We consider the linear differential equation with deviating argument of the form

x'(t) + p(t)x(r(t)) where p,

T

= 0,

(2.4.1)

= oo, and p may change

E C([t 0 , oo ), R), Tis nondecreasing, lim r(t) t-oo

signs infinitely many times as t-+ oo. We will use the following notations: {2.4.2)

where ri(t) is defined on

E; Clearly, lim ri(t) t-oo

Lemma 2.4.1. k ;:: 3 such that (i) lim b,. n-oo

= {t:

= oo,

ri- 1 (t);:: t 0 },

i

= 1,2, ....

(2.4.3)

i = 1, 2, ....

Assume that there exists a sequence {bn} and a positive integer

= oo,

rk(bn) < bn,

n

= 0, 1, ... ;

(ii) p(t) ;:: 0, r(t) < t fort E [rk(b .. ), b.. ], (iii) there exists mk E [0, 1) such that mk$

t

p(s)ds,

n

= 0, 1, ... ;

tE[rk- 2(b,.),b.. ],

n=0,1,2, ....

Jr(t)

Let x( t) be a solution of the differential inequality

x'(t)

+ p(t)x(r(t)) :50

(2.4.4)

such that x(t) > 0 fortE [r(k+Il(b.. ), b.. ]. Then (2.4.5)

41

Oscillations of First Order Delay Equations

i = 1, ... , k - 3, where (2.4.6)

i = 1, 2, ... ' k - 3.

Proof: If m,. = 0, then O:i = 0, i = 0, 1, ... , k- 3. Hence (2.4.5) holds. Now let us consider the case 0 < mk < 1. For any t E [r"- 2 (bn), r(bn)], there exists t• E [r"-2 (bn), bn] such that r(t*) ~ t < t• and

1 t•

p(s)ds

T(t•)

~ m,. =

1 t•

(2.4.7)

p(s)ds.

t

From (2.4.4), we have

x'(t) ~ -p(t)x(r(t)) ~ 0,

t E [r"(bn),bn]·

(2.4.8)

Integrating {2.4.4) from t to t• we have

x(t)

~ x(t*) +

1,.

p(s)x(r(s))ds.

{2.4.9)

We note that r"- 1 {bn) ~ r(t) ~ r(s) ~ r(t*) ~ t ~ r(bn) for s E [t,t*] and integrating (2.4.4) from r(s) tot we obtain

x(r(s))

~ x(t) +

1'

p(u)x(r(u))du

T(a)

~ x(t) + x(r(t)) = x(t) + x(r(t))

1' [1•

p(u)du

T(8)

p(u)du

T(u)

~ :z:(t) + x( r(t)) [m,.

-1•

-1•

p(u)du]

t

p(u)du].

{2.4.10)

Chapter 2

42 Substituting (2.4.10) into (2.4.9) we have

x(t)

~ x(t*) + 1'" p(s){ x(t) + x(r(t)) [mk -1• p(u)du] }ds

=x(t*)+mkx(t)+x(r(t))[m~-1'" ds 1•p(s)p(u)du].

(2.4.11)

By exchanging the order of the integral we have

1'" ds 1• p(s)p(u)du = 1'" du

1'"

p(s)p(u)ds = 1'" ds 1'" p(s)p(u)du,

and hence

1'• ds1• p(s)p(u)du =

~ [1'"1• p(s)p(u)du + 1'"1'" p(s)p(u)du]

Substituting this into (2.4.11) we obtain (2.4.12) Thus

x(t) ml ( )> x r(t) 2(1 - mk)

= ao,

t

(k

)

E (r - 2 (bn), r(bn)],

(2.4.13)

i.e., (2.4.5) holds for i = 0. For i = 1, t E [rk- 2 (bn), r 2 (bn)], there exists t* E [rk- 2 (bn),r(bn)] such that r(t*) :5 t < t* and (2.4.7) holds. By (2.4.13), we have

x(t*) x(r(t•)) > ao.

(2.4.14)

By the same method as in the above proof we can show that (2.4.13) holds. In view of the monotonic property of x we have

x(t*) > a 0 x(t) > a~x(r(t)).

(2.4.15)

Oscillations of First Order Delay Equations

43

Substituting (2.4.15) into (2.5.12) we obtain

x(t) > o:~x(r(t)) i.e., (2.4.5) holds fori

+ o:omA:x(r(t)) + !mix(r(t)) =

O:tz(r(t)),

(2.4.16)

= 1. By induction, we see (2.4.5) holds fori = 0, 1, ... , k-3. D

Lemma 2.4.2. In addition to the assumptions of Lemma. 2.4.1, further assume tha.t for some i E {0, 1, ... , k- 3}

(2.4.17)

Then (2.4.4) has no solution which is positive on [r"+ 1 (bn), bn]· Proof: If not, there is a. solution x(t) of (2.4.4) with x(t) > 0 fort E [r"+ 1 (bn), bn]· By Lemma 2.4.1 x(t) > o:;x(r(t)), t E [r"- 2 (bn),ri+ 1 (bn)] and x'(t) ~ 0, t E [r"(bn), bn]· Integrating (2.4.4) from r(t) tot fort E [r"- 2 (bn), ri+ 1 (bn)] we obtain

-x(t) + x(r(t))

~ f' p(s)x(r(s))ds ~ x(r(t)) f' p(s)ds. J..(t) J..(t)

Hence

l

t ( ) x(t) psds~1- ( ()), r(t) X T t

te [T A:-2( bn),r i+l( bn)·]

In view of that x(t) > o:;x(r(t)) fortE [r"- 2 (bn),ri+ 1 (bn)], we obtain (2.4.18) which contradicts (2.4.17).

D

Similarly, we have the following lemma. Lemma 2.4.3.

Assume the assumptions of Lemma. 2.4.2 hold. Then the

Chapter 2

44

differential inequality x'(t) + p(t)x(r(t)) ~ 0

(2.4.19)

has no solution x(t) with x(t) < 0 on [r"'+ 1 (bn),bn]· Combining Lemma 2.4.2 and Lemma 2.4.3 we obtain the following oscillation criterion.

Under the assumptions of Lemma. 2.4.2 every solution of Theorem 2.4.1. Eq. (2.4.1) has a.t least one zero on [r"'+ 1 (bn),bn]· Consequently, every solution of (2.4.1) is oscillatory. Remark 2.4.1. The first part of Theorem 2.1.2 can follow from Theorem 2.4.1. In fact, the assumptions of Lemma 2.4.2 are satisfied in this case. Example 2.4.1. Consider the equation

x'(t) + 0.8x(t- sint) = 0

(2.4.20)

where r(t) = t - sin t is nondecreasing. Let b be the real root of the equation

b=

f + sinb

which lies on the interval ( f, 1r). It is easy to see that bE ( f, ~1r). Let bn = 2n1r+ b, n = 0, 1, 2, .... Then r(bn) = 211" + f, r 2 (bn) = 2n?r + f - 1, r 3 (bn) = 2n?r + f- 1- cos 1 > 2n?r. Clearly r 3 (bn) < bn and r(t) < t fortE (r(bn), bn)· Let k = 3, then the condition (i) of Lemma 2.4.1 is satisfied and [' 0.8ds lr(t)

= 0.8sin t 2:: (0.4) v'2 = m3,

t E [r(bn), bn]·

On the other hand, we see that

1

T(bn)

T 2(bn)

O.Sds

= 0.8 > 0.63161 2:: 1- ao.

That is, all assumptions of Lemma 2.4.2 are satisfied. Therefore, by Theorem 2.5.1, every solution of (2.4.20) is oscillatory.

Oscillations of First Order Delay Equations

45

Example 2.4.2. Consider the equation

x'(t) +a sin~ x(t- 2;) = 0

(2.4.21)

e-li, t).

where a E Set k = 3, and

bn = 2(2n- 1)11". Then the assumptions of Lemma 2.5.1 hold

a :::;;

l

t

t-.!!:

s a sin 2 ds,

2

and

,~-·

~

s

Jbn-~ asin2ds=2a>1- 2( 1 -a), i.e., all assumptions of Lemma 2.4.2 are satisfied. Therefore by Theorem 2.4.1 every solution of (2.4.21) is oscillatory.

2.5. Equations with Positive and Negative Coefficients We consider the linear delay differential equation with nonpositive and nonnegative coefficients of the form

x'(t) + p(t)x(t- r)- Q(t)x(t- u) = 0 where p; Q E C{[to,oo),.R;.), r,u E {O,oo). The following is an oscillation criterion for (2.5.1). Theorem 2.5.1.

Assume that

T ~

u

~

0 and

p(t)=p(t)-Q(t+u-r)~O,

lim inf t-+oo

and for every A> 0

it

t-T

{¢0),

p( s )ds > 0,

t~t1 ~to,

{2.5.1)

Chapter 2

46 Then every solution of (2.5.1) is oscillatory.

In the next chapter we will prove a more general result, so we omit the proof here. From (2.5.2) we can derive some explicit conditions easily.

H the condition (2.5.2) is replaced by the explicit condition

Corollary 2.5.1.

li~inf [el' p(s)ds + t

t-1'

00

lt-u

Q(s + u- T)ds] > 1,

(2.5.3)

t-1'

then the conclusion of Theorem 2.5.1 is true. The following is a result for the existence of positive solutions of (2.5.1). Theorem 2.5.2. Assume that i) T > u ~ 0 and

p(t) = p(t)- Q(t + u- T)

~

0 (¢ 0),

t

~ t1 ~ t0 ,

I,';'

p(t)dt = oo, ii) iii) there exist T ~ t 1

and~·

sup{;.

exp(~·

t~T

> 0 such that

1'

t-,.

p(u)du)

(2.5.4) Then Eq. (2.5.1) has a positive solution which tends to zero as t-+ oo. To prove Theorem 2.5.2, we first prove the following lemma. Lemma 2.5.1. integral inequality

Assume that condition i) in Theorem 2.5.2 holds. H the

(2.5.5)

47

Oscillations of First Order Delay Equations

has a continuous positive solution z E C[t1- r, oo) with lim z( t) = 0. Then there t-+00 exists a positive solution x : [t 1 - r, oo) -+ (0, oo) of the integral equation

i~~.r Q(s + u)z(s)ds +

1:

p(s + r)z(s)ds

= z(t),

Proof: Choose T;?: t 1 so large that z(t) > z(T), for t 1 Define a set of functions as follows:

n ={wE C([t 1 and a mapping T on

(Tw)(t) =

-

t;?: t1.

r :::; t < T.

0:::; w(t):::; z(t),

r,oo),R+):

(2.5.6)

t;?: t1 - r}

n.

{ J/~; Q(s + u)w(s)ds + ft~rp(s + r)w(s)ds, (Tw)(T) + z(t)- z(T),

t;?: T (2.5.7)

t1- T:::; t < T.

Clearly, Tfl c n. Define a sequence of functions as follows:

zo=z,

Zn=Tzn-b

n=1,2, ....

It is easy to see that

. 0:::; Zn(t):::; Zn-l(t):::; · · ·:::; z(t), t;?: t1- T. Hence J~~ zn(t) = x(t) exists and x(t) is continuous and nonnegative for t ;:::

t1-

T.

By Lebesgue's dominated convergence theorem x(t) satisfies (2.5.6). Since x(t) > 0 fort E [tt - r, r), it follows from (2.5.6) that x(t) > 0 for all t ;?: t 1 - r. The proof is complete.

Proof of Theorem 2.5.2:

Set

z(t) = exp(-

.x·i:+r p(u)du).

(2.5.8)

Chapter 2

48

We shall show that (2.5.8) satisfies integral inequality (2.5.5). In fact, (2.5.4) implies that ;. exp (A +

•l~r p( u )du)

l~~cr Q(s + u --r)exp(;\* 1' p(u)du )ds 5, 1

for

t

?_ T.

(2.5.9)

It is not difficult to see the relations ;. exp(x· [' p(u)du) J1-r

=

roo

p(s + -r)exp(>.·

J1-2r

r'

p(u)du)ds

(2.5.10)

J5+r

and

l~~cr Q(s + u- r)exp( ).* 1' p(u)du )ds =

r-r-CT Q(s + u)exp(.x·j!5+r p(u)du)ds.

h-2r

(2.5.11)

Substituting (2.5.10) and (2.5.11) into (2.5.9) we obtain

roo

p( S

+ T )exp ().

lt-2r

+

*1

1

s+r

['-r-cr Q(s

p( U )du) ds +u)exp(.x• [' p(u)du)ds 5, 1, Js+r

lt-2r

or

p( s + T )exp (_x * ['' p( U )du) ds

roo

~+r

h-2r

+

1!-r-a

Q( s + 0" )exp (.x.

f''

Js+r

t-2r

p( u )du) ds 5, exp ( -

x·jt~ p( u)du)'

i.e.,

j oo

t-2r

p(s + r)z(s)ds +

jt-r-cr Q(s t-2r

+ u)z(s)ds 5, z(t- r),

t?. T.

49

Oscillations of First Order Delay Equations

The hypothesis of Lemma 2.5.1 is satisfied. Therefore the integral equation (2.5.6) has a positive solution x(t) with x(t) :5 z(t). By differentiation we obtain

x'(t) = Q(t)x(t- u)- p(t)x(t- r). 0

The proof is complete.

From Theorem 2.5.1 and Theorem 2.5.2 we obtain the following corollary. Corollary 2.5.2. If p > Q 2::: 0 are constants of Eq. (2.5.1) oscillates if and only if

->.p + pe>..;;r where

p=

- Qe>..;;a

> 0,

T

> u 2::: 0, then every solution

for all

)..

>0

(2.5.12)

p- Q.

Remark 2.5.1. H Q(t)

= 0, (2.5.4) becomes that there exists ).. • > 0 such that

sup { ;. exp ().. I~T

•f'

p( u )du) } :5 1.

(2.5.13)

1-r

In particular, we take)..* = e, then (2.5.4) leads to

1 1

1

p(u)du :5 -, e

1-r

for

t 2::: T.

(2.5.14)

Example 2.5.1. Consider the equation

x'(t) + z(t; 2) x(t- 2)- !x(t) = 0 where p(t)

z(~2 2 )

-

= z(~; 2 >, Q(t) = h T

(2.5.15)

t

t

= 2, u

= 0.

Thus p(t)

= p(t)- Q(t + u- r) =

1 ~ 2 > 0 fort 2::: 10 and f 17: p(t)dt = oo. It is easy to see that there exists

T 2::: 10 such that

l

lne p(u)du :5 - , 2e 1-r t

l

1

t

Q(s- r)ds :5 -, 2e 1-r

for

t 2::: T.

Choose )..* = 2e, then (2.5.4) holds. Hence Eq. (2.5.15) has a positive solution which tends to zero as t -+ oo. In fact, x(t) = is such a solution of (2.5.15).

t

Chapter 2

50

2.6. Equations with Several Delays 2.6.3. Equations with constant parameters

First we discuss the oscillation problem of the following equation n

x'(t) + LP;x(t- r;)

=0

(2.6.1)

i=l

where p; E R,

r; ;:::

0 are constants. A known result for it is

Theorem 2.6.1. equation

Eq. (2.6.1) is oscillatory if and only if its characteristic n

f(>.) = >. + LP;e-~r, = 0

(2.6.2)

i=l

has no real roots. This theorem is of theoretical interest. But this condition is not easy to verify. In the following we give a new necessary and sufficient condition for oscillation of (2.6.1) which leads to some sharp explicit conditions for oscillation. Let n

D = {i = {i1, ... ,ln)li;;:?: 0,

i = 1, .. . ,n,El; = 1}. i=l

For any i E D define

>.· L ....!.ln(ep;r;fl;) n

f(l) =

(2.6.3)

i=l r;

where i; = 0 implies that the corresponding term vanishes. (i) f(l) has a maximum at a point i 0 = (i~, ... ,i~) on D Theorem 2.6.2. which is uniquely determined by the condition that

>.;

1 = -ln(ep;r;/l;), T;

have the same value if l; =f; 0.

i = 1, ... , n

(2.6.4)

51

Ouillations of First Order Delay Equations

(ii) Eq. (2.6.1) is oscillatory if and only if f(£ 0 )

> 0.

Another version of Theorem 2.6.2 is Theorem 2.6.3. Eq. (2.6.1) is oscillatory if and only if there exists an f.* = (ii, ... ,i~) such that >.; =f Aj for some i,j E In = {1, ... , n} and f(i*);:::: 0 where>.;, i E In is defined by (2.6.4).

Corollary 2.6.1.

Assume that n

PITt

>

~

for

n = 1,

~p·r·

>1 L....J••-e

and

for

n

>_ 2.

i=l

Then every solution of Eq. (2.6.1) is oscillatory.

Proof: We only prove it for the case that n ;:::: 2. Choose 0 < a ~ 1 such that E7= 1 aep;r; = 1. Let li = aep;r;, i = 1, ... , n, and f.* = (li, ... , £~). Then i* E D, and >.; are different fori = 1, ... , n. Furthermore, 1

f(i*)

n

= aln- l:Pi;:::: 0. a i=l

According to Theorem 2.6.3, Eq. (2.6.1) is oscillatory.

0

From Theorems 2.6.2 and 2.6.3 we can derive some other explicit conditions for oscillation of (2.6.1). In the next chapter we will prove more general results than Theorems 2.6.2 and 2.6.3. Therefore we omit the proofs here. Remark 2.6.1. The following still is an open problem. Find an explicit necessary and sufficient condition for the oscillation of (2.6.1) which does not depend on solving an equation.

52

Chapter 2

2.6.4. Equations with variable parameters We consider the first order delay differential equations of the form n

x'(t) + LPi(t)x(t- ri(t)) = 0

(2.6.5)

i=l

where pi, Let

t-ri(t) --too ast --too,

TiE C([to,oo),R+),

f

i E In= {l, ... ,n}.

E C(R, R) and define

type

f

=lim sup tIn lf(t)l.

(2.6.6)

t-+oo

Here f(t) is allowed to vanish and we denote ln 0 = -oo. Clearly, a function f with type -oo tends to zero faster than any exponential functions as t --too. We start discussion with some basic lemmas.

Lemma 2.6.1.

If liminf { inf _!, ~ Pi(t)e.\ri(t)} t-+oo .\>0 ).. L.J

>1

(2.6.7)

i=l

and Eq. (2.6.5) has nonoscillatory solutions. Then every nonoscillatory solution of (2.6.5) has type -oo. Proof: Without loss of generality assume Eq. (2.6.5) has a solution x(t), ~ 0. From (2.6. 7) we may assume that to is so large that

t 2:: t 0

(2.6.8)

n

and hence EPi(t) > 0 fort 2:: t 0 • We define a sequence {tk},

k = 0, 1, 2, ... by

i=l

tk = sup{ t: minl~i~n(t- r;(t)) ~ tk-b i E In}, k 2:: 1, then tk 2:: tk-h k ~ 1. It is easy to see that {tk} is unbounded. Otherwise, there exists a constant A such that limk-+oo tk =A and hence Ti(Ai) = 0, i E In. But then (2.6.8) and hence (2.6.7) is not true. Therefore limk-+oo tk = oo. x'(t) From (2.6.5) we have x'(t) < 0 fort~ t 1 . Define u(t) = - X(t) > 0, t ~ t1.

-----------~-------

---- - - -

53

Oscillations of First Order Delay Equations

Then u(t)

= tPi(t)expjt i=l

u(s)ds,

t

~ t1.

(2.6.9)

t-r,(t)

We shall show that lim u(t) = oo. To this end, define .A 1 = inf u(t), and t~t 1

t-oo

n

Ak+l =

inf

t~tA:+l

LP;(t)exp(.Akr;(t)). i=l

(2.6.10)

Clearly, each Ak is well defined. It is easy to see that

= tPi(t)expit

u(t)

i=l

u(s)ds

t-r,(t)

n

~ LPi(t)exp(.AI Ti(t)) ~ A2

for

t ~ t2.

i=l By induction, we have u(t) 2:: Ak

t

for

~ tk,

k

= 1, 2, ....

Now we shall prove that lim Ak = oo. k-oo

We first claim that A1 > 0. Otherwise there is an arbitrarily large t such that u(s) 2:: u(t) > 0 for all s E [to,t]. Then from (2.6.9)

u(t) = tp;(t)exp i=l

it

u(s)ds

> to

~ tPi(t)exp(u(t)r;(t)) i=l

t-Ti(t)

which contradicts (2.6.8). From (2.6.8) and (2.6.10) there is an o: > 1 such that Ak+l 2:: o:Ak for k = 1, 2, ... , and hence Ak --+ oo as k --+ oo. Since x(t) exp(- fe: u(s )ds) and hence x(t) has type -oo. The proof is complete. 0

Lemma 2.6.2. Let f : R+ --+ (0, oo) be continuous. If there exist b, B > 0 such that f( t) / f( t - b) 2:: B for t ~ b, then there exists M > 0 such that

f(t)

~ M exp

(f;

lnB),

for t ~b.

(2.6.11)

Chapter 2

54 The proof is easy so we omit it here.

Lemma 2.6.3. Let p(t) ~ 0 and r(t) be continuous functions fort ~to ~ 0 and lett- r(t) be nondecreasing such that t- r(t) ~ oo as t ~ oo. Assume

lim inf t-+oo

it

p( s )ds > 0

(2.6.12)

t-r(t)

and lim inf r( t) t--+00

> 0.

(2.6.13)

Then the differential inequality x(t)[x'(t)

+ p(t)x(t- r(t))] 50

(2.6.14)

does not have any nonoscillatory solutions with type -oo. Proof: Let x(t) be a nonoscillatory solution of (2.6.14). Without loss of generality, we assume x(t) > 0 fort ~ t 0 • From (2.6.12) and (2.6.13) there are b, B > 0 such that for some t 1 ~ t 0

i

t t-r(t)

p(s)ds

~ B,

r(t)

~ b,

t

~ t1

(2.6.15)

which implies that

(2.6.16) (see Lemma 2.1.3). Since r(t)

~band

x(t) is nonincreasing we have

x(t)fx(t- b)~ By Lemma 2.6.2, type x

> -oo.

B2

4' t

~ t1.

(2.6.17)

0

Remark 2.6.1. The following example shows that the conditions in Lemma 2.6.3 are sharp.

55

Oscillations of First Order Delay Equations

Example 2.6.1. The equation

(2.6.18) has a solution x(t) (2.6.12) holds.

= exp(-

Theorem 2.6.4.

Assume that (2.6. 7) holds and

!t 2 ) with type -oo. We see that (2.6.13) fails and

(2.6.19)

Then every solution of (2.6.5) is oscillatory.

Proof: Let x(t) be an eventually positive solution of (2.6.5). Then x(t) > 0 and x'(t) :::; 0 for all large t. We first show that (2.6.20) If not, there exists a sequence {t k} such that lim t k = oo and Pi (t k) Ti (t k) ~ 0 k-->oo

as k ~ oo, i E In. If Pi(tk)ri(tk) = 0 for some tk and all i E In, then 1

n

XLPi(tk) exp [ri(tk)]:::; npo/'A--+ 0,

as

>..--+ oo.

i=l

Hence we may assume that max {Pi(tk)ri(tk), i E In} For each k, set aik = oo if Pi(tk)ri(tk) = 0 and aik = Pi(tk)ri(tk) > 0. From (2.6.19) we have

> 0 for all k = 1, 2, ....

T}t,.)

ln (2po/Pi(tk)) if

In view of the fact that Pi(tk)r,(tk) ~ 0 as k--+ oo, it is not difficult to see that aik ~ oo ask~ oo for each i E In. Now let A.(tk) =min {aibi = 1,2, ... ,n}. Then we have 0 < A.(tk) < oo and A.(tk) ~ oo ask~ oo. Thus 1

n

A.(t ) LPi(tk)exp(A.(tk), r,(tk)) k

i=l

Chapter£

56

which contradicts (2.6.7) and hence (2.6.20) holds. Next, if we let p(t) = p;(t), r(t) = r;(t) be such that Pj(t)rj(t) = max{p;( t)r;( t), i E In} for each t, then 0 ::; p(t) ::; Po, 0 :5 r(t) ::; ro. From (2.6.20), there is a constant c > 0 such that p(t)r(t) ;?: c for all large t and so p(t) ;?: c/ro and r(t);?: cfpo. From (2.6.5) we have n

0 = x'(t)

+ LP;(t)x(t- r;(t)) i=l

;?: x'(t)

+ p(t)x(t- r(t))

;?: x'(t) + .!:.x(t-.!:. ), ro Po

and so, by Lemma 2.6.3, type x > -oo. On the other hand, by Lemma 2.6.1, under (2.6.1) the nonoscillatory solution x has type -oo. This contradiction completes the proof of the theorem. 0 Considering that the functions p;(t) and r;(t) may be unbounded, we present the following oscillation criterion. Theorem 2.6.5.

Suppose that there exists a nonempty subset I of the set

In= {1, ... ,n} such that liminf u(t) t-oo

>0

(2.6.21)

and liminfft t-oo

LP;(s)ds > 0

(2.6.22)

t-u(t) iEJ

whereu(t) = min{r;(t), i E I}. H(2.6.7) holds, then every solution ofEq. (2.6.5)

is oscillatory.

57

Oscillations of First Order Delay Equations

Proof: Assume that (2.6.5) has a solution x(t) > 0 for t ~ to ~ 0, by Lemma 2.6.1, it is sufficient to show that type x > -oo. From (2.6.7) we may assume that t 0 is sufficiently large so that (2.6.8) holds. Then, from (2.6.5), x'(t) < 0 fort~ t1 ~to, where t 1 =sup {t : min(t- r;(t)) :5 to, i E In}· Since n

0 = x'(t) + LP;(t)x(t- r;(t)) i=l

t ~ t1•

~ x'(t) + x(t- u(t)) Ep;(t), iEI

By virtue of (2.6.2) and (2.6.22) and Lemma 2.6.3, we have type x > -oo. The 0 proof is complete. The following example shows that condition (2.6.19) or conditions (2.6.21) and (2.6.22) are important for Theorem 2.6.4 and Theorem 2.6.5 to hold. Example 2.6.2. Consider the equation

x'(t) + Pl(t)x(t) + P2(t)x(t -1) = 0. Choose a, b > 0 so that a= b/(1- e- 6) > 1. Let t 0 be such that to> // ln(1- e- 6 ) > 0. Pl(t) = a(g(t) -1), P2(t) =a exp(-g(t)), where g(t) = (1- exp( -b))exp(bt), t ~ t 0 • For any ,.\ > 0,

(2.6.23)

t ~ t0 ,

'nf g(t)- 1 + exp(..\- g(t))] ,[ . ,.\ =A- 1 +a1 t2:to

The function

/(..\) = g(t)- 1 + e;p(..\- g(t))

,

,.\ > 0

minimized at ..\ = g( t) and the minimum is 1. Hence n

-,.\ + inf 2:Pk(t)e.\r.(t) ~ ..\( -1 +a) > 0 t2:to k=l

Chapter 2

58

for all .\ > 0, and so condition (2.6.7) holds. On the other hand, it is easy to check that Eq. (2.6.23) has a positive solution

x(t) = exp(- exp(bt)),

t ~ t0 •

Furthermore, neither (2.6.19) nor (2.6.21) holds. Therefore nonoscillatory solutions with type -oo may exist (see Lemma 2.6.1). Corollary 2.6.2. Assume that i) (2.6.19) holds or both (2.6.21) and (2.6.22) bold, n

ii)

liminf

L: p;(t)r;(t) > ~.

(2.6.24)

t-oo i=l

Then every solution of (2.6.5) is oscillatory. In order to present a nonoscillation criterion we need the following lemma.

Suppose that there exists a nonempty set I

Lemma 2.6.4.

i) r;(t) > 0

fort~

to,

~

{In} such that

i E I, and

ii) l:p;( t) > 0 fort ~ to. iEJ

If y is an eventually positive solution of the differential inequality n

y'(t) + LPk(t)y(t- Tk(t)) $0,

(2.6.25)

k=l

then there exists an eventually positive solution x(t) of (2.6.5) with

x(t) $ y(t)

for all large t,

and lim x(t) = 0. t-oo

Proof: Since y is an eventually positive solution of (2.6.25) we have that eventually

y'(t) $- LP;(t)y(t- r;(t)) $- LP;(t)y(t) iEJ

iEl

59

Oscillations of First Order Delay Equations

and hence, because of i), we have the y'(t) < 0 eventually. Choose T sufficiently large. For any t;::: t;::: T integrating (2.6.5) from t to have t we

+

y(t);::: y(t)

i t

n

:l::>A:(s)y(s- TA:(s))ds A:=l

i A:=l t

>

t

n

~:::>A:(s)y(s- TA:(s))ds.

t

Because tis arbitrary we get

Define a set fl = {:r: E C([T,oo),J4): 0:5 :r:(t) :5 y(t),

t;::: T}.

For any :r: E fl, we define

t;::: T x(t) = { :r:(t), :r:(T) + (y(t)- y(T)), T :5 t :5 T. Then define a mapping T on ('T:r:)(t) =

n as follows:

ioo A:=l

tPA:(t)x(s- TA:(s))ds,

t;::: T.

t

Clearly, Tis well-defined, and ('T:r:)(t) :5 y(t) on t;::: T and hence 'Tfl Defineasequence{:r: n(t)}, n=0,1,2, ... asfollows: :r:o = y, Xn+l='T:r:n,

t;::: T, t;:::T,

n=0,1,2, ....

By induction, we see that 0 :5 Xn(t) :5 Xn-l(t) :5 · · · :5 y(t),

t;::: T,

c n.

Chapter 2

60

and hence x( t) = lim xn(t) exists on [T, oo ). Then we can apply the Lebesgue's n ..... oo dominated convergence theorem to show that x = T x, i.e.,

x(t) =

l oo

n

LPk(s)x(s- T~;(s))ds,

t

~

T.

A:=l

Obviously, lim x(t) t ..... oo

= 0. Also, we have n

x'(t) =- LPA:(t)x(t- Tk(t)),

t ~ T,

k=l

which means that x or x is a solution on [T, oo) of Eq. (2.6.5). Moreover, it is easy to see that x(t) $ y(t) fort~ T. It remains to prove that x is positive on [T, oo ). Clearly, x > 0 fort E [T, T) and x ~ 0 for t ~ T. If there is a t ~ T such that x(t) = 0, set

t* then x(t*)

= inf{f I x(f) = o,

= 0, x(t) > 0 fort x(t*) =

E [T, t*) and T;(t)

roo

lt

t•

t

~ T},

> 0,

i E I, t ~to. Hence

00

LP~;(s)x(s- Tk(s))ds > 0 k=l

contradicting that x(t*) = 0. Theorem 2.6.6. Suppose that there exists a nonempty set I ~ In so that i) and ii) in Lemma 2.6.4 are satisfied. Moreover, assume that for a sufficiently large T 0 ~to n

->.+sup LPk(t)exp(>.Tk(t)) $0,

>. > 0.

for some

(2.6.26)

t~To k=l

Then there exists a positive solution x of Eq. (2.6.5) satisfying

x(t) $ exp (->.t)

for alllarge t

Proof: The point To is sufficiently large so that t -

T~;(t) ~

(2.6.27)

t 0 , for t

~

T0 ,

61

Oscillations of First Order Delay Equations k E In. Set y(t)

= exp( -,\t), t;;:: to. Then n

n

y'(t)

+ LPk(t)y(t- Tk(t))

= -,\y(t) + LPk(t)exp(- -\(t- Tk(t)))

k=l

k=l

This means that y(t) is an eventually positive solution of (2.6.25). By Lemma 2.6.4, Eq. (2.6.5) has a solution x(t) satisfying x(t) $ y(t) = exp( -.\t) D eventually.

Corollary 2.6.3.

Consider the differential equation n

x'(t) + LPkx(t- Tk) = 0

(2.6.28)

k=l where Pk, Tk are positive constants, k E In. Then evezy solution of (2.6.28) is oscillatory if and only if n

-.\ + LPkexp (ATk) > 0

for all ,\ > 0.

(2.6.29)

k=l

Corollary 2.6.4. Assume that the conditions of Lemma 2.6.4 hold. Moreover, assume that there exist T ;;:: to and .\ > 0 such that

(2.6.30)

for a To ;;:: t 0

and .\

> 0.

Then there exists a positive solution x of Eq. (2.6.5) satisfying lim x(t) t-+oo

= 0 and

Chapter 2

62 eventually

Proof: Choose T0 so large that t- Tk(t) ~to fort ~To,

k E In. Set

It is easy to see that y is a positive solution of (2.6.25) So the conclusion of the corollary follows from Lemma 2.6.4. n

Suppose that

Remark 2.6.2.

l: Pk(t) >

0, for t ~ to, and for a sufficiently

k=l

large To~ to sup

t~To

{

max

k=1,2, ... ,n

l

t

t-rk(t)

(

n ) 1 LP;(s) ds } ~-.

i=l

e

(2.6.32)

Then (2.6.30) holds. Consequently, the conclusion of Corollary 2.6.4 holds. Corollary 2.6.5. Assume that the assumptions i) and ii) in Lemma 2.6.4 hold. Then every solution of (2.6.5) is oscillatory if and only if differential inequality (2.6.25) has no eventually positive solutions.

From this corollary we can derive some comparison results. With Eq. (2.6.5) we associate the equation n

x'(t) +

L q;(t)x(t- u;(t)) = 0.

(2.6.33)

i=l

Theorem 2.6.7. and

Assume that the assumptions i) and ii) in Lemma 2.6.4 hold,

(2.6.34)

63

Oscillations of First Order Delay Equations

eventually. Then every solution of Eq. (2.6.5) is oscillatory implies the same for (2.6.33). Proof:

If not, let x(t) be an eventually positive solution of (2.6.33). Then

eventually n

x'(t) =- L

q;(t)x(t- u;(t))

i=l n

:::; - LP;(t)x(t- u;(t)) i=l n

$- LP;(t)x(t -r;(t)). i=l

By Corollary 2.6.5, Eq. (2.6.5) has an eventually positive solution contradicting the assumption. D

Corollary 2.6.6.

If Eq. (2.6.5) has an eventually positive solution and (2.6.35)

eventually, and there exists a nonempty set I E In such that u;(t) > 0 for t ~ t 0 , i E I and l:q;(t) > 0 fort ~ to. Then (2.6.33) has an eventually positive solution also.

iE/

Proof: Let x(t) be an eventually positive solution of (2.6.5). Then eventually n

n

x'(t) = - LP;(t)x(t- r;(t)):::;- L i=l

q;(t)x(t- u;(t)).

i=l

By Lemma 2.6.4, Eq. (2.6.33) has an eventually positive solution. We then consider (2.6.5) for the case that lim r;(t) = t-oo

D

r; and t-oo lim p;(t) = p;.

Chapter It

64 The "limiting equation" of (2.6.5) is n

z'(t) + :EiJ;z(t- r;) = 0.

(2.6.36)

i=l

lim T;(t) Assume that T;, Pi E C([to,oo),R+), t-ooo Theorem 2.6.8. lim p;(t) =Pi· Then (2.6.36) is oscillatory implies the same for (2.6.5).

= r;,

Proof: By Corollary 2.6.3 we have -A+ :Ep; exp (AT;)> 0 for all

A> 0.

(2.6.37)

Assume the conclusion of the theorem is not true. Let x(t) be a positive solution of (2.6.5). Then for any e > 0 there exists aT?: to such that n

x'(t) + :EC.r;- e)x(t- r; +e)~ 0,

t?: T.

i=l

In view of Corollary 2.6.5, it follows that n

x'(t) + :E(p;- e)x(t- r; +e)= 0 i=l

has a positive solution. Hence by Theorem 2.6.1 the characteristic equation n

F(JL)

= -JL + :E(.P;- e)e"
i=l

has a positive root JL". From (2.6.37) there exists an m > 0 such that n

l:P'; exp (Ar;)?: A+ m

for all

A > 0.

i=l

Choose e > 0 such that n

JL"'(exp(JL*e) -1) +e l:exp(JL"'T;) < m. i=l

(2.6.38)

65

0Jcillations of FirJt Order Delay Equations

Then n

""~ L....JPi exp (JL *~) r;

< JL * + m

i=l

0

which contradicts (2.6.38). The proof is complete.

2.6.5. Equations with oscillating coefficients In the following we consider the equations of the form n

x'(t) + LPi(t)x(ri(t)) = 0

(2.6.39)

i=l

where Pi : R+ ~ R are locally integrable, r; : R+ ~ R are measurable. We have the following existence results for the positive solutions of (2.6.39) anywhere Pi and t- Ti(t) may change signs. Theorem 2.6.9.

Assume that either

p;(t) :;::: 0,

for t:;::: 0,

i E In

(2.6.40)

Ti(t) :5 t,

for t :;::: 0,

i E In.

(2.6.41)

or

Further assume that there exist a; E R, M;, locally integrable function q : R+ ~ R+ such that

p;(t) :5 Miq(t),

q(s)ds :5 a;,

['

E

t:;::: t 0 ,

In,

>.,

i E In

t 0 E R+ and

(2.6.42)

lr;(t)

and n

L

M;ea;>.-

i=l

Then Eq. (2.6.39) has a positive solution.

>. :5 0.

(2.6.43)

Chapter 2

66

Proof: We will prove this theorem with condition (2.6.41). The theorem with condition (2.6.40) can be proved by a similar argument. Set v(t)=inf{s:s~t, r;(e)~t, for(~s,

iEln}.

Let t1 ~ v(to)· Let C([t 0 ,oo),R) be a space of all continuous functions defined on [to,oo) with the Frechet topology. Let n c C([t0 , oo ), R) be the subset satisfying: a)

x(r;(t)) a.\ . x(t) $ e ' , t ~ t1, ~ E In

b) x(t) = 1, t E [to,tt]

c) exp(-),. ] 111 q(s)ds) $ x(t) $ exp(l:::':, 1 ea;.\ J:, [p;(s)J- ds), t ~ t1 where [p;( s )]- = max (- p( s ), o). Clearly, n is a nonempty, closed and convex subset of C([t 0 , oo ), R). Define an operator T : n - C([t 0 , oo ), R) by

(Tx)(t)=

{ ( j exp-

t

;~ p;(s)x(T;(s))) ()

t,

X S

,t~t1

t E [t 0 , tt].

1,

n

It is easy to see that Tf! ~ and T is continuous. On every compact interval of [to, oo), (T x)(t), X E n is uniformly bounded and equicontinuous. By Ascoli theorem cl{Tf!} is compact in the topology of the Frechet space C([t 0 , oo ), R). By the Schauder-Tychonoff fixed point theorem, there exists an X En such that Tx == x, i.e.,

x(t) = exp

(i

n

t

;"£ p;(s)x(r;(s))ds)

-

( )

X S

11

,

which implies that x is a positive solution of (2.6.39) on [t 1 , oo ). n

Corollary 2.6.7.

and

Assume LP;(t)

~

0, t ~ T for a sufficiently large T ~ t 0 ,

i=l

sup{~ax [' (tPi(s)) ds} $ ~. t~T •Ein lt-T;(t) e j=l

67

Oscillations of First Order Delay Equations

Then Eq. (2.6.39) has a positive solution. In fact, choose n

q(t) = l:p;(t) i=l

and ). = e. Then all assumptions of Theorem 2.6.9 are satisfied. This corollary follows from Theorem 2.6.9. In particular, if n = 1, from Theorem 2.6.9 we obtain the following results. Corollary 2.6.8.

Assume that either

r(t) :5 t,

1

1

1

P1(s)ds

:5-,

r(l)

e

1

1

or

p(t) ;::: 0,

1

p(s )ds :5 -, e r(t)

Then (2.6.39) has a positive solution on [to, oo ), where p 1 (s) =max (p(s ), 0). Corollary 2.6.9.

Assume that:

(i) there exist holds:

E R, M;, i E

a;

In,>., t 0

p;(t) :5 M;,

tp;(t) :5 M;, t lntp;(t)::; M;,

E

Rt

such that one of the following

.

01 r·(t) > - t ''

t ;::: t 0 ,

or

r;(t) ;::: o:;t, t ;::: t 0 ,

or

r;(t) ;::: t

01 ' ,

t ;::: to;

n

(ii) L:M;e 01 '>.-). :50; i=l

(iii) one of (2.6.40) and (2.6.41) holds. Then Eq. (2.6.39) has a positive solution. This corollary follows from Theorem 2.6.9 by taking q(t) = 1, q(t) =

q(t)

= tfot respectively.

t and

Chapter£

68

Theorem 2.6.10. Assume that there exist nonnegative numbers M;, a;, i E In, .X, t 0 E R+ and a locally integrable function q : ~ --+ 14 such that

I{'

IP;(t)l5 M;q(t),

q(s)dsl5 a;,

t 2:: to,

i E In,

lr;(t)

and n

LM;exp(a;.X)- .X 50. i=l

Then Eq. (2.6.39) has a positive solution.

The proof is similar to the proof of Theorem 2.6.9. We only point out that the functions in the set n satisfy

Corollary 2.6.10.

If for all sufliciently large t

l

1

t

lp(s)lds 5 -. e

r(t)

Then Eq. (2.6.39) has a positive solution.

Corollary 2.6.11.

If there exist constants p;, r;, i E In such that

It- r;(t)l 5 r;

IP;(t)l 5 p;, and n

.X= Lp;e~r; i=l

has a positive root, then Eq. (2.6.39) has a positive solution.

69

Oscillations of First Order Delay Equations

2.6.6. Advanced equations

Finally, we would like to mention that most oscillation criteria for Eq. (2.6.5) can be extended to the equations with advanced type of the form n

x'(t) = Ep;(t)x(t + T;(t)).

(2.6.44)

i=l

For instance, we present the following results. Theorem 2.6.11. Assume that p;, T; E C([t0 , co), R+), with T;(t) uniformly bounded, i E In and such that

liminf{ inf t-oo

~>O

p;(t)exp(..\T;(t))} > 1. ~~ L...., A

(2.6.45)

i=l

Then every solution of (2.6.44) oscillates. Proof: Suppose the contrary, assume x(t) Set

x'(t)

..\o(t) = x(t),

> 0,

t ;::: T, is a solution of (2.6.44).

t;::: T.

Then

Define a set

!l = {..\ E C([T,oo),R): ..\(t);::: 0, t;::: T} and a mapping T defined on

n

(2.6.46)

Chapter 2

70

Clearly, Tf! and

c n,

A1 (t);::: A2(t);::: 0 implies (TA 1 )(t);::: (TA2)(t);::: 0 fort;::: T,

Ao(t) = (TAo)(t),

t;::: T.

Ao(t);::: 0,

In view of the assumption, there exists To > 0 such that IT;(t)l $ To, Form (2.6.45) there exists C > 1 such that 1

inf

>.e(O,oo)

n

:\ ~::>;(t)exp(AT;(t)) ;::: C. i=I

In particular, taking A= 1, we have fort;::: T n

n

C $ 'Lp;(t)exp(r;(t)) $

ero

'Lp;(t). i:l

i51

Hence for t ;::: T

C n "p;(t) ;::: ~

ero

•=I

=M

> 0,

n

Ao(t) = (TAo)(t);::: 'Lp;(t);::: M, i=l

and so n

Ao(t) = (TAo)(t);::: TM = LP;(t)exp(MT;(t));::: MC, i=l

Repeating this procedure we obtain Ao(t);::: C"M-+ oo,

as

k-+ oo,

which contradicts (2.6.46). The proof is complete. Corollary 2.6.12.

Assume that T;(t) $To, liminf t-+oo

i E In and

p;(t)T;(t) > ! . ~ e ~ i=l

t;::: T.

i E In.

71

Oscillations of First Order Delay Equations

Tben every solution of (2.6.44) oscillates.

Theorem 2.6.12. Assume tbat p;, >. • > 0, T ~ t 0 sucb tbat

r; E

C([to,oo),R+), and there exists

n

->.*+sup I>;(t)exp(>.*r;(t)) :::; 0. t;::T i=l

Tben (2.6.44) bas a positive solution on (T, oo ). Remark 2.6.3. Finding conditions for the existence of oscillatory solutions of (2.6.5) is an open problem.

2. 7. Equations with Forced Terms We consider the forced differential equations of the form

x'(t) + f(t,x(ri(t)), ... ,x(rn(t))) = q(t).

(2.7.1)

Assuming here q E C([t 0 ,oo),R) and let Q(t) E C 1 ((t 0 ,oo),R) such that Q'(t) = q(t) fort~ t 0 • The following result provides an oscillation criterion.

Assume tbat

Theorem 2.7.1.

i) j, q, r;, i E In, are continuous, ii) lim r;( t) = oo, i E In, t-+oo

iii) fort

~

T

~

t.

ud( t, ul! u2, ... , un) > 0 whenever u1 u; > 0, i E In and for flxed t, nondecreasing witb respect to u;, i E In, iv)

1 1

00

00

f(t,Q+(ri(t)), ... ,Q+(rn(t)))dt =

f is

00

f(t, -q-(ri(t)), ... , -Q-(rn(t)))dt = -oo

(2.7.2)

Chapter 2

72 wbere Q+(t) = HIQ(t)l + Q(t)), solution of (2. 7.1) oscillates.

Q_(t)

= HIQ(t)l- Q(t)).

Then every

Proof: Suppose the contrary, let x(t), be an eventually positive solution of (2.7.1). Then

(x(t)- Q(t))'

= - f( t, x( Tt (t)), ... , x( Tn( t))).

(2.7.3)

In view of condition ii), (x(t)- Q(t))' < 0 eventually. Condition iv) implies that Q(t) always changes sign for sufficiently large t. Therefore there exists a sequence tk --+ oo such that x(tk) > Q(tk), and then

x(t)- Q(t) > 0 for t

~

lim (x(t)- Q(t)) = t--+oo

l

T*,

(2.7.4)

which implies that

exists. From (2.7.3) we obtain that for sufficiently large T

!roo f(t,x(rt(t)), ... ,x(rn(t)))dt < oo. From (2.7.4), x(t) > Q+(t)

fort~

(2. 7.5)

T. In view of the monotonic property of

f and (2.7.5) we obtain (2.7.6) which contradicts (2.7.2). Similarly, we can derive a contradiction when Eq. (2.7.1) has an eventually negative solution. The proof is complete. Example 2.7.1. We consider the equation x'(t) + px(t- 1r) = sint + pcost

(2. 7. 7)

where pis a constant with 0 < p7re < 1, then the homogeneous equation x'(t) + px(t- 1r) = 0

(2.7.8)

73

Oscillations of First Order Delay Equations

has a positive solution by Corollary 2.1.2. For (2.7.7) Q(t) =-cost+ psint, it is easy to see that (2.7.9) By Theorem 2.7.1, every solution of (2.7.7) oscillates. In fact, y(t) =-cost is such a solution. Remark 2.7.1. From the proof of Theorem 2.7.1, it is easy to see that condition iv) can be replaced by iv),

liminf Q(t) = -oo t-+oo

(2.7.10)

lim sup Q(t) = +oo. t-+oo

We now consider the case that lim sup Q(t) = M < oo t-+oo

liminf Q(t) = m t-+oo

(2.7.11)

> -oo.

For the sake of simplicity we consider the equation of the form x'(t) + p(t)x(r(t))

Theorem 2.7.2.

= q(t).

(2.7.12)

Assume that

p E C([to,oo),~), r, q E C([t 0 ,oo),R), limr(t) = oo. t-+oo

Set Q'(t)

and

= q(t),

Q changes sign on (T,oo), where Tis any number. If

(2.7.13)

74

Chapter~

Then every solution of (2. 7.12) oscillates. Proof: Suppose the contrary, x(t) > 0, t C: t 0 , is a solution of (2.7.12), then there exists t1 C: to such that x(r(t)) > 0 fort C: t 1. As in the proof of Theorem 2.7.1 we know that x(t)- Q(t) > 0 and (x(t)- Q(t))' < 0 fort C: t1. which implies that

1,,

00

p(s)x(r(s))ds
(2.7.14)

On the other hand (2.7.13) implies that

1,,

00

p(t)dt =co.

(2.7.15)

From (2.7.14) and (2.7.15) we obtain liminf x(t) = 0.

(2.7.16)

t-+oo

Set lim (x(t)- Q(t))

t-+oo

=l

C: 0.

(2.7.17)

We will show that l = -m. In fact, for any e > 0 there exists T C: t 1 such that l < x(t)- Q(t) < l

+ e,

and hence

-Q(t) < l+e,

t C: T.

Then -m

= -liminf Q(t) ~ l +e. t-+oo

(2.7.18)

On the other hand, there exists a sequence {tn} such that lim tn =co and n-+oo lim x(tn) = 0. From (2.7.17) we have

n-+oo

-Q(tn) > l- x(tn)

75

Oscillations of First Order Delay Equations

and hence -liminf Q(tn)

~f.

R-+00

and so -m ~f..

(2.7.19)

(2.7.18) and (2.7.19) imply that f.= -m. From (2.7.17), x(t)- Q(t)

> -m,

~

t

t1

and so x(t)

> [Q(t)- m]+, t

Choose t 2 > t 1 so large that r(t) x(r(t))

~

t1

fort~ t2.

~ t 1•

Then

> [Q(r(t))- m]+,

t ~ t2.

Substituting this into (2.7.14) we obtain

1

00

p(t)[Q(r(t))- m]+dt

< oo,

t2

which contradicts (2.7.13). Similarly, we can prove that (2.7.11) has no eventually negative solutions. The proof is complete. Example 2. 7 .2. Consider the equation x'(t) + p(t)x(t- '11") =-cost,

where p(t)=

{ ~t,

1.,.(--t), "

t E [0, ~] t E (,.2•'~~"1

and p(t + 2'11") = p(t) for all t ~ 0. In this case Q(t) = -sint and M = 1, m

= -1.

(2.7.20)

Chapter 2

76 Cleaxly, we have

ioo p(t)(1- Q(t- 1r)]+dt = oo, ioo p(t)[1 + Q(t- 1r)]+dt =

(2.7.21)

00.

By Theorem 2.7.2, every solution of (2.7.20) oscillates. We note that condition (2.7.2) does not hold for (2.7.20). In the next chapter we will allow a result for the existence of nonoscillatory solutions of (2.7.1).

2.8. Single Population Models with Delays 2.8.7. Delay logistic equation We consider the delay logistic equation with a constant caxrying capacity k

dx(t) = () ( )( _ x(t- r(t))) , k rtxt1 dt

(2.8.1)

where r, T axe positive continuous functions defined on [0, oo) and k is a positive constant. Denote r* = sup r( t). t>O

Motivated by the ~pplication of (2.8.1), we consider solutions of (2.8.1) with initial conditions of the type

x(s) =

~P(s)

is continuous on

where

IP ~ 0

[-r*, 0] and

~P(O)

> 0.

(2.8.2)

It is not difficult to show that solutions of (2.8.1) and (2.8.2) are defined for all t ~ 0 and remain positive fort ~ 0. A solution of (2.8.1) and (2.8.2) is said to be oscillatory about k if there exists a sequence {tn} such that lim tn = oo and k- x(tn) = 0, n = 1, 2, .... Set

n_.oo

y(t) =

x~t)

- 1,

t

~ -r•.

(2.8.3)

Then (2.8.1) becomes

y'(t) = -r(t)(1 + y(t))y(t- r(t)),

t > 0,

(2.8.4)

77

Oscillation.s of Fir.st Order Delay Equations

whose initial conditions are inherited from (2.8.2). By way of (2.8.3), the oscillation of x about k is equivalent to that of y (about zero) as in the usual sense. We note from the positivity of x(t) that 1 + y(t) > 0 fort ~ 0. We state some results on oscillation of Eq. (2.8.4) without proofs. Theorem 2.8.1. Assume that i) r, T E c(~.(O,oo)), and lim

t-oo

J0

(t- r(t))

= oo,

r(s)ds = oo. Then every solution of Eq. (2.8.4) is either oscillatory or converges to zero, eventually monotonically, as t-+ oo. ii)

00

Theorem 2.8.2.

In addition to the assumptions of Theorem 2.8.1 assume

r(s)ds >

liminflt t-r(t)

t-oo

~. e

Then every solution of Eq. (2.8.4) is oscillatory.

Theorem 2.8.3.

Assume there exists a

i

to~

0 such that

1

t

r(s)ds

t-r(t)

~-,

e

t

~to.

Then Eq. (2.8.4) has a nonoscillatory solution.

=

r > 0, r(t) If r(t) Corollary 2.8.1. Eq. (2.8.4) is oscillatory if and only ifrre > 1.

=

> 0, then every solution of

T

Let us consider now the delay logistic equation of the form

dx(t) dt where n is a positive integer,

= T

r

(t) (t) [ 1 _ x(tx

nr)]

k(t)

> 0, r, k E C(R+,(O,oo)).

(2.8.5)

Chapter£

78

The following result is concerned with the oscillation of solution about k. Theorem 2.8.4. Assume that i) k is r-periodic and positive and a= max k(t), t~O

1

00

ii)

b =min k(t), t~O

r(t)[a- k(t)]dt

a > b,

= oo

(2.8.6)

(2.8.7)

and

1

00

r(t)[k(t)- b]dt

= oo.

(2.8.8)

Then every positive solution x of Eq. (2.8.5) is oscillatory about k(t), i.e., there exists a sequence {tn} such that lim tn = oo and x(tn) = f3(tn), n = 1, 2, .... n-oo

Proof: Assume the contrary, and let x(t) be a solution of (2.8.5) such that x(t) > k(t) fort ~ t 0 > 0. Then x'(t) < 0, t ~ t 0 , and hence lim x(t) = l ~ a t-oo exists. We claim l = a. Otherwise, from (2.8.5) we see that for t ~ t 1 ~ t 0 x'(t) :5

~ (1- ~)r{t).

Noting that {2.8.7) and {2.8.8) imply that x(t) :5 x(t 1 )

+~ 2

(1- ~) lt,ft

contradicting that x(t) > k(t) From (2.8.5)

a

f0

00

r(t)dt = oo. We have

r(s)ds-+ -oo,

as t-+ oo

~b.

a ] :5 - 1 r(t)(a- k(t)) a r{t) [1- k(t) x'(t) :5 2 2

for all large t, which leads to a contradiction. Similarly, (2.8.5) has no positive solution x(t) with x(t) < k(t) eventually. The proof is complete.

79

Oscillations of First Order Delay Equations

The following provides a sufficient condition for the existence of a solution of (2.8.5) with x(t) > k(t) eventually. Theorem 2.8.5. Assume 0 < b < a< oo and ft'; r(t)dt < oo. Then Eq. (2.8.5) has a solution x( t) satisfying that x( t) > k( t) eventually. Proof: Choose T so large that

2b =

b* < k(t)
t ~T

and

1

00

T

rtdt
b*

{

b*

2(b*+2(a*+1))' (1+a*)(a*+1-b*)

-

}.

(2.8.9)

Let C B be the Banach space of all bounded continuous functions defined on R+ with the sup norm. Define a subset of CB as follows:

n

t

U = {x EX: a* $ x(t) $a*+ 1,

~

0}.

Clearly n is a nonempty, bounded, closed and convex subset of CB. Define a mapping T : U --. C B as follows:

(Tx)(t)= {

a*+ftoor(s)x(s)["'<~(ar> -l]ds, t~T 0 $ t $ T.

(Tx)(T), For any

X

E

n and t ~ T

a*+ 1 ) a"$ (Tx)(t) $a*+ (a*+ 1) ( --;;;- -1 Hence TU CU. For any Xt,X2

En, t

I(Txi)(t)- (Tx2)(t)i $1

~

00

00

lr(

r(s)ds $a*+ 1.

T we see that

r(s)[ixt(s)- x 2 (s)l

+ k!s) lxt(s)xt(s- nr)- x2(s)x 2 (s- nr)i] ds

(2.8.10)

80

Chapter 2 00

!5£ r(s) [11x 1 - x21i + : .. x1(s)!x1(s- nr)- x2(s- nr)l + : .. x2(s- nr)lx1(s)- x2(s)l] ds

!5 [1 +!(a*+ 1)]

£

00

r(s)ds1ix1- x21i

1

!5 2llx1 - x2ll· Hence

=sup I(Tx1)(t)- (Tx2)(t)J t~T

i.e., Tis a contraction on n. Therefore there is an X En so that Tx = x. It is easy to see that xis a solution of Eq. (2.8.5) and x(t) > k(t) fort ~ T. The proof is complete. 0

2.8.8. Generalize d delay logistic equation (I) We now consider the generalized delay logistic equation (2.8.11) where r, T E C(~,(O,oo)), r(t) !5 t, limr(t) = oo, k t-oo Make a change of variables: y(t) =

x(t)

k

> 0, a> 0, a =F 1.

-1.

(2.8.12)

Then (2.8.11) becomes

y'(t) = -r(t)(1 +y(t))y(r(t) )Jy(r(t))!a- l.

(2.8.13)

81

Oscillations of First Order Delay Equations

We are concerned with the existence of nonoscillatory solutions of (2.8.11) about k or the existence of nonoscillatory solutions of (2.8.13). First we establish a basic lemma. Lemma 2.8.1. A sufficient and necessary condition for the existence of a nonoscillatory solution of Eq. (2.8.13) is that there exist constant t1 and Ca such that the integral equation

.A(t) =

(1 + Caexp(- l>(s)>.(s)ds))iCal"'exp(lt r(s)>.(s)ds) + a) ir(t) r(s).A(s)ds). 1

(1-

X

r(t)

t1

(2.8.14)

has a continuous positive solution. Proof: If Eq. (2.8.13) has an eventually positive solution y(t), then there exists t1 > 0 such that

y(r(t)) > 0, Let >.(t) = - ~' of (2.8.14).

y'(t) < 0,

t;:: t1.

y(t 1 ) = C01 > 0, then >.(t) is a positive continuous solution

Similarly, if Eq. (2.8.13) has an eventually negative solution x which is larger than -1 then -1 < Ca < 0. Conversely, if (2.8.14) has a positive continuous solution >.(t), then

y(t) = Caexp ( -

Lt >.(s )r(s )ds)

is a nonoscillatory solution of Eq. (2.8.13). The proof is complete.

(2.8.15) D

We discuss the following two cases for a respectively i) 0
ii) a > 1. Theorem 2.8.6.

Assume that a E (0, 1). Then a necessary and sufficient

Chapter£

82

condition for every solution of (2.8.13) to be oscillatory is

Proof: i) Nece&sity.

l,.,

r(t)dt = oo.

(2.8.16)

1

r(t)dt

< oo,

(2.8.17)

H 00

then there exists t 0

~

0 such that

1

00

Co

r(s)ds

< -

1

a 1 :=

(2- et)e(1 + Ca)Ca

k,

where Ca is a positive number. Set T0 = infr(t). Let C([T0 ,oo),R) denote a locally convex of all cont>co tinuous function; defined on [ro, +oo) endowed with the topology of uniform convergence on compact subsets of [T0 ,oo). Define a subset n of ([T0 ,oo),R) as follows:

n=

{x E C([T0 ,oo),R): x(t) ~ 0,

lx(t)l ~ e(1 + Ca)c:-I, t ~ T0 }.

Clearly, n is a nonempty, bounded, closed and convex subset of C([To,oo),R). Define a mapping T on n as follows:

c:- 1 (1 + Caexp(- I:. r(s)x(s)ds)) (Tx)(t) =

{

X

exp((1- a) J;!') r(s)x(s)ds + J:(c) r(s)x(s)ds),

(Tx)(T), It is easy to see that Tfl c n. To prove T is a completely continuous operator on prove that the integral operator

( l

('Hx)(t) = exp (1- et)

r(t)

To

r(s)x(s)ds

+

1'

n,

it is sufficient to

r(s)x(s)ds

)

r(t)

is continuous on compact subsets of [T0 , +oo) and 'Hfl is relatively compact.

Oscillations of Fir&t Order Delay Equations

83

Assume {x;}, X En and x; - t X as i-t 00 which means that .lim x; ( t) = x( t) uniformly on any compact subset of [To, oo). In view of 1-+00

[' r(s)x(s)dx $ }To

ec~- 1 (1 +Cor)

( r(s)ds < oo }To

and using the Lebesgue's bounded convergence theorem we obtain

(1lx;)(t)

-t

(1ix)(t),

as

i-t

oo,

i.e., Tis continuous. From the integrability of r( s )x( s )ds, it is easy to see that 1l!l is equicontinuous and uniformly bounded. Then ?i!l is relatively compact. It follows that Tis a completely continuous operator on n. Using the Schauder-Tychonov fixed point theorem, there exists a A E n such that

J;:

(

x C~- 1 exp (1- a)

l r(t) r(s)A(s)ds + 1t To

)

r(s)x(s)ds ,

t?. T.

r(t)

By Lemma 2.8.1, Eq. (2.8.13) has a nonoscillatory solution. ii) Sufficiency. If (2.8.13) has an eventually positive solution, by Lemma 2.8.1 there exist it. Cor, and a continuous function A(t) satisfying

X

( i r(t)

c~- 1 exp (1- a)

A(s)r(s)ds +

1t

t,

?.

( i r(t)

c~- 1 exp (1- a)

t,

A(s)r(s)ds +

A(s)r(s)ds

r(t)

1t

A(s)r(s)ds

r(t)

Set

z(t) = exp(-

lt

(1- a)r(s)A(s )ds),

)

)

Chapter~

84

then

Integrating it we have

which implies that lim z(t) = -oo, a contradiction. t-+oo Similarly, we can show that Eq. (2.8.13} has no eventually negative solution y(t) with 1 + y(t) > 0. The proof is complete. Theorem 2.8. 7. Assume that o > 1. Then a necessary and sufficient condition for the existence of a nonoscillatory solution of (2.8.13) is that there exists a nonnegative continuous function A(t) such that

( l

exp (1- o)

T(t)

A(s)r(s)ds +

T

1t

A(s)r(s)ds

)

:5 mA(t),

t?:. T

(2.8.18)

T(t)

where m and T are some positive constants. Proof: i) Sufficiency. First we consider the case that Then there exist l, T and Ca > 0 such that

I;; r(s)A(s)ds

~00 A(s)r(s)ds < i (1 + Ca)c:- 1 < ...!.._ • mt

Define a mapping Ton C([To,oo},R+) as follows

J,

00

(Ty)(t) =

{

r(s)(1 + y(x))y 0 (T(s))ds,

t?:. T

(Ty)(T) + Caexp{- J;o r(s)A(s)ds) - Caexp{- I~ A(s)r(s)ds),

Obviously, Tis an increasing operator.

To :5 t :5 T.

< +oo.

85

Oscillations of First Order Delay Equation.s Set

Yo= OQexp(Yn+t

= Tyn,

n

£

.X(s)r(s)ds)

= 1, 2, ....

Then we see that

Yo(t);;::: Yl(t);;::: · · · ;;::: Yn(t);;::: · · · 2:: 0,

(2.8.19)

t 2:: To.

In fact,

y 1 (t) =

(Tyo)(t) $1 r(s)(1 + OQexp(00

X

$

o:exp(- a

o:(l + OQ)m

$ OQ

1:

.X(u)r(u)du))

£.-(•) A(u)r(u)du )ds

loo r(s).X(s)ds exp(- 1: .X(s)r(s)ds)

exp(- £r(s).X(s)ds) = Yo(t),

t;;::: T.

Combining this together with the increment ofT we obtain (2.8.19). Then lim Yn(t) = y(t) ;;::: 0, t;;::: To exists. n-oo

According to the Lebesgue's convergence theorem

ftoo r(s){1 + y(s))yQ(T(s))ds,

y(t) =

t;;::: T

{ y(T) + OQexp(- J; .X(s)r(s)ds) 0

- OQexp(-

JJ. .X(s)r(s)ds)

To$ t $ T.

It is easy to see that y( t) > 0 on [To, T) and hence y( t) Therefore y is a positive solution of Eq. (2.8.13) on [T, oo ).

We now consider the case:

>

0 for all t ;:: T0 •

ft': r(s).X(s)ds = oo. Using (2.8.18) we see that

Chapter 2

86 for sufficiently large t

f1

00

~

T

r( u )(1 + Caexp(- J; r(s )>.(s )ds )c::- 1 exp( -a J;(t) r(s )>.(s )ds) )du exp(- J;r(s)>.(s)ds)

:5 (1 + c .. )c::- 1 m < ~ < 1. Hence there exists a sufficiently large T1 such that

1 r( 00

u) ( 1 + c..exp ( -

lu

£

:5 exp ( -

r(s )>.( s )ds)

r( s )>.( s

)ds) ,

c::- exp (-a J:(u) r( s )>.( s )ds)) du 1

t

~ Tt.

By using the similar method to the previous one, we can show that (2.8.13) has a nonoscillatory solution. ii) Necessity. If (2.8.13) has an eventually positive solution then from Lemma 2.8.1 there exists a continuous positive function >.(t) such that

>.(t) = (1 + c .. exp( X

-1:

>.(s)r(s)ds))

c::- exp((1- a) ltr(t) r(s)>.(s)ds + lt

r(s).\(s)ds)

1

r(t)

1

~ c::- 1 exp((1- a) ir(t) r(s)>.(s)ds + lt

r(s)>.(s)ds).

r(t)

t1

(2.8.20)

.

Let m = cd-~ Then (2.8.20) implies (2.8.18). If (2.8.13) has an eventually negative solution, then

( i

>.(t) ~ (1-IC.. I)IC::-tlexp (1- a)

h

r(t)

>.(s)r(s)ds +

1t r(t)

)

>.(s)r(s)ds ,

where !C.. I < 1. Thus (2.8.18) is true also. The proof is complete. In the following we derive some explicit sufficient conditions for the existence of nonoscillatory solutions of (2.8.13).

87

Oscillations of Fird Order Delay Equations

If a > 1 and

Corollary 2.8.1.

(1- a)

l

r(t)

r(s)ds

+

1'

r(s)ds :5 A,

t

~ t0 ~

0,

(2.8.21)

r( t)

It

where A is some positive number. Then Eq. (2.8.13) has anonoscillatory solution. In fact, (2.8.21) follows from (2.8.18) by taking ~(t)

= ~ > 0.

Remark 2.8.1. H

loo

r(t)dt

< oo

or {

00

Jo0

r(t)dt = oo,

and

1'

r(s)ds :5 A,

r(t)

t

~ t0 ~ 0.

Then (2.8.21) is satisfied. Example 2.8.1. Consider

y'(t) =- _2_1 (1 + y(t))y 2 ( v't).

t+

(2.8.22)

Since

-

1

.,fi

to

ds

--+ 1 +S

j'

ds +S 1 ,fi

--=ln(1+t)-ln(1+t+?'i)+ln(v't;+1) :5ln (y't; + 1),

the condition (2.8.21) is satisfied. By Corollary 2.8.1, Eq. (2.8.22) has a nonoscillatory solution. In fact, y(t) = is such a solution.

t

2.8.9. Generalized delay logistic equation (II) Here we discuss the equation

x'(t) = r(t)x(t) 1 - x(t- r(t)) 1- cx(t- r(t))

(2.8.23)

88

Chapter 2

where c E (0, 1), r, T E C(R+, J4), r(t) :5 t, lim (t- r(t)) = oo. When c = 0, t--+oo (2.8.23) reduces to (2.8.1). From an ecological point of view, we will restrict our attention to the bounded positive solutions of (2.8.23). Denote

Eo= {t- r(t): t- r(t) :50, t

~

0} U {0}.

The initial conditions for (2.8.23) which we consider are c- 1 > x(8) = 4'(8) ~ 0,

8 E Eo,

cp(O) > 0

(2.8.24)

where cp(8) is continuous on E 0 • Assume that r(t) ~ T > 0 for all t ~ 0. It is known, by the method of steps, the initial problem (2.8.23) and (2.8.24) always (locally) has a unique positive solution. Make the change of variables 1- x(t) y(t) = 1- cx(t) ·

Then (2.8.23) and (2.8.24) reduce to

y'(t) =- r(t) (1- y(t))(1- cy(t))y(t- r(t)), 1-c

(2.8.25)

and

y(8) = [1- cp(8)]/[1- c4'(8)],

8 E E0 .

(2.8.26)

Obviously, -oo < y(8) :5 1, y(O) < 1. Thus, the study of the oscillatory behavior of solutions of (2.8.23) about 1 is equivalent to the study of the oscillatory behavior of solutions of (2.8.25) with respect to zero. It is easy to see that the solution of (2.8.23) and (2.8.24) is positive, which implies that the solution of (2.8.25) and (2.8.26) satisfies 1 - y(t) > 0 and 1 cy(t) > 0. We call a solution of a differential equation global, if it exists for all t ~ 0. In the following, we will consider global solutions of (2.8.23). For this we first give a sufficient condition to guarantee the existence of global solutions of (2.8.23). Theorem 2.8.8. Assume that t - r(t) is continuous and nondecreasing, limt--+oo(t- r(t) = oo, and r(t) ~ T > 0 fort~ 0. Assume further that

Oscillations of First Order Delay Equations

89

r(t) 2: r 0 > 0 is continuous,

0'

=sup t~O

{it

r(8) d8}

t-r(t)

is finite, e"' < c-I, and that 0 < cp(8) < c- 1 for 8 E [-T(O), 0) and cp(O) < c- 1 e-"'. Then the solutions x(t) of (2.8.23) are bounded, and limsupx(t) ~ e"'. t-+oo

Proof: Since r(t) 2: 0 ~ t ~ t, we have

T

> 0 fort 2: 0, there exists at> 0 such that t = r(t). For

[' 1 - x( 8 - r( 8)) ) x(t) = x(O)exp ( Jo r(8) 1 _ cx( 8 _ r( 8 )) d8 ·

Since 0 ::; cp(8) < c-I,

(2.8.27)

1- u(8- r(8))/(1- cu(8- r(8))) ::; 1, thus for

O~t~t

1 - x( 8 - r( 8 )) d8 < [' r(8)d8 < f' r(8) 1cx(8- r(8)) - } -

}0

0'.

(2.8.28)

0

Obviously, (2.8.27) and (2.8.28) imply that

x(t) ~ x(O)e"' < c- 1 , Hence we have shown 0 ~ x(t) t. Then for t ~ t1, we have

0 ~ t ~ t.

for

< c- 1 for -r(O)

~

t

~

t. Assume that t 1 -r(t1 ) =

Thus

x(t)

~ x(to)exp1 1 r(8)d8. to

From this we have

x(t- r(t)) 2: x(t)exp(

-it

t-r(t)

r(8)d8)

Chapter£

90 and hence

x(t- r(t))

~ x(t)e_.,..

This implies for t ~ t :5 t1 1- x(t)e_.,. ) x'(t) ~ r(t)x(t) ( 1 _ cx(t)e-... .

Since r(t)

~

ro

> 0, we see that all positive solutions of (2.8.29)

with initial values less than 6, for 6 > 0, e.,.< 6 < c- 1 e... , will be bounded by 6 and have e.,. as their limit. Therefore, forE~ t ~ tt, 0 ~ x(t) < c- 1 < c- 1 e.,.. By repeating this argument (assume that h > t 1 is such that t 2 - r(t 2 ) = t 17 we can prove that (2.8.29) holds for t 1 ~ t ~ t 2 • Define ti+ 1 - T(t;+t) = t;, then t;-+ +oo as i-+ +oo), we see x(t) is bounded and satisfies (2.8.29) for all t ~ t, thus the solution x(t) of (2.8.23) is bounded by x(O)e.,. < c- 1 and lim sup x(t)

~e.,..

t-oo

This completes the proof of the theorem. The corresponding result for Eq. (2.8.25) is the following.

Theorem 2.8.9. Assume that r(t) and T(t) are the same as described in Theorem 2.8.8, and the initial function y(8), 8 E [-T(O), 0] satisfies y(8) ~ 1, y(O) > (1 - c-le-.,.)/(1 -e-.,.). Then the solution y(t) of (2.8.25) is bounded, and liminfy(t) t-oo

~

(1- ea)/(1- ce.,.).

Remark 2.8.2. As we mentioned before, when c = 0 (2.8.23) reduces to the logistic equation. In general, from the ecological point of view c is small, so that equation (2.8.23) is close to the logistic equation, and hence e.,. < c- 1 holds naturally.

91

Oscillations of First Order Delay Equations

In the following we will consider only global solutions of (2.8.23).

Assume that r(t) and r(t) are positive and continuous, Theorem 2.8.10. lim (t- r(t)) = oo, Jo"', r(t)dt = oo. Then eve.ry global solution of (2.8.23) is t-+oo either oscillato.ry with respect to 1 or tends to 1 as t -+ oo. Proof: By Theorem 2.8.8, a global solution x(t) of Eq. (2.8.23) satisfies 0 < x(t) < c- 1 • Then x'(t) < 0 fort> T*, where T*- r(T*) > T, and hence lim x( t) = p. 2::: 1 exists.

(2.8.30)

t-+oo

If p. > 1, then we have 1-p.

x'(t) :5 r(t)p. -1 - - , - cp.

for

t 2::: T*,

which leads to limx(t) = -oo which contradicts (2.8.30). Therefore p. = 1. Similarly, we can prove if 0 < x(t) < 1, fort 2::: T, then lim x(t) = 1. t-+oo Next, we would like to discuss oscillation for Eq. (2.8.25). Theorem 2.8.11.

Let r,

T

E C(~,(O,oo)), with lim (t- r(t)) = oo, t-+oo

liminf {'

r(8)d8

t-+oo lt-r(t)

> 1- c . e

Then eve.ry solution of Eq. (2.8.25) oscillates. Proof: Let y(t) be a negative solution of (2.8.25), then

y'(t)

+ 1r~~

y(t- r(t)) 2::: 0.

(2.8.31)

By Theorem 2.1.1 this is impossible under the assumptions of the theorem. Now let y(t) be a positive solution of (2.8.25). Then y'(t) < 0 and y(t) -+ 0 as t -+ oo by Theorem 2.8.10. Dividing (2.8.25) by y(t) and integrating from t- r(t) tot we have ln y(t

CD(t)) = 1 ~

Y

[' C lt-r(t)

r(s)(1- y(s))(1- cy(s)) y(s

C~(s)) ds (2.8.32)

Y

S

Chapter!

92

Let w be defined by w(t) = y(t- r(t)) . y(t)

Clearly w(t)

~

1. From (2.8.32) we have

ln w(t) = w(e) 1- c

r

lc-..(t)

r(8)(1- y(s))(1- cy(8))d8

(2.8.33)

where e E (t- r(t), t). Similar to Lemma 2.1.3 we can prove that w is bounded. Taking the liminf on both sides of (2.8.33) we obtain ln R.

-JJ-

~

,

where R. = lim inf w( t). Since t-+oo the theorem. Theorem 2.8.12. that

. f -1li mm 1-+oo 1- c lnt t

1' ( r

t-.. (t)

8)d8

(2.8.34)

$ ! , (2.8.34) contradicts the assumptions of e 0

Suppose r(t) is bounded above, and there is a t 1 > 0, such

i

t

t- .. (t)

1-c r(9)d9 $ - e

for t

~ t1.

Then Eq. (2.8.25) has a bounded nonoscillatory solution on [t,oo). Proof: Assume r(t) $ M for all t ~ 0. Let BC denote the space of all bounded continuous functions defined on~. with the supernorm. Let n c BC denote the subset satisfying: is Lipschitzian and nonincreasing on

(i)

y(t)

(ii)

lip

y(t)IR+ $ Me(1- a)/(1- c)

(iii)

y(t)

= 1- a,

(iv)

e y(t- r(t)) ~ y(t)e ~ y(t- r(t)),

t E [0, t],

R+

a E (0, 1) t ~ t1

(v) (1- a)exp(- -1 e [' r(9)d9) $ y(t) $ 1- a, -c}c,

t

~ t1

O$cillation$ of Fir$t Order Delay

1-a IY'(t)I:5Me 1 _c'

(vi)

93

Equatio~

t~t1

tER+,

where lip y(t)IR+ denotes the Lipschitzian constant of y over~· Let

t E [0, t1]

1 - a,

Yo(t) = {

t

(1- a)exp(- l~c fc, r(8)d9), t ~ t1.

Then y 0 (t) E {}and Sis nonempty. It is not difficult to see that {}is convex and compact. Define a mapping T: n-+ BC as follows: 1

(Ty)(t)

={

tE[O,ti]

a

(1--

~)exp(-

l:c

J,t, r(8))

x (1- y(8)) (1- cy(9))

11 ( 11~;~ 11 >) d8, t ~ t 1 •

It is easy to see that Tfl c n, and T is completely continuous. Hence, by the Schauder-Tychonov fixed point theorem, we conclude that T has a fixed point y in S. Obviously, y* is a positive, bounded solution of Eq. (2.8.25) on [t, oo ). The proof is complete.

=

=

Corollary 2.8.2. Ifr(t) r, T(t) T, rand Tare positive constants. Then every solution of (2.8.25) is oscillatory if and only if rre > 1 - c.

2.9. Notes Lemma 2.1.1 is obtained by Koplatadze and Canturija [97]. Lemma 2.1.2 is due to Kwong [101]. Lemma 2.1.3 is taken from Yu, Wang, Zhang and Qian [184], also see Jian [92], Erbe and Zhang [44]. Lemma 2.1.4 is adopted from [92], also see Elbert and Stavroulakis [33]. Theorem 2.1.1 is obtained by Koplatadze and Canturiza [97], a special case is by Ladas and Ladde, see [110]. Theorem 2.1.2 is based on [92] and [101], and related work can be found also in [184]. Theorem 2.1.3 and Theorem 2.1.4 are taken from Zhou [219], Theorem 2.1.5 is new. The results in Section 2.2 are taken from Li [124]. Theorem 2.3.1 and 2.3.2 are new, Theorem 2.3.3 and 2.3.4 are taken from Gyori [72]. The materials of the Section 5.4 are based on [184]. Theorem 2.5.1

94

Chapter£

and 2.5.2 are taken from Zhang and Gopalsamy [203], also studied by Ladas and Qian [106]. Theorem 2.6.1 is obtained by Tramov and Ladas, etc., by a different method, see [110 or 76]. Theorems 2.6.2 and 2.6.3 are extracted from Huang [84]. The proposition is established by many authors such as Gyori [74], Kwong [101], Li [126], Philos [150], Chen and Huang [15] and Chuanxi, Ladas, Zhang and Zhao [25]. Theorems 2.6.4 and 2.6.5 are based on Chen and Huang [15]. Corollary 2.6.2 is obtained by Hunt and Yorke (85] using a different method. Lemma 2.6.4, Theorem 2.6.6, and Corollary 2.6.4 are taken from Philos [150]. Corollary 2.6.5 is obtained by several authors, Wei [170], Gyori and Ladas [76]. Theorem 2.6.7 is based on Lalli and Zhang [117]. Theorem 2.6.8 is an improved form of a result by Gyori and Ladas [76]. Theorems 2.6.9, 2.6.10 and Corollaries 2.6.7-2.6.10 are taken from Nadareishvili (139]. An oscillation criterion for Eq. (2.6.5) with oscillating coefficients can be found in Yu, Wang, Zhang and Qian [184]. The first study of the oscillation of differential equations with advanced type is due to Zhang and Ding [199], and Zhang (202]. The corresponding results for differential inequalities with advanced type is due to Onose, see (76]. Theorems 2.6.11 and 2.6.12 are found in Zhou [217]. Theorem 2.7.1 is from Zhang [191], also see Erbe and Zhang (44]. Theorem 2.7.2 is a new result by Yu and Zhang. Theorems 2.8.12.8.3 are taken from Zhang and Gopalsamy (208]. The global attractivity of the delay logistic equation is studied by Zhang and Gopalsamy [207]. Theorems 2.8.4 and 2.8.5 are obtained by J.S. Yu and B.G. Zhang. Theorems 2.8.6 and 2.8.7 are due to Li [125], also see Aiello [4]. Eq. (2.8.23) without delay as an ecological model of the single species is posed by Cui and Law [27]. Theorems 2.8.8 2.8.12 are taken from Kuang, Zhang and Zhao [70], in which some results for the existence of periodic solutions and the global attractivity of solutions are obtained. There are many important results for oscillation and nonoscillation of nonlinear delay differential equations in the literature. Some of then can be found in the monographs by Ladde, V. Lakshmikantham and Zhang [110], Gyori and Ladas [76]. Some further results for delay ecological equations can also be found in the monograph by Gopalsamy [51].

3 Oscillation of First Order Neutral Differential Equations

3.0. Introduction In general, the theory of neutral delay differential equations is more complicated than the theory of delay differential equations without neutral terms. For example, Snow (see also Gyori and Ladas, Chapter 6, [76]) has shown that even though the characteristic roots of a neutral differential equation may all have negative real parts, it is still possible for some solutions to be unbounded. In this chapter, we will present a systematic study for the oscillation theory of first order NDDEs, which contains some recent results and consequently is a useful source for researchers in this field. In Section 3.1 we consider the linear neutral delay differential equations with constant parameters, where the "characteristic equation" method plays an important role. Some necessary and sufficient conditions for the characteristic equations to have no real roots are presented. In Sections 3.2 and 3.3 we consider the neutral delay differential equations

95

Chapter 9

96 with variable coefficients of the form

(x(t) + p(t)x(r(t)))'

+ q(t)x(t- u(t))

= 0

(3.0.1)

where q(t) ;::: 0 and -1 :5 p(t) :5 0, p(t) := -1, p(t) :5 -1, or p(t) > 0, respectively. Since the behavior of solutions of Eq. (3.0.1) is closely dependent on the range of p(t), we deal with the case that -1 :5 p(t) :5 0 in Section 3.2, and the other cases in Section 3.3. Some sharp conditions for all solutions to be oscillatory and for the existence of positive solutions are presented. In Section 3.4, we present a comparison result for oscillation. In Section 3.5, we consider Eq. (3.0.1) in the unstable case. In Section 3.6, we discuss a class of sublinear neutral delay differential equations. In Sections 3. 7 and 3.8, we deal with the neutral delay differential equations with positive and negative coefficients of the form

(x(t)- c(t)x(t- r))'

+ p(t)x(t- r)- q(t)x(t- u) = 0.

Oscillation criteria, criteria for the existence of positive solutions and results of linearized oscillation are given. In Section 3.9, we are concerned with the neutral delay differential equations with a nonlinear neutral term. In particular, the equation

(x(t)

+ px

0

(t- r))' + q(t)xft(t- u) = 0

is considered, where a =f. 1, f3 =f. 1. The deviating arguments in the neutral differential equation (3.0.1) can be of various types. Usually, we investigate the following cases:

1) r(t) :5 t,

u(t);::: 0,

2) r(t) :5 t,

u(t) :50,

3) r(t) ;::: t,

u(t);::: 0,

4) r(t);::: t,

u(t) :50,

and the mixed type. In this chapter we mainly discuss case 1), which corresponds to the neutral delay differential equations. But the methods and techniques employed here can be used for equations of other types.

Oscillation of First Order Neutral Equations

97

3.1. Characteristic Equations We first state a basic result on the oscillation of the neutral delay differential equation with constant parameters

(x(t)+ where p;, r;, and q;,

~p;x(t-r;))' + ~q;x(t-u;),

u;

are given real numbers, and r;,

t

~to

u; ~ 0,

(3.1.1)

j E Im, i E Ik.

Theorem 3.1.1. Every solution of Eq. (3.1.1) is oscillatory if and only if its characteristic equation m

k

>. + >. :~:::>;exp( ->.r;) + j=l

L q; exp( ->.u;) = 0,

(3.1.2)

i=l

has no real roots. The above proposition can be extended to higher order NDEs. From Theorem 3.1.1 the characteristic equation (3.1.2) plays an important role in the investigation of the oscillation and asymptotic behavior of solutions of Eq. (3.1.1). But to determine if (3.1.2) has a real root is quite a problem itself. In the following we derive other necessary and sufficient conditions for the oscillation of Eq. (3.1.1) which can be easily applied to get some explicit sufficient conditions. For sake of convenience, we consider m

(x(t)- P1x(t- r1)- pzx(t- rz))'

+ L q;x(t- u;) = 0

(3.1.3)

i=l

where Pi (j = 1,2,), q;, u; (i = 1,2, ... ,m) E R+, T; > 0 (i = 1,2). From Theorem 3.1.1 Eq. (3.1.3) is oscillatory if and only if its characteristic equation

F(>.)

= >.(1- p1e-.Xr, -

m

pze-.Xr2 )

+ L q;e-.Xu; = 0 i=l

has no real roots, or, F(>.)

> 0 for all>.

E R.

(3.1.4)

Chapter 9

98

From (3.1.4) we see that if p 1 + p 2 = 1 then every solution of Eq. (3.1.3) is oscillatory. For the case that 0 < p 1 +p 2 < 1, we define a subset D of the space .e1 and a function f on D as follows:

D = {t = (t;;k): (ijk) E /1, t;;k:::: 0,

~1 t;;k

= 1}

and

f( t ) =

'<"'

LJl

. ZT1

+

(k t;;k') - 1 T2

+ Uj

i k-i 1n ( e cikp1p 2 q;

r·tr1 + (k - t.) T2 + u; l/t;;k )

(3.1.5)

where / 1 ={(ijk): j=1, ... ,m; k=0,1, ... ; i=O, ... ,k},

~1

=

L:

,and

(ijk)Elt

in (3.1.5) t;;k = 0 implies the corresponding terms vanish. Here Ct = i!(f~i)! . It is easy to see that the series in (3.1.5) is convergent in D and then f(t) is well-defined on D. In fact, fortE D '<"'

LJl

.

zTl

t;;k')

+ (k -

t T2

+ ui

In (e cikP i1 P2k-i q;

r· + (k tT1

.)

t T2

+ u j l) < oo

by the comparison test and the definition of D. Also

the right hand side is convergent for t E D.

Assume 0 < Pl + P2 < 1. Then i) f(t) achieves its maximum at a point t 0 = (t?;k) on D which is completely detennined by the condition that

Theorem 3.1.2.

has the same value for all (ijk) E Ill t?ik =f. 0. ii) Eq. (3.1.3) is oscillatory if and only if f(t 0 ) > 0.

Oscillation of First Order Neutral Equations

99

For the case that p 1 + P2 > 1, we define the following three subsets Db, b = 1,2,3, of the space £1, and three functions hb(t) defined on Db, b = 1,2,3, · 1y: (u' · t h e expansiOns · b elow c;-(k+ ) = (-k-1)(-k-2) (-le-i) respective vve h ave m i! ··· 1 i,k=1,2, .... ) D1 = {S = (S;;~c): (ijk) E 12, S;;~cC~(k+l)[(k + i

+ 1)-r1- i-r2- u;] 2:: 0,

E2S;;1c = 1}, D 2 = {S = (S;;~c): (ijk) E 12, S;;~cC~(k+ 1 )[(k E2S;;~c

= 1},

D 3 = {S = (S;~c): (j, k)

e 13 ,

h (S) 1

= E2

(k

S;;~c

+ i + 1)7"1 -

i7"2 -

S;~c[(k

+ i + 1)-r2- i-r1- u;] 2:: 0,

+ 1}-rt- u;] 2:: 0,

EaS;1c = 1},

O"j

; x ln ( eC -(lc+l)P 1-(k+i+l) p;2 q; [(k + z. + 1) T1

-

. - u; ]/Sij/c ) , z-r2 (3.1.7)

h (S) = E 2 2 (k + i X

S;;1c

+ 1)7"2- i7"1- O"j

; ; -(k+i+l) q; [(k + z. + 1) 1"2 ln ( e C -(k+1)P1P2

.

- ZT1 -

u; ]/Sijk ) , (3.1.8)

(3.1.9) where12={(ijk): j=1, ... ,m; k, i=1,2, ... }, 1a={(jk):j=1, ... ,m, k = 0, 1, ... }, E2 = L: , L:a = L: , where S;;1c = 0 or S;k = 0 imply that (ijk)E/2

(jk)Ela

the corresponding terms vanish. Different from the series in (3.1.5), we cannot guarantee that all the series in (3.1.7) - (3.1.9) are convergent in Db, instead, as shown in the Remark 3.1.1, we see there exists at least one series which is convergent on Db. Theorem 3.1.3.

Assume p 1

+ P2 > 1.

Then

i) for some b = 1,2, or 3, hb(t) has a maximum at a point S(b) = (S~Ji), b = 1, 2, or s< 3 > = (Sm) on Db, which is completely determined by one of

Chapter 3

100 the conditions that 1

(k

+ i + 1h X 1n

ir2- ui

; -(k+i+I) ; [(k (c -(k+l)Pl p 2 qj + t. + 1)T1

.

- tT2 - U j

]/S(l)) ijk , (3.1.10)

1

(k

+ i + 1)r2 X 1n

ir1 -

Uj

; ; 2-
-

Uj

]/S(2)) ijk , (3.1.11)

or

(3.1.12)

has the same value for all (ijk) E I 2 or (jk) E [ 3 , such that

s}fi =f:. 0,

d=

1,2, orS}!l =f:.O, and(k+i+1h -ir2-ui =fO, or(k+1h -u; =f:.O. ii) Eq. (3.1.3) is oscillatory if and only ifO < hb(S< 6>) < oo for some b = 1, 2, or 3. Theorems 3.1.2 and 3.1.3 can be applied to get a series of sufficient conditions for oscillation of Eq. (3.1.3). To use Theorem 3.1.2 we need to find a t• E D such that f(t'") > 0, whereas, to use Theorem 3.1.3, we need firstly to choose a suitable function hb which is convergent on Db, and then find aS* E Db such that hb(S'") > 0. The following corollaries are derived from Theorems 3.1.1 and 3.1.3, where m )1/m q= ( q; ,

II

i=l

Corollary 3.1.3. Assume Pt + P2 < 1. Then each one of the following is sufficient for Eq. (3.1.3) to be oscillatory:

(3.1.13)

Oscillation of First Order Neutral Equations 00

ii)

mq

k

101

.

2:: 2:: Cip~p~-'[iTI + (k- i)T2 + u];::::

~,

k=O i=O

Corollary 3.1.4. Assume P1 i) Let T1 f. T2, and E3 ( 2pl ) - ( k+l) qi [(k

+ P2 > 1.

+ 1) T1 -

-ln u i ] e (.,- •• 1

llP2 )((k+l)r1 -a,·]

= 1.

(3.1.14)

Then Eq. (3.1.3) is oscillatory if and only if (3.1.15)

Otherwise, each one of the following is sufficient for Eq. (3.1.3) to be oscillatory: " L.J'

ii)

ci-(k+l)Pl-
-

. tT2

Uj

or mq

l >_ -, 1

(3.1.16)

e

(ijk)E/2

"' L.J c;-(k+l}Pl-
. + 1)TI

.

t

-

tT2 -

I;= { (ijk): G~(k+l) > 0, (k + i + 1)Tl -

iT2- O"j

> 0,

(ij)EJ2

U

l> 1 _ e

(3.1.17)

where

(k + i + 1)T2 - iT1 -

O"j

J2 = {(ik): G~(k+l) > 0, (k+ Z.+1) T2

-

· tT1 -

U

> 0,

and

)k+ 2i+l (k + i + 1)T2- iTt- u·1 ( Pl . . P2 ( k + t + 1 )TI - t T2 - u i

> 1}

(k + i + 1)Tl- iT2- U > 0, >O

an

d

(Pl)k+ 2i+l(k+i+1)T2-iTI-0" (k . . P2 + z + 1)Tl - tT2 - u

iii) (3.1.16) and (3.1.17) bold ifp1 and p2 , respectively.

T1

and

T2

> 1}

exchange their positions

102

Chapter 8

It is easy to see that 12 I 0 (empty set) and has infinitely many terms provided PI > P2, or PI = P2 and TI > r2. To prove Theorem 3.1.2 we need the following lemmas.

Lemma 3.1.1.

Assume 0
GI(,\) = ,\ +

m

E qie-A"; (1- PI e-Ar, - P2e-Ar•)-I'

(3.1.18)

i=I

and let,\ .. < 0 be such that Pie-A"r, Ao E ( ,\ .. , oo) such that

+ p2e-A"r• = 1.

Then there exists a

(3.1.19)

and GI(.\o) is the minimal value ofG1(.\) in (,\•,oo). Proof: Noting that G 1 ( oo) = oo and GI (,\ * + 0) = oo, the conclusion is obvious 0 by the continuity of G 1 (.\). Lemma 3.1.2. let

Assume 0 < Pl

+ P2 <

1, Ao is defined by Lemma 3.1.1, and

Then

i) t 0 = (t~jk) ED, ii) (3.1.6) has the same value for all (ijk) E /1, t?ik

I

0,

iv) f(t 0 ) is the maximum value of f(t) on D. Proof: By (3.1.19) there exists a neighborhood A on ,\ 0 such that

103

Oscillation of First Order Neutral Equations

G1(.X) =.X+

m

oo

i=1

k=O

L q;e-~.,.i L(p1e-~r

1

+ J>2e-~1"2)k

Thus

Since G' (.Xu) = 0 we have

i.e., ~1t~jk = 1, or t 0

for£?;,.

f. 0.

e D.

From (3.1.20)

Furthermore by (3.1.5) and (3.1.21)

~

f(to) =

{ 1

£?;,.

ir1+(k-i)r2+u;

x ln(C~p~p~-iq;[ir1

+

i?;~o

ir1+(k-i)r2+u;

+ (k- ih + u;]/~;,.)}

(3.1.22)

= ~ 1 { C~p~p~-iq;e-~o[irt+(k-i)T"2+cri) + .Xoi~;,.}

Fort=

(t;;~o)

> ~1

-

eD

•

ZT1

= f(t).

from (3.1.22) we have

+(k t~;" ') - I T2

; ; k-i q; [.tT1 + (k -z.)T2 +u ·l/t o.. ,. ) + Uj 1n ( e0 kP1P2 1 IJ D

104

Chapter 9

Proof of Theorem 3.1.2: The first part is shown by Lemma 3.1.2. Assume Eq. (3.1.3) is oscillatory. Thus F(A) > 0 for F defined by (3.1.4) and all A E R. In particular, m

F(Ao)

= >.o(1-ple-.>.or1 -P2e-.>.or2) + Lq;e-.>.o"'i

>0

j=l

where Ao satisfies 0 < Ple-.>.0 '"1 + P2e-.>.o.-, < 1 by Lemma 3.1.1. Hence m

GI(Ao)

= Ao + Lq;e-.>.o"'i (1- Ple-.>.o.-1- P2e-.\or2)-1 > 0. j=l

From Lemma 3.1.2 we have f(t 0 ) = G 1(>. 0) > 0. On the other hand, if f(t 0 ) > 0, by Lemma 3.1.2, G 1 (>. 0 ) > 0. Hence G1(A) > 0 for all A E (>.'",oo), where A* is given by Lemma 3.1.1. If Eq. (3.1.3) is not oscillatory, then there exists a A1 E R such that m

F(At)

= >.1(1- Ple-.>.

1 - p2e-.>.1'"2)

1 '"

+ Lqie-.>.1"'i

= 0.

(3.1.23)

j=l

Obviously A1 < 0 and thus 0 < p 1e-.>.1n (3.1.23) gives us that

+ P2e-.>.1'"2 <

1, i.e., A1 E (A*,oo).

m

0

= A1 + Lq;e-.>.1"'i (1- P1e-.>.1n - P2e-.>.1'"')-1 =

G 1(>. 1)

j=l

which contradicts G1(A)

> 0 on (A*,oo).

0

To prove Theorem 3.1.3 we need the following lemmas. Lemma 3.1.3.

Assume Pl

+ P2 > 1. Define m

G2(1J)

= -I'+ Lq;e-P"'i (p1e-P'"1 + P2e-P'"2 -

1) -1,

j=l

and let IJ* > 0 be such that p 1 e_,..,.1 !Jo E ( -oo, IJ*) such that

+ p2 e_,..,. = 1. 2

Then there exists a

(3.1.24)

105

Oscillation of First Order Neutral Equations

and G2{Po) is the minimum value of G2(p) in {-oo, p*). 0

Proof: Similar to Lemma 3.1.1. Assume p 1 i) la.e-l'o(r,-rt) < 1 and

Lemma 3.1.4.

p 0 is defined by Lemma 3.1.3. Let

'

P1

ii)

+ P2 > 1,

sg! = C~(k+l)P~(J:+i+ 1 )p~qj((k + i + 1)r1 -

ir2 - O"jjel'o[(k+i+l)r1-i1'2-CTjJ,

(ijk) E 12.

Then i) S< 1> =

(sgl) E D1,

ii) (3.1.10) has the same value for all (ijk) E 12, iii) h1 (S< 1>)

sm =/:

0,

= G2(Po),

iv) h 1(S< 1>) is the maximal value of h1(S) on D1. Proof: By (3.1.24) and condition i), there exists a neighborhood U of po such that 0 < (p1e-""1 + P2e-""2)- 1 < 1, and ~e-"("2 -rt) < 1 for p E U. m

G2(p)

= -p + Lqie-""'; (p1e-""1 + P2e-""2)-1 [1- (p1e-""1 + p 2 e-""2)- 1]-1 j=1 m

= -p

+L j=l

co

L qie-1'"'; (p1e-""1 +P2e_,.,.,) -(1:+1) k=O

(3.1.25) P-(J:+l)p;q·e"[(J:+1)r1-ir,-cr;] ~ Ci -- _,_,... + "-'2 • 2 1 -(k+1) 1

{3.1.26)

The rest of the proof is similar to that of Lemma 3.1.2 and hence is omitted. 0 Lemma 3.1.5. Let the conditions of Lemma 3.1.4 hold ifp1 and P2, r 1 and are replaced by s~J!. Then the 7'2 exchange their positions, respectively, and

sZ!

Chapter 9

106

conclusion of Lemma 3.1.4 holds if s(l>, D~, h~, and (3.1.10) are replaced by D 2 , h2 , and (3.1.11), respectively.

s< 2>,

Lemma 3.1.6. Assume p 1 i) l!.a.e-l'o(r.-rt) = 1 and Pt

+ p 2 > 1,

p. 0 is defined by Lemma 3.1.3. Let

'

ii) S~!) = (2pi)-(k+l)qj[(k+1)TJ-O'j]e~' 0 [(k+l)rt-D'i], (jk) E 13.

(3.1.27)

Then

i)

s< 3> = (SW) E D3,

ii) (3.1.12) has the same value for all (jk) E 13, S~!)

f 0,

iii) h3(S< 3>) = G2(P.o),

iv) h3 (S< 3 >) is the maximum value of h 3 (S) on

D3.

Proof: From (3.1.25) and condition i) (3.1.28) The rest of the proof is omitted.

Remark 3.1.1. There exists at least one of hb which is well-defined on Db (b = 1, 2, 3). In fact, there is at least one expansion of (3.1.26), its dual form, and (3.1.28), of G 2 (p.), which converges at p. = p.0 • Therefore, at least one of hb(s< 6>), (b = 1, 2, 3), has a finite value and satisfies hb (S< 6>) = G 2 (p. 0 ). Proof of Theorem 3.1.3: The first part is shown by Lemmas 3.1.4- 3.1.6. Assume Eq. (3.1.3) is oscillatory. Then F(p.) > 0 for all p. E R. In partieular, m

F(p.o) = P.o(1- Pte-l'ort- P2e-l'or•)

+ Lq;e-I'OD'i > 0 i=l

where p. 0 satisfies (3.1.24). Hence m

G2(P.o)

= -p.o + Lqie-l'oD'i (Pt e-l'ort +P2e-l'or2 j=l

1) -1

> 0.

107

Oscillation of First Order Neutral Equations

By Remark 3.1.1 there exists b = 1, 2, or 3, such that hb(S(b)) = G2(!-Lo). Therefore, 0 < hb(S(b)) < oo for some b = 1, 2, or 3. On the other hand, if 0 < hb(S(b)) < oo for some b = 1, 2, or 3, by Remark 3.1.1, G2(fLo) > 0. Hence G2(fL) > 0 for all fL E ( -oo,fL*). Assume Eq. (3.1.3) is not oscillatory. Then there exists a ILl E R such that m

F(fLl) = /Ll (1- Ple-,.,r,- P2e-"'r2 )

+ Lqie-l'ta;

=

0.

j=l

Obviously, /Ll

> 0, and thus Pl e-"' r, + P2e-~'' r 2 > 1, i.e.,

/Ll E ( -oo, fL*), and

m

G2(/Ll) = -ILl

+ Lqie-"'a; (ple-"' r, + p2e-"' r

2 -

1) -l = 0,

j=l

contradicting that G 2(fL)

> 0 on ( -oo, fL*).

0

Proof of Corollary 3.1.3: i) Choose

such that l:1tijk = 1. By (3.1.13) we have 0

< c ~ 1. Hence

i k-i qic 1n -1 > ~ c;kP1P2 f(t *) = .c..1e _ o. c

Noting that t• does not make (3.1.6) have the same value for (ijk) E I, we see t* =f. t 0 • So f(t 0 ) > 0. By Theorem 3.1.2, Eq. (3.1.3) is oscillatory. The proofs for ii) and iii) are similar. In fact, for ii) we choose

0 Proof of Corollary 3.1.4:

i) The conditions of Lemma 3.1.6 are satisfied, where fLo = _l_lnl?.!.. P2 rt -r2

Chapter 9

108

Hence from (3.1.9) and (3.1.27)

Thus h 3 (S< 3 >)

> 0 if and only if (3.1.15) holds.

ii) Without loss of generality we may assume the left hand side of (3.1.16) is finite, for otherwise we can replace I2 by its suitable subset in (3.1.16) such that the above assumption holds. Choose S'!'.k IJ

i [(k +'· + 1)Tt = { eC i-(lc+t)Pt-(Hi+I) P2q; 0,

such that 'E 2 Si;~c From (3.1.7)

= 1, i.e., s• = (Si;~c) E D2.

h I (s *)

u; ]c, (.l:J"k) E I*2 (ijk) E !2\!2

By (3.1.16) we have 0 < c :5 1.

i In ~ 1 > e ci-(k+t)Pt- p2q;c - o.

" L.J'

=

·

-tr2-

(ijlc)E/2

From (3.1.8) and the definition of !2

ln

[( PI) k+2i+l (k(k + i. + 1)r2 1)

+I +

P2

ir1 .

u;] > 0.

Tt - lT2 - Uj

By Remark 3.1.1 we get that there exists a b = 1 or 2 such that h6(S< 6>) < oo. Noting that S* does not make (3.1.10) have the same value for (ijk) E 12 , hence hb(S< 6>) > hb(S*) ~ 0. By Theorem 3.1.3, Eq. (3.1.3) is oscillatory. iii) The proof here is similar and so is omitted. 0 If we let p 1

= p,

P2

= 0,

r1

= r, then Eq.

d dx [x(t)- px(t- r)]

(3.1.3) becomes

m

+ ~q;x(t- u;) = J=l

0,

(3.1.29)

109

Oscillation of First Order Neutral Equations and the above results can be reformulated as follows. Define two sets

D 1 ={t=(t;k):t;k~O,j=1, ... ,m;

k=0,1, ...

;L Lt;k=1, j=l m

oo

}

k=O

D 2 = { S = (S;k): S;k((k + 1)r- u;};:::: 0, j = 1, ... ,m; k = 0, 1, ... ;

L j=l m

oo

}

L:S;k=1 . k=O

On D 1 we define a function

on D 2 we define a function m

oo

h(s)=L L

S jk

J=l =0 (k + 1)rl - u 3· .

ln(ep-(k+llq;((k+1)r-u;)/S;k)·

k

Theorem 3.1.4. Assume 0 < p < 1. Then i) f(t) has a maximum at a point t 0 = (t~k) on D 1 which is determined by the condition that

has the same value for all j = 1, ... , m, k = 0, 1, ... , t~k ii) Eq. (3.1.29) is oscillatory if and only if f(t 0 ) > 0.

:f:. 0.

Theorem 3.1.5. Assume p > 1. Then i) h(S) has a maximum value at a point SO = (SJk) on D2, which is determined by the condition that

Chapter 3

110

has the same value for all j = 1, ... , m, k = 0, 1, ... , SJk f- 0, (k+1)T-l7jf-0. ii) Eq. (3.1.29) is oscillatory if and only if h(S0 ) > 0.

Corollary 3.1.5. Assume 0 < p for Eq. (3.1.29) to be oscillatory: m

oo

I: I: pkqi(kr + l7j) 2::

i)

< 1. Then each of the following is sufficient

~'

j=I k=O 00

I: pk(kr + u) 2::

ii) mq

~'

k=O

m

oo

I: I: pkqj,

iii) let a=

•

a;k = k~.:~., j = 1, ...

j=l k=O

m

oo

E I:

~ ln(ea(kr

,m, k = 0, 1, ... , and

'

+ u;)] 2:: 0.

j=l k=O

Corollary 3.1.6. Assume p > 1. Then each one of the following is sufficient for Eq. (3.1.29) to be oscillatory:

i)

there is a k; for each j = 1, ... , m, such that (k;

ii)

there is a k0 such that ( k0 00

mq

E

k=ko

+ 1)r -

+ 1)r- l7j

> 0, and

l7 > 0, and

p-(k+I)[(k + 1)r -u] 2:: ~·

3.2. Equations with Variable Coefficients (I) 3.2.1.

We consider the linear neutral differential equations of the form

(x(t)- p(t)x(t- r))'

+ q(t)x(t -u(t)) = 0,

t 2:: to.

(3.2.1)

Clearly, the oscillatory behavior of solutions of (3.2.1) depends on the range of p. In this section, we discuss the case that 0 ~ p ~ 1.

111

Oscillation of First Order Neutral Equations

The following result reduces the oscillation problem of NDE (3.2.1) to a corresponding problem of delay differential inequality of type (2.1.1).

Assume that

Theorem 3.2.1.

i)

T

E (O,oo),

p, q, u E C([to,oo),R+) such that 0

1

00

~

q(t)dt

= oo

and

~

lim (t- u(t))

t-oo

p(t)

~

1,

= oo;

ii) there exists a positive integer N such that the differential inequality N-1

y'(t)+q(t)[1+p(t-u(t))+ ..

·+IT p(t-u(t)-ir)]y(t-u(t)) ~ 0 (3.2.2) i=O

has no eventually positive solutions. Then every solution of (3.2.1) is oscillatory. Proof: Assume the contrary, and let x(t) be an eventually positive solution of (3.2.1). Set

z(t) = x(t)- p(t)x(t- r).

(3.2.3)

Then z'(t) ~ 0. If z(t) < 0 eventually, then x(t) ~ p(t)x(t- r) ~ x(t- r) eventually, which implies that x(t) is bounded. Hence z(t) is bounded, and lim z(t) = e exists. From (3.2.1), we obtain that t-oo

1

00

q(t)x(t- u(t))dt < oo.

to

This together with condition i) derives that liminf x(t) = 0. Hence there exists t-oo a sequence {tn} such that lim tn = oo and lim x(tn) = 0. Since n-+oo

n-+oo

+ r)- z(tn)) = n~~ (x(tn + r)- (p(tn + r) + 1)x(tn) + p(tn)x(tn- r))

0 = lim (z(tn n-oo

we see that

Chapter 9

112

which implies that lim p(tn)x(tn- r) = 0 and hence n--+oo

R.

= n--+oo lim (x(tn)- p(tn)x(tn- r))

= 0.

This contradicts the assumption. Therefore z( t) > 0 eventually. Obviously, x(t) > z(t) for all large t. Considering that z(t) is decreasing we have

x(t) = z(t) + p(t)x(t- r)

2:: [ 1 + p(t) + .. · +

!!

N-1

]

p(t- ir) z(t)

(3.2.4)

for all large t. Substituting (3.2.4) into (3.2.1) we obtain

z'(t) + q(t) [ 1 + p(t- u(t))

+ · ·· +

!!

N-1

]

p(t- a(t)- ir) z(t- a(t)) 50

D

which contradicts assumption ii).

Remark 3.2.1. Some sufficient conditions for Eq. (3.2.2) to have no eventually positive solutions have been presented in Theorems 2.1.1 and 2.1.2. In the following we shall give a nonoscillation result for Eq. (3.2.1).

Theorem 3.2.2. and T E (O,oo), u(t) T ;::: t 0 such that

Assume that p E

C([t 0 ,oo),~),

q E C([t 0 ,oo),(O,oo)),

= u E [O,oo). Further assume that there exist A* E (O,oo) and

sup [p(t- u) ( q(t) ) e.\"r t~T q t- T

+ ,1

q(t)e.\""] 5 1.

A*

Then Eq. (3.2.1) has a positive solution on [T, oo ).

(3.2.5)

Oscillation of First Order Neutral Equations

113

Proof: First we claim that the integral equation

w(t)

tc:} )

= p(t- u) q t

r

w(t- r)exp(

+ q(t)exp 1~, w(s)ds,

t 2::: T

r' w(s)ds) lt-T + m,

(3.2.6)

possesses a positive continuous solution on [T, oo ), where m =max{ r, u }. To this end, set w1 (t) = 0, t 2::: T, and fork= 1,2, ...

p(t- u) 9 (~~)T)wk(t- r)exp(J,'_T wk(s)ds) Wk+t(t)

={

+q(t)exp(J,'_, wk(s)ds), Wk+t(T+m),

t 2::: T

+m

(3.2.7)

T:::;t
Clearly, by (3.2.7) and (3.2.5), w 2 (t) > 0 = w 1 (t) fort 2::: T and w2(t):::; ..\* fort 2::: T. Assume for some positive integer k, Wk- 1 (t) :::; Wk(t) for t 2::: T. Then by (3.2.7) and (3.2.5), wk+ 1 (t) 2::: Wk(t) fort 2::: T, and

Wk+l (t) :::; p(t - u) q(:~)r) ..\ * e>.•T + q( t)e>.•" :::; ..\*sup [p(t- u) t;::T

q(t) e>.•T + _!_q(t)e>.•,] :::; ..\*. ..\*

q(t- r)

By induction we see that for t 2::: T

Then it follows that lim wk( t) = w( t) exists and is positive on [T, oo ). By taking k-oo limits on both sides of (3.2. 7) and by using the Lebesgue's monotone convergence theorem we see that w(t) is a solution of (3.2.6) on [T,oo). We also see that the sequences {wk(t)}f:1 converge uniformly on [T, T+m]. Then it follows by (3.2.6) that w(t) is a continuous function on [T,oo). Set z(t) = exp(- J;w(s)ds). Then w(t) =- £ill Z(t} and z'(t) < 0 fort 2::: T. Thus (3.2.6) reduces to

z'(t) = p(t- u) (q(t) ) z'(t- r)- q(t)z(t- u). q t- r

(3.2.8)

Chapter 9

114

Now define x(t) = - z'(t + u)/q(t + u). It is easy to see that x(t) is a positive solution of Eq. (3.2.1) on [T, oo ). 0 Theorem 3.2.3.

Assume that

i) p, q E C((to,oo),R), (O,oo);

0 :5 p(t) :5 p

<

r E (O,oo) and u(t) := u E

1,

ii) there exists p. > 0 such that

+ jq(t)!e""" :5p,,

p(t)p.e~'r

t;:::: to.

(3.2.9)

Then Eq. (3.2.1) has a positive solution on [t 0 ,oo). Proof: Set p(t)

= p(t 0 ),

t :5 t 0 , and

t + m- to ( ) - - - - q to , to - m :5 t :5 to q(t) = { m 0, t :5 t 0 - m where m = max{r,u}, thenp,q are well defined on Rand i) and ii) hold on R. Define a set of functions on R as: X={>.:>. E C(R,R), I.X(t)l :5p, fort;:::: t 0 where the norm of >. is II.Xll =

-

m, .>.(t)

=0

fort :5 t 0

-

m}

sup I.X(t)le- 2 ~-'t. Then X is a Banach space. t>to-m

Define a mapping on X as follow;;

('T.X)(t) = q(t) [ exp Clearly, ('T >. )( t)

!~.,. >.(s)ds + ~

= 0 for t :5 t 0 -

m.

U

p(t- u- jr)exp

j~u-ir >.(s)ds].

(T >. )( t) is continuous and

I(T>.)(t)! :5 jq(t)l [exp !~.,. l.X(s)lds

+~

U

p(t- u- jr)exp

:5lq(t)!(e"
j~.(s)!ds]

Oscillation of First Order Neutral Equations

115

i.e., T X C X. We show that T is a contraction on X. In fact, for AI, A2 E X, and t;::: to-m

I(TAI)(t)- (TA2)(t)l :5 q(t) [lexp

/,~u A1(s)ds- exp i~u A2(s)dsl

+~pi lexp ( i~u-ir AI (s )ds) :5 lq(t)le"u {

exp (

i~u-ir A2(s )ds) !]

i~u IA2(s)- AI(s)ids

+ ~pieipr i~u-ir IAI(s)- A2(s)lds} :5 lq(t)IIIAI- A211e"u { =

i~u e2~'"ds + ~pieipr i~u-ir e2~'"ds}

e~pt lq(t)IIIAI- A2lle"u ~

[1-

e-2pu

+ fpieipr(l- e-2p(u+ir))] i=I

e2pt

:5 2 ~ lq(t)li!AI- A2lle"u /(1- pe"r) :5

~e 2 " 1 i!AI- A2ll·

Hence

liT AI- TA2II =

sup I(TAI)(t)- (TA2)(t)le- 2111 :5 iiiAI- A2ll·

t~to-m

By Banach contraction theorem, there is a A EX such that T A= A, i.e.,

[

t

00

i-1

r

]

A(t)=q(t) exp lt-u A(s)ds+ ~]lp(t-a-jr)exp lt-u-ir A(s)ds.

Set y(t) = exp(- ft:-m A(s)ds). Then y(t) > 0 and

y'(t)

= -q(t) [y(t- a)+~)] p(t- a- jr)y(t- a-ir)].

(3.2.10)

Chapter 9

116

Denote 00

x(t)

i-1

= y(t) + 'E II p(t- jr)y(t- ir). i=l j=O

Then x(t) > 0 and y(t) = x(t)- p(t)x(t -r). Hence x(t) is a positive solution of (3.2.1). 0 Corollary 3.2.1.

Assume that p, q E C([to,oo),R),

0 $ p(t) $ p < 1,

lq(t)l :::; q,

u(t)

= u > 0,

t 2:: to,

(3.2.11)

and the equation

(x(t)- px(t -r))' + qx(t- u) = 0 has an eventually positive solution. positive solution.

(3.2.12)

Then Eq. (3.2.1) also has an eventually

In fact, (3.2.12) having an eventually positive solution implies that the characteristic equation of (3.2.12)

has a real root x•. It is easy to see that A* must be negative. Set 11. = -A*. Then ll.Pe"r + qe~'u = 11., and hence (3.2.9) holds. Therefore, the conclusion of Corollary 3.2.1 follows.

3.2.2. We consider the neutral differential equation

(x(t)- x(t- r))' + q(t)x(t- u) = 0,

t 2:: t 0 ,

(3.2.13)

Oscillation of First Order Neutral Equations

whereqEC([t0 ,oo),R+),

r, uE(O,oo),

117

m=max{r,u}.Set

= q(t), q1(t) = tq(t)

1

q(u)du,

qn(t) = tq(t)

1

qn-l(u)du,

qo(t)

00

00

= 1,2, ...

n

(3.2.14)

provided the right hand side of (3.2.14) exists. Theorem 3.2.4.

Assume that either

1 lto

00

or there exists an integer l

1 lto

q(t)dt

= oo,

(3.2.15)

> 0 such that q1 , ••• , qt-1 are defined, and

00

sq(s) /.

00

•

qt- 1 (u)duds

= oo.

(3.2.16)

Then every solution of (3.2.13) is oscillatory. Proof: Assume the contrary, and let x(t) be an eventually positive solution of (3.2.13). Set

z(t) = x(t)- x(t- r).

(3.2.17)

Then it is easy to see that z(t) > 0 eventually. Hence there exists M > 0 such that x(t) ~ M,

t

~

T ~to.

(3.2.18)

Substituting (3.2.18) into (3.2.13) we have

z'(t) + Mq(t) $ 0.

(3.2.19)

If (3.2.15) holds, (3.2.19) leads to

lim z(t) = -oo,

t-oo

(3.2.20)

Chapter 9

118

a contradiction. If

roo q(t)dt < oo,

(3.2.21)

fto from (3.2.19) we obtain

z(t) 2:: M

1

00

q(s)ds,

t;::: T.

(3.2.22)

Hence

x(t) 2:: x(t- r) ;::: M (

;::: M [ t

+M

1

00

q(s)ds

roo q( s )ds + roo q( s )ds + ... + roo

},

ft-r

~

T] 1 q(s)ds, 00

ft-{['~T]-l}r

q( s )ds)

t 2:: T,

where [·] denotes the greatest integer function. Hence there exist M1 T1 2:: T such that

> 0 and

(3.2.23) Substituting {3.2.23) into (3.2.13) we have

If lim t-+oo

roo sq( s) /.• Jr

00

q( u )duds = oo,

(3.2.24)

we obtain (3.2.20). Otherwise

Repeating above procedures f-1 times we get a contradiciton with (3.2.16).

0

Oscillation of First Order Neutral Equations

119

As an example we consider the equation

(x(t)- x(t- r))'

+t

0

x(t- 0') = 0.

(3.2.25)

From Theorem 3.2.4 we have the following result. Corollary 3.2.2.

If a > -2, then every solution of (3.2.25) is oscillatory.

In fact, there exists an integer n > 2 such that 1

a> n - -2+-. Let i

= n,

(3.2.26)

then (3.2.16) holds. Corollary 3.3.2 follows from Theorem 3.2.4.

Remark 3.2.2. We will see from Theorem 3.2.6 that if a < -2, then (3.2.25) has a bounded positive solution. This shows that condition (3.2.26) is sharp. Example 3.2.1. Consider the equation ( xt -xt-1 ) ,

()

(

1

( ) =0 . ) + 2(t- 1)Vt ( Jt + Vt=l) xt-1

(3.2.27)

x(t) = .Jt is a solution of (3.2.27), which implies a > -2 is the best possibility for oscillation of (3.2.25). Example 3.2.2. Consider the equation

(x(t)- x(t- 1))' + ___.;_ x(t- 1) = 0, tln t

t

~ 2.

(3.2.28)

It is easy to see that (3.2.16) with i = 1 is satisfied. Therefore every solution of (3.2.28) oscillates by Theorem 3.2.4. We note that (3.2.15) does not hold here. In the following we present a necessary and sufficient condition for the existence of a bounded positive solution of Eq. (3.2.13). Theorem 3.2.5.

Eq. (3.2.13) possesses a bounded positive solution if and

Chapter 3

120

only if

L: },roo . q(t)dt < 00

(3.2.29)

00.

to+•r

i=O

Proof: St£fficiency. By (3.2.29) there exists T ~to such that fort~ T

f: 1~ q(t)dt s

1.

q(s)ds,

t~T

(3.2.30)

t+tr

i=O

Define a function

f1

00

H(t) =

{

(t- T

+ r)H(T)/r,

T- r S t < T

t < T-

0,

(3.2.31)

T.

It is easy to see that HE C(R,R+)· Let 00

y(t)

= 2: H(t- ir),

t ~ T.

(3.2.32)

i=O

Then y E C([T, oo), R+)· Condition (3.2.30) implies that 0 < y(t) S 1 From (3.2.31) we have

y(t)=y(t-r)+H(t),

fort~

T.

t~T+r.

Define a set X as follows: X= {x: x E C([T,oo),R),

0 S x(t) S y(t),

t

~

T}.

The set X is considered to be endowed with the usual pointwise ordering

<·

Oscillation of First Order Neutral Equations

121

Then for any A C X there exists inf A and sup A. Define a mapping T on X as follows:

(Tx)(t) = {

x(t- r) + ft00 q(s)x(s- u)ds,

t2::T+m

(T X )(T + m) (T+~f;{hm)

T :5 t :5 T

+ y( t)(1 - T~m ),

+ m.

It is easy to see that

(Tx)(t) :5 y(t- r) +

1

00

q(s)y(s- u)ds

:5 y(t- r) + H(t) = y(t),

t 2:: T

+m

and

(Tx)(t):5y(t)

for

te[T,T+m],

i.e., T X ~ X. It is obvious that T is continuous and nondecreasing. Therefore, by Knaster's fixed point theorem there exists an x EX such that Tx = x. That is,

x(t) = x(t- r) +

1

00

q(s)x(s- u)ds,

t;::: T

+m

which implies that x(t) is a bounded positive solution of (3.2.13) on [T + m, oo ). Necessity. Let x(t) be a bounded and positive solution of Eq. (3.2.13). Set z(t) = x(t)- x(t- r). Then z'(t) :5 0. As in the proof of Theorem 3.2.4, z(t) > 0 for t ;::: t 0 • Hence x(t) > x(t- r), t 2:: to, and there exist M > 0 and t 1 2:: t 0 such that x(t) 2:: M, t 2:: t 1 • From (3.2.13), z'(t) :5 -Mq(t), t 2:: t 1 + u. Integrating it from t to oo for t 2:: t1 + 0' we have

x(t) 2:: x(t- r) + M

1

00

q(s)ds.

Then

x(t1 +

n joo

0'

+ nr) 2:: x(t1 + u) + M ~

t,+u+ir

q(t)dt.

122

Chapter 3

Since x(t) is bounded, from the preceding inequality we obtain that

~ 00

i=l

100

. q(s)ds

< oo.

0

t,+.,.+•r

The following result will lead to an equivalent version to condition (3.2.29). To this end we consider the second order delay equation x"(t) + q(t)x(t- r) = 0.

(3.2.33)

Theorem 3.2.6. Assume that q E C([t 0 , oo ), R+ ), three propositions are equivalent

r

> 0. Then the following

(i) Every bounded solution of Eq. (3.2.33) is oscillatory;

(ii)

f 1';tq(t)dt = oo;

Proof: (i) ==> (ii). It is sufficient to show that Eq. (3.2.33) has a bounded positive solution if

1

00

tq(t)dt

< 00.

(3.2.34)

to

In fact, from (3.2.34), there exists T > t 0 such that J;' tq(t)dt ~ t· We introduce the Banach space X of all bounded continuous functions x : [to, oo) ~ R with the sup norm Uxll = sup{lx(t)i: t E [to,oo)}. We consider the subset n of X defined by

!l={xEX:

l~x(t)::=;2,

t2::to}.

(3.2.35)

It is obvious that n is a bounded, closed and convex subset of X. We define the operator 7 on n by 1 + J;(s- T)q(s)x(s- r)ds

(Tx)(t)

={

+ (t- T) J00 q(s)x(s- r)ds, 1

(Tx)(T),

t 2:: T to:::; t:::; T.

(3.2.36)

123

Oscillation of First Order Neutral Equations In view of (3.2.35) we have 1

~ (Tx)(t) ~ 1 + 2(

£

00

(s- T)q(s)ds)

~ 1 + t ~ 2,

i.e., Tn ~ n. Let X}, X2 be elements of n. Then

I(Txi)(t)- (Tx2)(t)l

~

£

(s- T)q(s)lx1(s- T)- x2(s- T)lds

+ (t- T)

1

~ 1lx1- x21l

00

q(s)lx1(x- T)- x2(s- T)ids

(£

~tllx1-x21l,

(s- T)q(s)ds

+ (t- T)

1

00

q(s)ds)

t~T.

Hence

That is, T is a contraction operator on Tx=x,or

n. Then there exists an X

E

n such that

1 + J;(s- T)q(s)x(s- T)ds

x(t) =

{

+(t- T) J100 q(s)x(s- T)ds,

t

to~

x(T1),

(3.2.37)

~T,

t

~

T.

which implies that x(t) is a bounded positive solution of (3.2.33). (ii) ==> (i). If not, let x(t) be a bounded positive solution of Eq. (3.2.33). Then x"(t) ~ 0, x'(t) > 0 and x(t) > 0 eventually. Thus lim x(t) = o: > 0 t-+oo exists, because x(t) is bounded. Integrating (3.2.33) twice we have

x(t)

~

£

(s- T)q(s)x(s- T)ds

+ (t- T)

1

00

q(s)x(s- T)ds

(3.2.38)

where Tis a sufficiently large number. Then condition (ii), implies that lim x(t) = t-+oo oo which contradicts the boundedness of x.

Chapter 9

124

(i) => (iii). It is sufficient to show that Eq. (3.2.33) has a bounded positive solution if

f: 1

00

q(t)dt

•

<

(3.2.39)

00.

to+•r

i=O

From (3.2.39), there exist T ;::: to such that

L: ~00 . q(s)ds $ 00

T+tr

i=O

Let X and

n be the same as before.

(Tx)(t) = {

1+

J: f

00 6

1 2T

Define an operator T on

q(u)x(u- r)duds,

1,

For any

X

E

(3.2.40)

•

n as follows:

t 2:: T to$

t $ T.

n we have 1 $ (Tx)(t) $ 1 + 2

< 1+2 = 1

J: 1

00

£1 00

+ 2 L: oo

00

q( u )duds

~T+(i+l)r

i=O

.

i=O

J.oo q(u)duds

T+tr

•

roo

+ 2r L: }1 00

$ 1

q(u)duds

.

q( u )du $ 2,

t 2:: T.

T+tr

Therefore Tn s;;; n. For any Xl,X2 En, it is not difficult to see that IITxlT x2ll $ llx1 - x2ll, i.e., T is a contraction. Hence there exists a fixed point x E n such that T x = x. It is easy to see that x( t) is a bounded positive solution of Eq. (3.2.33).

!

(iii) => (i). If not, let x(t) be a bounded positive solution of Eq. (3.2.33). Then x"(t) $ 0, x'(t) > 0, x(t) > 0 eventually. Integrating (3.2.33) we have x'(t)

2::

1

00

q(u)x(u- r)du,

t;::: T,

(3.2.41)

Oscillation of First Order Neutral Equations

125

where T is a sufficiently large number. Integrating (3.2.41) from t - r to t we have

x(t)

~ x(t- r) +

lt 1

00

t-T

~ x(t- r) + r

1

00

q(u)x(u- r)du,. t

~ x(t- r) + ar

~ T + r. (3.2.42)

Hence there exists a> 0 such that x(t)

x(t)

q(u)x(u- r)duds

•

~a fort~

1

00

T. From (3.2.42) we have t

q(u)du,

~ T + r.

By induction

x(T + (n + 1)r) ~ x(T) + ar

n+l

roo

i=l

T+ar

L J'l

.

q(u)du.

Since x(t) is bounded, it follows that

which contradicts (iii).

0

Combining Theorems 3.2.5 and 3.2.6 we obtain the following result. Theorem 3.2.7. only if

Eq. (3.2.13) possesses a bounded positive solution if and

roo tq(t)dt <

ito

(3.2.43)

00.

Example 3.3.2. Consider

1) (x(t)- x(t- 2) ) , + t (t 4(t_ 3)(t _ 2) x(t- 2) = 0, 2

t

~

4.

(3.2.44)

126

Chapter 9

It is easy to see that (3.2.43) holds. Therefore, by Theorem 3.2.5, Eq. (3.2.44)

has a bounded positive solution. In fact, x(t)

= t-;t

is such a solution.

We have a similar result for the more general form of neutral differential equations m

(x(t)- x(r(t)))'

+ Lq;(t)f;(u(t))

= 0,

t 2. to.

(3.2.45)

i=l

Theorem 3.2.8.

Assume that

i) r, u; E C([to, oo ), R), r is increasing, t - r• lim u;(t) = oo, i = 1, ... , m, where r• > 0;

~

r(t) < t fort

~ t0,

and

t.-oo

ii) q;EC([t 0 ,oo),R+),

i=1, ... ,m;

iii) /; E C(R, R), /; is nondecreasing, xf;(x) > 0 for x =/: 0, i = 1, ... , m. Then Eq. (3.2.45) has a bounded nonoscillatory solution if and only if

f

oo

m

t l:q;(t)dt < oo.

to

(3.2.46)

i=l

3.2.3. We return to the equation (3.2.1). Theorem 3.2.9.

i) p(t) ~ 1,

Assume that

q(t) > 0,

J;' q(t)dt =

ii) liminf J/+r-a (q(s)/p(s t.-oo

oo and r > u;

+ r- u))ds > 0;

iii) there exists a positive and continuous function H(t) such that

liminf t-oo

f t

t+r-a

H(s)ds > 0,

liminf{q(t)/p(t + r - u)H(t)} > 0; t.-oo

Oscillation of First Order Neutral Equations iv) there exists T

+r

~ t0

.m f

{

t?;T,.X>O

+

127

such that

jt+r H( s)ds) ( jl+r-a )}> )H( )exp .X H(s)ds

q(t)H(t + r) (' exp" p(t + T - u)q(t + r)H(t)

(t) .X ( q_ pt+r u

t

1

1.

t

(3.2.47)

Then every solution of (3.2.1) is oscillatory. Proof: Assume the contrary, and let x(t) be an eventually positive solution of Eq. (3.2.1). Set z(t) = x(t)- p(t)x(t- r). From condition (i) we have z(t) < 0 and z'(t) < 0 fort~ t 1 ~ t 0 • Eq. (3.2.1) becomes

z'(t)- p(t- u) tt) ) z'(t- r) + q(t)z(t- u) = 0, q t- T

t

~ t0,

or equivalently

z'(t)

=

p(t

q(t)

+ r- u)q(t + r)

z'(t

+ r)

q(t)

+ p(t +r-u )z(t+r-u), Set .X(t)H(t)

= z'(t)fz(t),

for

t

~ t 1.

t~t 0 -r.

(3.2.48)

Then

(3.2.49)

and .X(t) > 0 fort ~ t1. Substituting (3.2.49) into (3.2.48) we have

(Jl+r .X(s )H(s )ds) q(t) (jt+r-a .X(s)H(s)ds), (3.2.50) + p(t + u)exp q( t) ( jt+r-a ) 1 >p(t+r-u)exp .X(s)H(s)ds,

.X(t)H(t) = p(t +

q(t) u)q(t + r) .X(t + r)H(t + r)exp

7 _

7 _

1

1

1

t~t -r.

128

Chapter 9

By Lemma 3.3.2 (see Section 3.3) we have

liminf t-oo

l t+T-cr >..(8)H(s)d8 < oo. t

It is not difficult to see that liminf >..(t) t-oo

= >.. 0 E (O,oo).

From (3.2.47) there exists a a E (0, 1) such that . f

0 t;?!:~.).>O

{

q(t)H(t + T) (>.. p(t+T-u)q(t+T)H (t)exp q(t)

+ \1 ( "' p

t +T

_

( )H( ) exp >..

t

U

lt+T H( 8)d8) t

lt+T-cr H(8)d8 )}> 1.

(3.2.51)

t

There exists a t 2 ~ max{t 1 ,T} such that >..(t) > a>.. 0 , into (3.2.50) we have

q(t)H(t + T) ( >..(t)H(t) > a>..o ( )( pt+T-uqt+T ) exp a>..o q(t)

(

+ p(t + T _ u) exp a>..o

t

~

t 2 • Substituting it

lt+T H(8)d8 ) t

lt+T-cr H(8)d8) , t

t ~ t2• (3.2.52)

Thus, for

8 ~ t2

>..(8) > inf { a>..op( t;?!:t2

q(t)H(t + T) ( t + T - U )q (t + T )H() t exp a>..o

lt+T H(8)d8 ) t

)} + p(t + T _ u)H(t) exp ( a>..o lt+T-cr t H(8)d8 . q(t)

Taking inferior limits in

8

we have

q(t)H(t + T) ( >..o ~ t;?!:t2 inf { a>..o ( )( )H() exp a>.. 0 p t +T - U q t +T t q(t)

+ p(t + T _ u)H(t)

(

exp a>..o

lt+T H(8)ds ) t

lt+T-cr H(8)d8 )}. t

(3.2.53)

Oscillation of First Order Neutral Equations

129

Letting .X 1 = a.Xo in (3.2.53) we have

. f { a 1n t~t2

q(t)H(t + r) exp ('"1 p(t+r-u)q(t+r)H(t) + \

1 p

"1

(

q(t) ( )H( ) exp .X1 t + T - 0' t

lt+r H( s )ds) t

lt+r-
Since t 2 ~ T and .X 1 > 0. (3.2.54) contradicts (3.2.51).

(3.2.54)

0

From Theorem 3.2.9 we can obtain different sufficient conditions for oscillation of Eq. (3.2.1) by different choices of H(t). For instance, if we choose H(t) = q(t)jp(t + T - u) or H(t) = 1, then (3.2.47) becomes {

inf

t~T,.X>O

1

p(t + 2T- u)

1 + \ exp(g.X

exp ( .X

lt+r-
"

t

lt+r t

q( s) ds ) p(s + T - u)

q(s) ) } ( ) ds > 1, ps+r-u

(3.2.55)

or inf

t~T,.X>O

{ ( pt+

q(t~ ( ) e.Xr +.X ( q(t) 0' q t + T p t+ T -

)

T -

Since e"' ~ ex for x

e.X(r-
> 1.

(3.2.56)

0'

> 0, (3.2.55) and (3.2.56) lead to the following corol-

laries. Corollary 3.2.6.

In addition to condition (i) of Theorem 3.2.9 further assume

that lim inf { q( t) / p( t + t-+oo

T -

0')} > 0

(3.2.57)

and liminf { t-+oo

1

p( t + 2T -

+ e 0')

lt+r-
q(s)

p( S

+T

Then every solution of Eq. (3.2.1) is oscillatory.

-

0')

ds

}

> 1.

(3.2.58)

Chapter 9

130

In addition to condition (i) of Theorem 3.2.9 and (3.2.57) Corollary 3.2. 7. further assume that liminf{ t-oo

q(t) p(t + T - u)q(t + r)

+

e(r- u)} > 1. q(t) p(t + T - u)

(3.2.59)

Then every solution of Eq. (3.2.1) is oscillatory.

Assume that

Theorem 3.2.10.

(i)

T

p(t) ~ 1,

> u ~ 0,

q(t) > 0,

J:'; q(t)dt = oo, and

liminf{q(t)/p(t t-oo

(ii) there exists sup{ ( t2:T p t

.x• > 0 and T

+T

q(t~

-

(7

( qt

+T

~to

) e>."r

+ T - u)}

> 0;

such that

+ ,1 A•

) e>."(r-rr)} < 1. (3.2.60)

( q(t) p t +T -

-

(7

Then Eq. (3.2.1) bas a positive solution on [T + r- u, oo). Proof: At first, we show that the integral equation

.X(t) =

q(t) ) .X(t + r)exp ) ( ( pt+r-uqt+r q(t)

)exp ( + pt+r-u

Jt+r .X(s)ds t

Jt+r-u .X( s )ds

(3.2.61)

t

has a positive solution. To this end we define a sequence {Ak(t)} as follows: .X 1 (t)

=

0,

t

~

T,

and

Ak+l(t)

( q(t) ).Xk(t)exp ) ( = p(t +r-uqt+r

Jt+r Ak(s)ds) t

) .Xk(s)ds , ( q(t) )exp (Jt+T-
t ~ T. (3.2.62)

131

Oscillation of First Order Neutral Equations

Noting (3.2.60) we have .X 1 (t)

( q(t) ) ~ .'*-. , = 0 < .X 2 (t) = pt+r-a

t 2:: T.

Assume that Ak-l(t) ~ Ak(t) ~ ).* for t 2:: T and some positive integer k. From (3.2.62) we have Ak(t) ~ Ak+J(t) fort 2:: T, and .Xk+ 1 (t)

~

)exp(.X*(r).X*exp(.X*r) + ( q(t) q(t~ ( ( pt+r-a pt+r-aqt+r

a))~).*

fort 2:: T. Therefore lim Ak(t) = .X(t) exists and).* 2:: .X(t) > 0 fort 2:: T. Taking k-oo the limits in (3.2.62) as k -+ oo and using the Lebesgue's convergence theorem we obtain that .X(t) is a solution of (3.2.61). Set

z(t)

= -exp

1:

.X(s)ds < 0,

t 2:: T.

(3.2.63)

Then z 1(t) = .X(t)z(t), and (3.2.61) becomes

q(t) I q(t) 1 )z(t+r-a). ( )z(t+z)+ pt+r-a ) ( ( z(t)= pt+r-aqt+r

(3.2.64)

Let

x(t) = -

z 1(t +a) ) > 0, ( q t +a

t 2:: T- a.

(3.2.65)

Then (3.2.64) becomes

(x(t)- p(t)x(t- r)) 1 + q(t)x(t- a)= 0. That is, we have found a positive solution x(t) of Eq. (3.2.1).

D

3.2.4.

=

=

a > 0, r > 0, q E p f=: 1, a(t) Assume that p(t) Theorem 3.2.11. 00 C( [to, oo ), R+) and j 0 q( t )dt < oo. Then Eq. (3.2.1) bas a positive solution.

Chapter 9

132

Proof: Let BC be the Banach space of all bounded continuous functions on [t 0 ,oo) with the super nonn. i) Consider the case that 0 < p < 1. Let t" be so large that t* - T ~ t 0 , t* - u ~ t 0 and q( s )ds $ ~ . Set f2 = {x E BC: 1 $ x(t) $ 2, t ~ t 0 }. Then n is a bounded, closed and convex subset of BC. Define a mapping T: n-+ BC as follows:

J,':'

(Tx)(t) = { 1- p + px(t- r) (Tx)(t"), Clearly, T is continuous and Tn

+ J1

00

q(s)x(s- u)ds,

t ~

t0 $

c n.

For

Xt, Xz

E

t"

t $ t*.

n and t ~ t*

I(Tx1)(t)- (Txz)(t)i $ plx1(t- r)- x2(t- r)l

+

1

00

q(s)lx1(s- u)- xz(s- u)lds

1-p 1+p Sllx1- x21!(p+ - 2- ) = - 2-llxi- xzll. Hence

1 + p/2 < 1 implies that T is a contraction. Then there is a Tx = x. It is easy to see that x(t) is a solution of (3.2.1) on [t",oo).

X

E

n that

ii) Consider the case that p > 1. Lett* be so large that t*- a~ to and ft':'+r q(s)ds $~·Define a subset n of BC and a mapping Ton n as follows:

p-1 fl={xEBC: - 2 -$x(t)$p,

t~to}

and

(Tx)(t) = {

p -1

+!. P

(Tx)(t"),

x(t + r)- l. f 100 +r q(s)x(s- a)ds, t ~ t* P

to$ t $ t*.

133

Oscillation of First Order Neutral Equations It is easy to show that

Tn c

n and

for xl, X2 E n, i.e., Tis a contraction on n. Then there is a X E n such that Tx = x, or x(t) is a positive solution of (3.2.1) on t ~ t•. The rest can be discussed similarly. We only give an outline for those. iii) The case that -1 < p:::; 0. Choose t• so that t• - r ~ t 0 , t• - u ~ t 0 and ft~ q( s )ds :::; ~. Define

n = {x E BC:

2( 1 + p) t > t } <1 ( 1 + p) 2 -< x(t) 1 -p -p , _o

and (Tx)(t)

={

1 + p + px(t- r) + ftoo q(s)x(s- u)ds, t

~

t•

to :::; t :::; t•

(Tx)(t*),

iv) The case that p = -1. Choose t• so that t• + T - u ~to and ft~+r q(s )ds :::; f!={xEBC: 1:::;x(t)::=;2,

t. Define

t~t 0 }

and

(Tx)(t) = {

1+

E ft':(~ii'"_l)r q(s)x(s- u)ds,

t

~ t•

i=l

(T x )( t*),

v) The case that p < -1. Choose t• so that t• + T

-

t0

u ~ to and ft~+r q( s )ds :::; -

n={xECB: ( 1 +p) 2 <x(t)< 2(1 +p) - p-1, (p-1)p-

:::;

t :::; t•.

Ef-. Define

t~to}

Chapter 9

134

and

(Tx)(t) = {

1+ 1

+ 1x(t + r)- 1 ft':r q(s)x(s- a)ds,

t;::: t*

P

P

P

t0

(Tx)(t*),

Assume that a > r > 0, p(t)

Theorem 3.2.12. and the equation

y'(t)

+

= p < 0,

:::;

t:::;

t*.

q E C([t 0 , oo), R)

Q(t) y(t- (a- r)) = 0 1-p

(3.2.66)

is oscillatory. Then Eq. (3.2.1) is oscillatory, where Q(t) =min {q(t), q(t- r)}. Proof: Otherwise, there is an eventually positive solution x(t) of (3.2.1). Set

z(t)

= x(t)- px(t- r),

y(t)

= z(t)- pz(t- r).

and

Then eventually z(t)

> 0,

z'(t) < 0, and y(t) > 0,

y'(t) < 0. We see that

y'(t) = z'(t)- pz'(t- r) = -q(t)x(t- a)+ pq(t- r)x(t-

T-

a)

:::; -Q(t)z(t- a). On the other hand

y(t- (a- r)) = z(t- (a- r))- pz(t- a)

:::; (1- p)z(t- a). Thus the inequality

Q(t) 1 y(t- (a- r)) y (t):::;- 1-p

135

Oscillation of First Order Neutral Equations

has a positive solution, which implies that Eq. (3.2.66) has a nonoscillatory so0 lution. This contradicts the assumption.

3.3. Equations with Variable Coefficients (II) 3.3.5.

We discuss the neutral delay differential equations of the form n

(x(t)- p(t)x(t- -r))' + q(t)

IJ lx(t- u;)j

0 '

sign x(t- u;) = 0

(3.3.1)

i=l

where p, q e C([to,oo),R), T, u; e (O,oo), a; ~ 0, and E7=1 a; = 1, i = 1, 2, ... ,n. Denote m = max{-r,u;, i = 1, ... , n}. If n = 1, (3.3.1) reduces to

(x(t)- p(t)x(t- -r))' + q(t)x(t- u) = 0.

Lemma 3.3.1. there exists a t•

(3.3.2)

Assume q is positive and p is bounded and nonnegative, and such that

~ t0

p(t* +n-r) $1,

n = 0,1,2, ....

(3.3.3)

Let x(t) be an eventually positive solution of Eq. (3.3.1) and set

z(t) = x(t)- p(t)x(t- -r).

(3.3.4)

Then eventually z(t) > 0 and z'(t) < 0. Proof: It follows from (3.3.1) that z'(t) < 0 eventually. It remains to show that z(t) > 0 eventually. Otherwise, z(t) is eventually negative. Thus there exists a sufficiently large T such that z(t) $ -d < 0 fort ~ T, where dis a positive constant. Hence

x(t) $ -d + p(t)x(t- -r) for t

~

T.

In particular,

x(t*+(n+N)r)$-nd+x(t*+(N-1)r),

n=1,2, ...

Chapter 3

136

if t* + Nr ~ T. Hence x(t) can not be eventually positive. This contradiction 0 proves the lemma.

Lemma 3.3.2.

Assume a-> 0, q E C([t 0 ,oo),(O,oo)), A E C([to-u,oo),(O,oo))

satisfying

liminf ['

lt-u

t-oo

q(s)ds > 0

(3.3.5)

and

(3.3.6) Tben liminf t-oo

t

lt-a

(3.3.7)

.\(s)ds < oo.

Proof: Define

Q(t)

=

lt

q(s)ds,

t

~to.

to

= +oo, and Q(t) is strictly increasing. is well defined, strictly increasing, and lim Q- 1 (t) = +oo. (3.3.5) implies that lim Q(t) t--+<X>

Then Q- 1 (t)

t--+<X>

(3.3.5) implies that there exist c > 0 and T1 c

Q(t)-Q(t-o-)~'2

for

~ t0

such that

t~T1

and thus

Set

A(t) = exp(-

£

.\(s)ds).

(3.3.8)

Oscillation of First Order Neutral Equations

137

(3.3.6) implies that A'(t) :5 -q(t)A(t- u),

t

~to.

(3.3.9)

By Lemma 2.1.3 A~~)) is bounded above under the condition (3.3.5). Then (3.3. 7) is true. 0 We are now ready to prove the following result. Theorem 3.3.1. hold, and either

In addition to the assumptions of Lemma 3.3.1 assume (3.3.5)

or liminf{ inf [ fipa'(t- u;)exp(.A t-+oo

A>O

i=l

t q(s)ds)

lt-r

+ ±exp(.A~ a; 1~~. q(s)ds)]} > 1.

(3.3.11)

Then every solution of (3.3.1) is oscillatory. Proof: First we assume (3.3.10) holds. Without loss of generality, assume that Eq. (3.3.1) has an eventually positive solution x(t). Let x(t) > 0, x(t- m) > 0, fort~ T1 ~ t 0 . Then by Lemma 3.3.1, z(t) > 0, z'(t) < 0 fort~ T1 , where z(t) is defined by (3.3.4). From (3.3.1), we have n

z'(t) = -q(t)

II xa'(t- u;) i=l n

= -q(t)

fi [z(t- u;) + p(t- u;)x(t- u;- r)]"'' i=l

138

Chapter 9 = -q(t)

IT z '(t- u;) + n

0

i=l

(t) n q ITp 01 (t- u;)z'(t- r). q(t- r) i=l

(3.3.12)

%'(t) Set >.(t) = -Z(i)· Then (3.3.12) reduces to

(3.3.13)

It is obvious that >.(t) > 0 fort 2:: T1 • From (3.3.13) we have

where u. = ~n {u;}, a= ~n {a;}. In view of Lemma 3.3.2, we have 1$a$n 1$a$n liminf t-+oo

it

t-cr.

>.(s)ds < oo

which implies that liminf >.(t) < oo. Now we show that liminf >.(t) > 0. In fact, if t-oo t-+oo liminf >.(t) = 0, then there exists a sequence {tn} such that tn 2:: T1, lim tn = oo t--+oo n-oo and >.(tn) :-:;; >.(t), fortE [T1, tn]· From (3.3.13), we have

>.(tn) 2:: >.(tn) ttn) ) q

tn-T

ITp

01

(tn- u;)exp(> .(tn)r)

i=l

+ q(tn)ex p( >.(tn) ~ a;u;). Hence

Oscillation of First Order Neutral Equations

139

which contradicts (3.3.10), and therefore, 0

< liminf A(t) = t-oo

h

<

(3.3.14)

oo.

From (3.3.10), there exist a E (0, 1) such that

f{· f[Iln

. a 1. 1mm t-oo

m

A>O

i=l

p "''(t ' -a; ) (tq(t) )e AT

+ q~)

q

exp

-

7'

(A~ a;a;)]} > 1.

(3.3.15)

In view of (3.3.14) we assume that A(t)

> ah, t;::: T2.

(3.3.16)

Substituting (3.3.16) into (3.3.13) we obtain

for t ;::: T2

+ m.

Hence

Set A*= ah. Then

A*;::: aliminf {A* (q(t) ) llp"';(t- a;)exp(A*r) t-oo q t - 7' i=l

which contradicts (3.3.15) and completes the proof of this theorem under condition (3.3.10).

Chapter 3

140

If (3.3.11) holds, we let ..\(t)q(t) =

fi t

..\(t) ;::: .\(t- r)

+ exp (

ti!2_

-Z(tf .

p"; (t- u,)exp ( a;

l~a;

. \(

Then (3.3.12) becomes

l~r

.\(

s )q( s )ds) (3.3.17)

s )q( s )ds) .

By Lemma 3.3.2, we know that liminf t-+oo

r

..\(s)q(s)ds <

(3.3.18)

00.

lt-a

From (3.3.5) and (3.3.18), we conclude that liminf .>.(t) t-+oo

< oo. From (3.3.17)

>.(t) ;::: 1 and so 0 < liminf >.(t) == h < oo. t-+oo

From (3.3.11), there exists a E (0, 1) such that

By a similar argument to the first part of the proof, we reach a contradiction. D

In addition to the assumptions of Lemma 3.3.1 assume

Corollary 3.3.1. (3.3.5) holds and liminf{ inf t-+oo

>.>0

[p(t- u) qt(t) t-

r

) eAr+

~q(t)eM]} A

> 1.

(3.3.19)

Then every solution of (3.3.2) is oscillatory. Example 3.3.1. Consider (x(t)- (2 + sint)x(t -1r))'

+ (3 + 2sint)x(t-

i)

= 0,

t;::: 1r.

(3.3.20)

It is easy to see that (3.3.19) holds. Therefore, every solution of (3.3.20) is oscillatory. In fact, x(t) sint is such a solution.

=

141

Oscillation of First Order Neutral Equations

It is easy to obtain explicit conditions for oscillation of Eq. (3.3.1) from Theorem 3.3.1. The following is an example. Corollary 3.3.2. (3.3.5) holds and

In addition to the assumptions of Lemma 3.3.1 assume

(3.3.21)

Then every solution of (3.3.1) is oscillatory. The following is a nonoscillation result for Eq. (3.3.1). Theorem 3.3.2.

Assume that

(i) p,q e C((t0 ,oo),R), p(t) 2:0, and 0 :5 q(t) :5 M, where M is a constant; (ii) There exists a positive number a such that fort 2: to

0 :5 p(t)e 0 ' :5 c < 1

(3.3.22)

and

(3.3.23)

Then Eq. (3.3.1) has an eventually positive solution which approaches zero exponentially. Proof: Let BC denote the Banach space of all bounded and continuous real valued functions defined on (to - m, oo) with the sup norm. Let n be the subset of BC defined by

fl={yeBC:0:5y( t):51

for

t2:t 0 -m}.

Chapter 3

142

Now we define operators 1i and

(1iy)(t)

={

12 on n as follows:

p(t)eorry(t- r),

t

~to

!;(1iy)(to) + (1- !; ), to - m::; t::; to; n

exp(a

1: a;u;) ft

00

q(s)

i=I

nyor (s- u;)ds, n

(12y)(t) =

X

exp(-a(s- t)j

1

t

~to

i=l

to -

!;(12y)(to),

m ::;

t ::; to.

In view of (3.3.22) and (3.3.23) we have 1iYI +12Y2 En for every pair YI, Y2 En, and 1i is a contraction on n. It is easy to see that lft(12y)(t)1 ::; N for some positive constant N. Hence 12 is completely continuous on n. By Krasnoselskii's fixed point theorem, there exists a y E n such that 1jy + 12Y = y. That is, for t ~to

y(t) = p(t)eorry(t- r)

at

+ exp( and for t 0

-

m ::; t

a;u;)

1

00

q(s)exp[-a(s- t)]

n

yor 1 (s- u;)ds,

< to t y(t) = - y(t 0 ) to

It follows that y(t) > 0 for all t

~ t0 -

x(t) = p(t)x(t- r) +

l

t + (1- -) > 0.

to

m. Set x(t)

oo

= y(t)e-ort. Then

n

q(s)

t

II xor (s- u,)ds, 1

t

~to.

i=l

Consequently, n

(x(t)- p(t)x(t- r))' + q(t)

IJ xor (t- u;) = 0, 1

t ~ t0

i=l

I.e. x(t) is a positive solution of Eq. (3.3.1) and x(t) approaches zero exponentially. D

143

Oscillation of First Order Neutral Equations

Remark 3.3.1. When p(t) (3.3.23) becomes

= p,

pe"'r

q(t)

+ fexp a

= q are both positive constants, condition

(at a;a;) : ;

(3.3.24)

1

i=l

for some a > 0. From Theorems 3.3.1 and 3.3.2 condition (3.3.24) provides a sufficient and necessary condition for the existence of a nonoscillatory solution of Eq. (3.3.1). Example 3.3.2. Consider the equation (3.3.25) where q(t) = Zettl•(t-l)11 7•(t-z)t/a , t ~ 3. It is not difficult to see that all assumptions of Theorem 3.3.21 are satisfied. Therefore, Eq. (3.3.25) has a nonoscillatory solution which tends to zero as t - t oo. In fact, x(t) = te- 1 is such a solution. Corollary 3.3.3.

Assume that there exist positive constants p and q such that (3.3.26)

and that the linear equation

(x(t)- px(t-

r))' + qx(t-

taw;)=

0

(3.3.27)

has a nonoscillatory solution. Then Eq. (3.3.1) also has a nonoscillatory solution. In fact, from Theorem 3.1.1, (3.3.27) having a nonoscillatory solution implies that the characteristic equation of (3.3.27)

). - >.pe-).r

+ q exp ( - >.

t a;a;)

= 0

(3.3.28)

•=1

has a negative real root >.. Let a = ->. > 0, then (3.3.24) is satisfied. The conclusion of the corollary follows from Theorem 3.3.2.

Chapter 9

144

The following result is for Eq. (3.3.1) with oscillating coefficients p(t) and

q(t). Theorem 3.3.3. Assume that p E C 1 ([t 0 ,oo),R), q E C([t 0 ,oo),R) and there exists a positive number p. such that (3.3.29)

Then Eq. (3.3.1) has a positive solution fort;::: t 0 • Proof: Set

t;::: to

p(t) PI(t) = { t-t~±r p(to),

0,

t0

p'(t),

P2(t) = {

t-t~±r

to-T$ t $to -

m-

T

$ t :5 t 0

-

r,

t ;::: t 0

p'(to), to - T :5 t $to

0,

to -

q(t),

t 2: to

t-t~±r q(to),

to-T :5 t $to

0,

t 0 - m- T $ t $to-T.

m -

T $ t $ to - r,

and

Pa(t)

Then p;, i

={

=1,2,3, are continuous on [to-m- r,oo). From (3.3.29) we have P.IPI(t)le"'r

+ IP2(t)le"'r + IPa(t)l exp(p.

t a;u;)

$ p.,

•=1

t 2: to-m- T.

(3.3.30)

We introduce the Banach space X of all bounded continuous functions x: [to-m- r,oo)-+ R with the norm llxll = sup jx(t)je-11t where 71 > 0 t;:::to-m-T

Oscillation of First Order Neutral Equations

145

satisfies the following inequality IPl(t)le"r(e-"r + ~) + e"r IP2(t)l + .!IP3(t)lexp(p. ~

~

~

t

for

ta;u;) ~1

:5

~to-r.

~ (3.3.31)

We consider the subset 0 of X as follows

n ={.X EX:

I.X(t)l :5 p.,

t

~to-m-

r}.

Clearly, f1 is a bounded, closed and convex subset of X. Define an operator Ton

n as follows:

.X(t- r)p1(t) exp(J,'-r .X(s)ds) + P2(t) exp(ft'-r .X(s)ds)

+ P3(t) exp(:E a; ft-tr· .X(s)ds), i=l • t

n

(T.X)(t) =

to - m - r :5 t :5 t 0

0,

t ~to-r

r.

-

In view of (3.2.41), we have 1(7-X)(t)l :5 P.IPI(t)le"r

+ IP2(t)le"r

+ IP3(t)lexp(p.

t a;u;)

:5 p. for

which shows that T maps 0 into itself. Next, we show that T is a contraction on and t ~to-r

t

n. In fact,

~to-r for any AI' A2 E

n

I(T.XI)(t)- (7.X2)(t)1 :5IPI(t)II.XI(t- r)

exp(l~r .X1(s)ds)- .X2(t- r) exp(l~r .X2(s)ds) I

+ IP2(t)ll exp(l~r .X1(s)ds) -

exp(l~r .X2(s)ds) I

+ IP3(t)llexp( ~a; l~tr; .X1(s)ds) -

exp(

t

a;

l~tr; .X2(s)ds) I

Chapter 3

146

::::; IPt(t){e"rl>.t(t- r)- >.2(t- r)l

+ J.Le"r l~r 1>-t(s)- >.2(s)lds)

+ e"riP2(t)lj~r 1>-t(s)- >.2(s)lds + IP3(t)l exp(J-1. ~ a;u;) ~a; 1~"' ::::; { e"riPt(t)le'IT [e-'IT

+ ~(1- e-'lr)] + ~e"riP2(t)l(l- e-'IT)e'IT

+ ~IP3(t)l exp(J-1. ~a;u;) ::::; e"t{ IPt(t)le"r ( e-"r

1>-t(s)- >.2(s)lds

~a;(l- e-""')e"1}11>.1- >.2ll

+ ~) + e~r IP2(t)l

+ ~IP3(t)1 exp(J-1. ta;u;)}ll>-t- >.211 TJ

•=1

Hence

IIT>.t- T>.2ll =

sup

I(T>.t)(t)- (T>.2)(t)le-" 1

t<=:t 0 -m-r

i.e., T is a contraction on Q. Therefore there exists a >. E Q such that T >. = >.. That is,

>.(t- r)pl(t)exp( fLr >.(s)ds) n

+ P2(s)exp(fLr >.(s)ds)

t

+ P3(t)exp(L: a; ft-cr· >.(s)ds), t::?: to-r, i=l

>.(t) =

I

0,

to - m - r ::::; t ::::; t 0

-

r.

147

Oscillation of First Order Neutral Equations According to the definition of p;, we have

>.(t)=>.(t-r)p(t)exp(i~r >.(s)ds) +p(t)exp(i~r >.(s)ds) + q(t)exp (~a; l~u,. >.(s

)d.s),

t

~to.

Set x(t) = exp(- f 110 >.(s)ds). Then we can find that x(t) is a positive D solution of Eq. (3.3.1) on [to, oo ). Example 3.3.3. Consider 2sint )' ( x(t)---x(t-1) t-1

2cost + 1-1-t x(t-1)=0.

(3.3.32)

It is easy to see that (3.3.29) with p. = 1 is satisfied for t ~ T, where T is a sufficiently large number. Then Eq. (3.3.32) has a positive solution on [T, oo). In fact, x(t) = t is such a solution of Eq. (3.3.32).

3.3.6. We now consider the neutral delay differential equations with several delays of the form n

(x(t)- px(t- r))' + Lq;(t)x(t- a;)= 0,

t ~ t0

(3.3.33)

i=l

where p E [0, 1], q; (i = 1,2, ... ,n) E C([t 0 ,oo), R+) and r;,a; (i = 1, 2, ... ,n) E (0, oo ). Denote m =max{ r, a1, ... , an}· Lemma 3.3.3.

Assume that liminf J.11_ cr,. q1 (s)ds > 0 and t-+oo qi(t-r)$qj(t),

t~t 0 +m,

j=1,2, ... ,n.

(3.3.34)

Let x(t) be an eventually positive solution of (3.3.33) and set z(t)

= x(t)- px(t- r).

(3.3.35)

Chapter 3

148

Then eventually z(t) > 0,

z'(t) < 0, and n

z'(t)- pz'(t- r)

+ Lq;(t)z(t- u;)

~ 0.

(3.3.36)

i=l

Proof: As in the proof of Theorem 3.2.1, it is easy to see that z(t) > 0, z'(t)

<0

eventually. From (3.3.34) and (3.3.35) we have n

z'(t)- pz'(t- r) =- L[q;(t)x(t- u;)- pq;(t- r)x(t- r- u;)] i=l n

~ - L

q;(t)z(t- u;)

i=l

eventually. Theorem 3.3.4. Assume that the assumptions of Lemma 3.3.3 hold. Further assume that for all p. > 0, and = r, 0"}, 'O"n

e

li~~f { pe~'r

0.

0

n + f.p.1 ~ e~'"'

it+l t

q;( s )ds

}>

1.

(3.3.37)

Then every solution of Eq. (3.3.33) is oscillatory. Proof: Assume (3.3.33) has an eventually positive solution x(t). By Lemma 3.3.3 there exists aT;::: to such that z(t) > 0, z'(t) < 0, and (3.3.36) holds fort ;::: T. Let w(t) =t;::: T. Then w(t) > 0, t;::: T, and (3.3.36) becomes

Z:N?,

w(t);::: pw(t-

r)exp(i~r w(s)ds) + tq;(t)exp(l~.,, w(s)ds)

(3.3.38)

fort;::: r + m. We now define a sequence of functions {w k( t)} for k = 1, 2, . . . and t ;::: T, and a sequence of numbers {p.k} fork= 1, 2, ... , as follows:

Oscillation of First Order Neutral Equations and for k = 1, 2, ... , t

~ T

149

+ km

= pwk(t- T) exp(l~T WA:(s)ds)

WH 1 (t)

+ tq;(t) exp(l~.r; Wk(s)ds) and p. 1

= 0, and fork=

Jlk+l = inf

t~T

(3.3.39)

1,2, ...

{ PJlkellkT min t=r,trt,···•trn

n lt+t } + £1 :~:::el'ktrk q;(s)ds .

t

i=l

(3.3.40)

We claim that

i) 0 = Jll < Jl2 < ... , ii) WA:(t) ~ w(t) fort~ T + (k -1)m and k = 1,2, ... , iii) J,t+t Wk(s )ds ~ Jlk fort~ T + (k + 1)m, l = 1, 2, ... , and l = T,aJ, ... ,an. In fact, i) and ii) follow from (3.3.39) and (3.3.40) by induction. Clearly iii) is true for k = 1. Assume iii) is true for some k. Then (3.3.39) and (3.3.40) imply that fort~ T + km, l = T,a1, ... ,an

t

f1

(1s-T Wk(8)d8 ds n lt+t (18 ) + 18 q;(s) exp s-tr; wk(8)d8 ds

lt+t t Wk+J(s)ds = f1 lt+t t Wk(S- T) exp 1

8

)

t

Hence iii) holds. Let p.* = lim Jln· From (3.3.37) and (3.3.40) there exists an a > 1 such n-+oo that JlA:+I ~ ap.k, k = 1, 2, ... , which implies that p.* = oo. In view of ii) and iii) lim J,t+trt w(s)ds = oo. From the definition of w, we t-+oo have

z

(

z(t)

t + al

) = exp

lt+trt t

w(s)ds.

Chapter 9

150

Thus

r

z(t)

t2-~z(t

+ u1)

(3.3.41)

= oo.

From (3.3.33) n

z'(t) =- Eq;(t)x(t- u;) $ -q1(t)x(t- u 1 ) i=l

By Lemma 2.1.3, z(~(t)t) is bounded above which contradicts (3.3.41).

Assume there exists a p* > 0 and a T

Theorem 3.3.5. l=T,u, ... ,un

~

0

to such that for

(3.3.42)

Then (3.3.33) has a positive solution on [T + m, oo ). Proof: First we claim that the integral equation

l

t

n

lt

v( t) = pv( t - T)exp t-r v( s )ds + ~ q;( s )exp ,_,7 , v( s )ds has a positive solution on [T + m, oo ). To this end set v 1 ( t) k = 1,2, ...

= 0,

(3.3.43)

t

~

T, and for

pvk(t- T)exp(JLr Vk(s)ds) Vk+t(t) =

{

+ E:=I q;(t)exp(f/-.,., vk(s)ds), Pk+t(t)

t~T+m

(3.3.44)

T$t
where {Pk} are given function sequences satisfying

i) Pk E C 2 ([T, T k = 1,2, ... ;

+ m), [0, oo)) with p~ ~ 0 and p~ ~ 0,

t e [T, T

+ m),

o~cillation

of Fir~t Order Neutral

Equation~

151

ii) fh(t) = 0, t E [T, T + m- r], /h(T + m) = vk(T + m), and f3k(t) are increasing in k for t E [T + m - r, T + m) and k = 1, 2, ... ; iii) for every e = r, al, ... 'an and t E [T +m-e, T + m), k = 1, 2, ...

1T+m f3k(s)ds :5 1T+m+i vk(s)ds. t+l

t

Clearly, VI (t) :5 v2(t) :5 .... By induction we will show that fork= 1, 2, ... andf=r,al>···•an 1 {t+l

R.}t

vk(s)ds:5p.*,

t ?:. T.

(3.3.45)

In fact, (3.3.45) is true for k = 1. Assume (3.3.45) holds for some k. Then from (3.3.42) and (3.3.44) we have for t ?:. T + m, = r, a 1, ... , an

e

r Vk(8)d8)ds 7.p },rt+l Vk(s- r) exp ( la-r 1

+ 7.

t; ltr'+l r

qi(s) exp

( ls-u; r vk(8)d8)ds

(3.3.46) For every f = r, a1, ... ,an and t E [T+m-f, T+m) from (3.3.46) and condition iii) for {/3k} we have

From the monotonic property of /h+ 1(t) with respect to t we see that (3.3.45) also holds fortE [T, T +m-e), e = T, a], ... , an.

152

Chapter 9 Let v(t)

= k-oo lim Vk{t). Then v(t) = 0,

on [T+ m- r,T+ m), and fort~ T and 1

t+l

t lt

t E [T, T+m-r],

v(t) is increasing

.e = r,ui,··. ,un

v(s)ds ~ p.•.

Taking limits ask-+ oo on both sides of (3.3.44), from the Lebesgue's monotone convergence theorem, we see that v(t) satisfies (3.3.43) fort~ T+m. It is easy to see that v(t) is well defined on [T, oo). Furthermore, froii?. condition (i) of {.Bk} we know that v(t) is continuous on [T,T + m], and from (3.3.40) v(t) is continuous on [T, oo ). Thus v(t) is a continuous positive solution of (3.3.43) on [T + m, oo ). Set

Then x(t) is a positive solution of (3.3.33).

Corollary 3.3.4.

0

If n

~::.:>'""' i=l

rt+l

},

(3.3.47)

q;( s )ds

t

is nondecreasing in t for .e = r, u 1 , ••• , Un. Then (3.3.33) is oscillatory if and only if for all p. > 0 and .e = r, u1. ... ,un 1

t~~ { pel'r + lp.

t; e,...,., },rt+l q;(s)ds }> n

1.

(3.3.48)

Corollary 3.3.5. If there exist i E In = {1, ... , n} and .e > 0 such that liminft-oo q;(s)ds > 0 and T = m 0 l, u; = m;l for some integers m;, i = 0, 1, ... , n. Furthermore

J,'_.,.,

q;(t)

= g;(t) + h;(t),

i

i

= 1, 2, ... , n,

where g;( t) are l-periodic functions with ftt+l g;( s )ds = gj, h;{ t) are non decreasing with lim h;(t) = hi, i = 1, 2, ... , n. Then (3.3.33) is oscillatory if and t-oo

153

Oscillation of First Order Neutral Equations only if for all p.

>0

For the case that q;(t) becomes

=0

and

h;(t)

= hi,

= 1, 2, ... , n,

(3.3.49)

(3.3.50)

this implies that Eq. (3.3.33) with constants parameters is oscillatory if and only if its characteristic equation has no real roots. If h;(t) = 0, i = 1, 2, ... , n. Eq. (3.3.33) becomes n

(x(t)- px(t- r))'

+ Lg;(t)x(t- a;)= 0

(3.3.51)

i=l

where g;(t) are £-periodic functions. Then (3.3.51) is oscillatory if and only if its "characteristic equation" n

A(1- pe->.r)

+ Lgie->. = 0

(3.3.52)

17 ;

i=l

t J/+l

has no real roots, where gi = g;(s )ds, i = 1, 2, ... , n. By employing the above technique in Eq. (2.7.1) the condition (2.7.3) in Theorem 2.7.1 can be improved to assuming that for all A> 0 and j = 1, 2, ... , n n 1 liminf { - - " ' " ' t.-oo

Corollary 3.3.6.

ATj(t)

£;:

ft+r;(t) t

p;(s)e>.r;(s)ds

If r; ( t) are non decreasing for i n

ft+r;(t)

liminf L t-oo

Then Eq. (2. 7.1) is oscillatory.

i=l

t

}

> 1.

(3.3.53)

= 1, 2, ... , n, and

1 p;(s)ds > - . e

(3.3.54)

Chapter 9

154

From Theorem 3.3.4 we derive some explicit conditions for oscillation of Eq. (3.3.33). Theorem 3.3.6. Assume that the assumptions of Lemma 3.3.3 bold. Furthermore, for f.= r, O'J, ••• ,un (3.3.55)

Then every solution of Eq. (3.3.33) is oscillatory.

> 0, and then Eq. (3.3.33)

Proof: We shall show that (3.3.37) is true for all p. is oscillatory by Theorem 3.3.4. In fact

q;(s)ds )e "''(1- pe"r)- 1 f -1 L (1jt+l n

1

/1- i=I

t

1 =-

LL n

00

(

f1

jt+l q;(s)ds)

/1- i=l k=O

pkep(kr+u;)

t

Thus (3.3.55) implies that 1 liminft--.oo p.

-f. q;(s)ds )e~-' 17'(1- pe"r)- 1 > 1. L (1jt+l n

i=l

Hence (3.3.37) holds for all p.

t

> 0 and f.

= r, u 1 , ... , u n. The proof is complete. 0

For the case p Theorem 3.3. 7. i) p ~ 1, ii)

T

>

O'j,

~

1, we can prove the similar results using above techniques.

Assume that

q;(t) :5 q;(t- r), fori= 1, 2, ... , nand all large T; t+ r-maxc:r; n i

= 1,2, ... ,n, liminfJ 1 t-<X>

2

L: q;(s)ds > 0;

i=l

155

Oscillation of First Order Neutral Equations

= r,T- u;, and i = 1,2, ... ,n

l

iii) for all fl.> 0,

it+l

1 1 n liminf { -epr + -l::>P(r-a;) p

t--+oo

lpfl i=l

q;(s)ds

}>

1.

(3.3.56)

t

Then every solution of (3.3.33) is oscillatory. Theorem 3.3.8. i) p

~

1,

Assume that n

liminf ).*

> 0 and T

1 •r sup { -e>. t>T

-

q;(t) > 0,

T

> u;,

= 1, 2, ... , n;

i

i=l

t-oo

ii) there exist

l:::

P

~to,

for f.=

T, T -

n • 1 l:e>. + --. (r-a;)

fp).

u;,

it+l

i = 1, 2, ... , n

q;(s)ds

}:s;

1.

(3.3.57)

t

.

•=1

Then Eq. (3.3.33) bas a positive solution on [T, oo ). Corollary 3.3.4. Assume that there exist l > 0 and positive integers m 0 ,m 1 , ••• ,mn such that T =mol, u; = m;l and mo > m;, i = 1,2, ... ,n, and q;(t) are £-periodic functions. Then every solution of (3.3.33) is oscillatory if and only if its "characteristic equation"

>.(1- pe->.r) +

n

L qie->.a; = 0

(3.3.58)

i=l

bas no real roots, where

i

Remark 3.3.2.

= 1,2, ... ,n.

The following condition is sufficient for (3.3.56) to hold: for

f=T,T-O"j, i=1,2, ... ,n 00

11t+l

lie~fLL i n

i=l k=O

(

t

q;(s)ds

) p

1

k+ 1

1

((k+1)r-u;)>;·

(3.3.59)

156

Chapter 9

3.4. Comparison Results We consider neutral delay differential equations of the form n

(x(t)- px(t- r))'

+L

q;(t)x(t- u;(t))

= 0,

t;::: to.

(3.4.1)

i=l

In the following we present a necessary and sufficient condition for oscillation of all solutions of Eq. (3.4.1) by a comparison method.

Theorem 3.4.1.

Assume that

(i) p, -r are constants, 0 ~ p < 1, -r > 0; (ii) u;EC(R+,R+), lim(t-u;(t))=oo, 1-+00

i=1, ... ,n;

(iii) q; E C(~, R+), i = 1, ... , n. Then every solution of Eq. (3.4.1) is oscillatory if and only if the differential inequality n

(x(t)- px(t- r))'

+L

q;(t)x(t- u;(t)) ~ 0

(3.4.2)

i=l

has no eventually positive solutions. Let us now compare Eq. (3.4.1) with the equation n

(x(t)- px(t- -r))'

+ Lq;(t)x(t- u;(t))

= 0,

t;::: t 0 •

(3.4.3)

i=l

Theorem 3.4.2. Assume that all the assumptions of Theorem 3.4.1 hold. Further assume that 0 ~ p:::; p < 1, Mt);::: q;(t);::: 0, i = 1, ... , n. Then the oscillation of Eq. (3.4.1) implies the oscillation of Eq. (3.4.3). Since we will establish more general results in Chapter 5, we omit the proofs here. We now consider a more general neutral differential equation n

(x(t)-px(t--r))' +Lq;(t)x(t-u;(t))+F(t,x(gt(t)), ... ,x(gm(t))) = 0. (3.4.4) i=l

157

Oscillation of First Order Neutral Equations

Theorem 3.4.3. In addition to all assumptions of Theorem 3.4.1 we assume g; E C(~,R), FE C(R+ x Rm,R) and F(t,Yl>····Ym)Yt ;:: 0 whenever y 1 y; > 0, j = 1, 2, ... , m. Then the oscillation of Eq. (3.4.1) implies the oscillation of Eq. (3.4.4). Proof: Hnot, assume Eq. (3.4.1) is oscillatory, and Eq. (3.4.4) has an eventually positive solution x(t). Then x(t) satisfies the differential inequality n

(x(t)- p:c(t- r))'

+L

q;(t)x(t- u;(t)) $ 0.

(3.4.5)

i=l

By Theorem 3.4.1, Eq. (3.4.1) has a nonoscillatory solution, which contradicts 0 the assumption. For example, we consider the mixed neutral differential equation m

n

(x(t)- px(t- r))' + L q;x(t- u;) + L r;x(t + 6;) = 0,

(3.4.6)

j=l

i=l

where 0 < p < 1, r, q;, u; are positive constants, r;,C; are nonnegative constants, j = 1, 2, ... , m. According to Theorem 3.4.3 we have the following conclusion. If every solution of n

(x(t)- px(t- r))' + L q;x(t- u;) = 0

(3.4.7)

i=l

is oscillatory, then every solution of Eq. (3.4.6) is also oscillatory. Using Theorem 3.4.1 we obtain the following "Linearized oscillation" result for the nonlinear equation n

(x(t)- px(t- r))' + Lqd;(x(t -u;)) = 0. i=l

Theorem 3.4.4. Assume (i) p E (0, 1), r, q; E (O,oo), u; E (O,oo), i = 1, ... ,n; (ii) /;EC(R,R), x/;(x)>Ofor:cf:O, i=1, ... ,n;

(3.4.8)

Chapter 9

158

(iii) there exists an e > 0 such that f;~u) 2: 1 for lui $ e, i = 1, ... , n. Then every solution of (3.4.8) is oscillatory if every solution of the "linearized equation" (3.4. 7) is oscillatory. Proof: Let x(t) be a positive solution of (3.4.8). Then we have lim x(t) = 0, t-oo

and n

(x(t)- px(t- r))'

+L

q;x(t- cr;) $ 0.

i=l

By Theorem 3.4.1, (3.4.7) has a nonoscillatory solution. This contradicts our assumption. Remark 3.4.2.

Theorem 3.4.4 is true for (3.4.8) with variable coefficients n

q;(t) E C(R+,R+), i

= 1, ... ,n, satisfying j 10 I: q;(t)dt = oo. 00

i=l

We then compare the oscillation of two neutral delay differential equations

(x(t)- p(t)x(t- r))'

+ q(t)x(t- cr) = 0

(3.4.9)

(x(t)- p(t)x(t- r))'

+ q(t)x(t- cr) = 0.

(3.4.10)

and

Theorem 3.4.5. 0; p(t) ~ 1, q(t)

> 0,

Assume that p, p, q, q E C([t 0 ,oo),R+), f 1";q(t)dt = oo; and

r > cr 2:

Then (3.4.9) is oscillatory implies that (3.4.10) is oscillatory. Proof: If not, let x(t) be an eventually positive solution of (3.4.10). Set z(t) x(t)- p(t)x(t- r). Then z(t) < 0, z'(t) < 0 eventually, and

'( ) Z

q(t)

1

q(t)

t = p(t +r-crq ) (t +r ) Z ( t + T) + pt+r-cr ( )z( t + T

-

CT ).

=

159

Oscillation of First Order Neutral Equations Set .X(t)

= :(w .Then .X(t) > 0 fort;:: T .X(t) =

p( t + T

+ ;::

(

-(t) q ( /(t + r) exp U )q t + T

-

(t) q ) (

pt+r-sqt+r (t)

it+T .X(s)ds t

1t+T-I1 .X( s )ds it+T .X(s )ds ) .X( t + r) exp

q( t) exp p(t+r-u)

+ pt+r-u ( q ) Define .X 0 (t) = .X(t),

and

t

t

exp

it+T-11 .X(s)ds.

(3.4.11)

t

t;:: T, and for n = 0, 1, 2, ... and t;:: T

An+l(t) = p(

q(t)

) ( /n(t t+r-uqt+r

q(t)

+ pt+r-u ( ) exp

+ r) exp

it+T An(s)ds t

(3.4.12)

it+T-11 An(s)ds. t

Clearly, 0 $ An(t) $ An-l(t) $ · • · $ Ao(t) Hence lim An(t) n-oo

= .X(t),

= .X*(t) exists and .X*(t) ;:: 0, t ;:: T.

t;:: T.

By the Lebesgue's domi-

nated convergence theorem to (3.4.12) we obtain that

.X*(t) =

it+T .X*(s)ds

q(t)

(t ) ( ).X*(t + r) exp p +r-uqt+r . q(t) + p(t+r-u)

exp

t

it+T-11 .X*(s)ds. t

(3.4.13)

Set

y(t)

= -exp

£

.X*(s)ds < 0.

(3.4.14)

Then y'(t) = .X*(t)y(t), and (3.4.13) becomes

'( ) _ q(t) 1 q(t) y t - p(t+r-u)q(t+r)y(t+r)+ p(t+r-u)y(t+r-u).

(3.4.15)

ChapterS

160 Let x(t) =- ~g::?

> 0. Thus (3.4.15) reduces to

(x(t)- p(t)x(t- r))' This means that Eq. assumption.

+ q(t)x(t -u) = 0.

(3.4.15) has a positive solution x(t), contradicting the 0

3.5. Unstable Type Equations We consider neutral differential equations of the forms

(x(t)- px(t- r))'- q(t)x(g(t)) = 0,

t ~to

(3.5.1)

and

(x(t)- px(t- r))'- q(t)x(g(t)) = F(t), where p ~ 0, r lim g(t) = oo.

> 0, q E C([to,oo),%),

t ~ t0 •

(3.5.2)

g, FE C([to,oo),R), g(t)::; t and

t-+oo

We first show some properties of nonoscillatory solutions of Eq. (3.5.1). Lemma 3.5.1. Let x(t) be a positive solution of (3.5.1). Then x(t) is of one of the following types of asymptotic behavior: (a) lim x(t) = 0, t-+oo

(b) lim x(t) = t-+oo

ei

(c) lim x(t) =

0,

t-+oo

00.

To prove this lemma we shall use the following facts: Lemma 3.5.2.

Let x, z E C([t,oo),R) satisfy

z(t) = x(t)- px(t- r),

t

~

t0

+ max{O, r },

where p, r E R. Assume that X is bounded on [to,oo) and lim z(t) = t-+oo Then the following statements hold: (i) Ifp = 1, then£= 0; (ii) Ifp ;f ±1, then lim x(t) exists. t-+oc

e exists.

161

Oscillation of First Order Neutral Equations The next lemma is for the first order linear difference equation x(t + 1) = p(t)x(t) + y(t),

(3.5.3)

x(to) = xo

where to and t are integer-valued, p andy are given functions with p(t) :/= 0 for all t. The corresponding homogeneous equation to (3.5.3) is

z(t + 1) = p(t)z(t),

Lemma 3.5.3. Let p(t) solution of Eq. (3.5.3J is

:f. 0 and y(t)

(3.5.3)'

z(to) = zo.

be given fort= t 0 , t 0 + 1, .... Then the

t-1

z(t) = z(to)

IT p(s),

t =to+ 1, to+ 2, ...

•=to to-1

where

IT

p(s)

= 1; and the solution of Eq. (3.5.3) is

•=to

(3.5.4)

to-1

where

E

p(s)

= 1.

•=to

Proof of Lemma 3.5.1: Set z(t) = x(t)- px(t- T), then z'(t) ~ 0. First we assume that z(t) > 0 eventually. Then lim z(t) = l t-+oo

~

co. If

l =co, then x(t) ~ z(t)-+ co, as t-+ co, i.e., (c) holds. If l < co and x(t) is bounded, Lemma 3.5.2 implies that lim x(t) exists t-+oo

when p :f. 1, i.e., either (a) or (b) holds. When p = 1, x(t) ~ x(t- T) + ~ ~ · · · ~ x(t- nT) + n · ~ -+co as n-+ co. This is impossible since x(t) is bounded. H l
which contradicts the boundedness of z. For p as the above. For p > 1, we have

= 1, we have the same argument

x(t) ~ px(t- T) ~ · · · ~ pnx(t- m)

Chapter 9

162

which implies that (c) holds. lim z(t) = l.::; 0. Next we consider the case that z(t) < 0 eventually. Then t-oo If p f 1 and xis bounded, by Lemma 3.5.2 either (a) or (b) holds. If p = 1 and z(t) -t l. < 0 as t -too, then l. x(t)::; x(t- r) + 2"

.

Applying this inequality repeatedly we obtain that lim x(t) impossible. If p

lim z(t) = 0, we define = 1 and t-oo

z(t)

=

This is

(t- to+ r)z(to)/r, to-r::; t $to

t $ t0

0, Clearly,

= -oo.

t~to

z(t), {

t-oo

-

r.

z is continuous on R and [.!:fa-+1) x(t)=

L

(3.5.5)

z(t-ir).

i=O

Hence x(t) is monotonic, and the conclusion of Lemma 3.5.1 holds. We now consider the case that x(t) is unbounded and z(t) -t l. $ 0 as t-+

00.

If p E (0, 1], then x(t,.)

x(t) = max t
-t oo as n

-+

oo. This is impossible

since z(t,.) ;::= x(t,.)(1- p) and z(t)is bounded. If p > 1, from (3.5.4), let x(t) be an unbounded solution, then every solution of x(t) = px(t- r) + w(t) takes the form x(t) = x(t) + w(t), where w(t) is some solution of w(t) = pw(t- r). Clearly, w(t) -+ oo as t -+ oo and hence x(t) -+ oo as t-+ oo. Theorem 3.5.1. Based on the range of p we have tlw conclusions: (i) Ifp ;::= 0, Eq. (3.5.1) has a positive solution x(t) satisfying (b) or (c); (ii) Ifp = 1, Eq. (3.5.1) has an unbounded solution x(t) satisfying (c); (iii) Ifp > 1, Eq. (3.5.1) has an unbounded solution x(t) which tends to infinity exponentially; q(t)dt = oo, Eq. (3.5.1) has an unbounded positive (iv) IfO $ p < 1, and solution, and eve.ry bounded solution of Eq. (3.5.1) is either oscillatory or tends to zero as t -+ oo.

J,";

163

Oscillation of First Order Neutral Equations

Proof: Choose a positive continuous function H(t) such that

i

oo

q(t)H(t)dt = oo

(3.5.6)

to

and

lim{ t--+oo

q(t) exp(J1: q(s)H(s)ds)

}=o.

(3.5.7)

Define a function v by

(3.5.8) Let BC be the Banach space of all bounded and continuous functions y : [to, oo)-+ R with the sup norm. Define a subset Q of BC as follows: Q = {y E BC: 0 $ y(t) $ 1,

t0 $ t

< oo}.

Clearly Q is a bounded, closed and convex subset of BC. Now we define a mapping T on Q as follows: p

(Ty)(t) = {

v~~l)

+ vhJ J;q(s)v(g(s))y(g(s))ds + 2 vltl'

~(Ty)(T)+(1-~),

where Tis sufficiently large so that t-

p fort~

v(t- r) v(t)

1

t~T

(3.5.9)

t 0 $t$T, T ~to,

g(t)

~to,

['

v(t)

~

1, and

1

+ v(t) lr q(s)v(g(s))ds $ 2

(3.5.10)

T. In fact, from (3.5.7) and (3.5.8) it is easy to see that

v(t- r) -+ 0 and v(t)

ft: q(s )v(g( s ))ds v(t)

-+ 0

as

t-+oo

which shows that (3.5.10) is true for large t. Thus we have TQ

c n.

Let YI and

Chapter 9

164

n.

Y2 be two functions in

Then

v(t- r) I(Ty2)(t)- (Tyi)(t)l ~ p v(t) IYz(t- r)- Yl(t- r)l 1 [' + v(t) lr q(s)v(g(s))lY2(g(s)) -yl(g(s))ids

and

1!TY2- Ty21! =sup I(Ty2)(t)- (Tyi)(t)i t?_ta

which shows that T is a contraction on that Ty = y. That is

y(t) = {

pv<:(~j>

n.

Hence there is a function y E

n such

+ vtt) J;q(s)v(g(s))y(g(s))ds + 2 v1(t)' t ~ T

,fy(T) + (1- ,f),

t 0 ~ t ~ T.

Obviously y(t) > 0 fort 2:: t 0 • Set x(t) = y(t)v(t). Then

r' q(s)x(g(s))ds + 1 , t;::: T. x(t)- px(t- r) = lr 2

(3.5.11)

Therefore, x(t) is a positive solution of (3.5.1) fort;::: T. This proves (i), (ii) and the first part of (iv). In the case that p > 1 we have

x(t)

~

px(t- r) 2:: · · · ~ pnx(t- nr),

or

x(t) ~ x(to)exp (JL(t- to)),

for

t ~to,

¥

where Jl = > 0, which shows that (iii) is true. In order to prove the second part of (iv), we let x(t) be a bounded positive solution of Eq. (3.5.1) and define z(t) = x(t)- px(t- r). Then lim z(t) = t-eo

e

165

Oscillation of First Order Neutral Equations exists. If£> 0, then x(g(t)) ~£fort~ T1 ~ t 0 • Hence

z(t)- z(T1 ) = [' q(s)x(g(s))ds

lr,

~£

[' q(s)ds.

lr,

J;,

q( s )ds ---+ oo as t ---+ oo, we have a contradiction. In view of (3.5.11) Since we assume that p ::j; 0. Now for p E (0, 1) it is impossible that £ < 0, thus 0 lim z(t) = 0 and hence lim x(t) = 0. This completes the proof. t-oo

t-.oo

Theorem 3.5.2. Assume that p E [0, 1), positive integer k such that

u'(t)

+ q(t) L"

1

i+l

u(g(t)

ft'; q(t)dt =

oo, and there exists a

+ (i + 1)r) ~ 0

(3.5.12)

i=O p

has no eventually negative solutions. Then every bounded solution of (3.5.1) is oscillatory. Proof: From the proof of Theorem 3.5.1, we see that if x(t) is a bounded positive solution of (3.5.1), then z(t) < 0 eventually. Also by induction we see that 1 " x(t-r)>-2: i+lz(t+ir). i=O p

Substituting it into (3.5.1) we obtain

z'(t) ~ -q(t)

L"

1

i+l

z[g(t)

+ (i + 1)r]

i=O p

which contradicts the fact that (3.5.2) has no eventually negative solutions. Example 3.5.1. Consider

(x(t)- ~x(t- 1))' = q(t)x(t- 10)

(3.5.13)

where q E C([to, oo ), R+ ). If the delay differential equation

z: 8

y'(t) + q(t)

i=O

2i+ 1 y(t- (9- i)) =

o

Chapter 9

166

is oscillatory, then every bounded solution of (3.5.13) is oscillatory. In fact, we may choose k = 8. Then all assumptions of Theorem 3.5.2 hold. Theorem 3.5.2. Assume that p < 0, p =f:. -1, and every bounded solution of (3.5.1) is oscillatory.

f

1";

q(t)dt = oo. Then

Proof: Let x(t) > 0 be a bounded solution. Set z(t) = x(t)- px(t- r), then z'(t) ~ 0. Thus lim z(t) =f. exists and f.> 0. From (3.5.1) we have t->oo

1

00

q(t)x(g(t))dt

< oo.

to

In view of the assumption we have that liminf x(g(t)) sequence {tn} such that lim tn n-oo

z(g(tn))- z(g(tn)-

= oo and

t-oo

= 0. Then there exists a

lim x(g(tn)- 2T)

n~oo

= 0. Noting that

r)

= x(g(tn))- px(g(tn)- r)- (x(g(tn)- r)- px(g(tn)- 2-r)),

and letting n --. oo we have liminf (x(g(tn))- (p + 1)x(g(tn)- r)) n-oo

If p

= 0.

< -1, then liminf x(g(tn)) n~oo

=0

and

liminf x(g(tn)- r) = 0, n~oo

and so liminf z(tn) = lim z(t) = 0, contradicting that f.> 0. t-+oo

n-+oo

If -1 < p < 0, we have liminf((l n-oo

+ p)x(g(tn)- r)- px(g(tn)- 2r)) = 0

which implies that lim z(g(tn)- r) n-oo

= 0, contradicting that f.> 0 again.

Example 3.5.2. Consider (x(t)

+ ~x(t- 27r))' =

~x(t- ~11").

(3.5.14)

Oscillation of First Order Neutral Equations

167

It is easy to see that the conditions in Theorem 3.5.2 are satisfied. Therefore,

every bounded solution of (3.5.14) is oscillatory. In fact x(t) = sint is such a solution. For the case that p = -1, we need the following lemma. Lemma 3.5.1. Let p < 0. Assume that x E C([to, oo), R+), x(t)- px(t- r) is increasing, and x(t) - px(t - r) ~ f. > 0, for t ~ T ~ to. Then for every t 1 E [T, oo) the set

E = {t I t1 $ t $ t1 satisfies mes (E) ~ r, where ,8 =min measure.

+ 2r, {f,

-px(t- r)

~

,8}

-~}, and mes (E) denotes Lebesgue

In Chapter 4, we will prove a more general result, so we omit the proof here.

JE

Theorem 3.5.4. Let p = -1. Assume q(t)dt = oo for every closed subset E of [to, oo) whose intersection with every interval of the form [t, t + 2r], to $ t < oo, has a measure not Jess than T. Then every bounded solution of (3.5.1) is oscillatory. Proof: Let x(t) be a bounded positive solution of (3.5.1). As shown before,

1

00

q(t)x(g(t))dt < oo.

to

Obviously, the conditions of Lemma 3.5.1 are satisfied. The intersection of the set E ={tIT$ t < oo, x(t- r) ~ f} with the interval [s,s + 2r], s E [T,oo), has a measure not less than r. Therefore,

["" q(t)x(g(t))dt

jT

~

r q(t)x(g(t))dt ~ ~2 jEr q(t)dt =

jE

This contradiction proves the theorem.

00.

D

Now we briefly discuss an oscillatory property of the forced equation (3.5.2).

Chapter 9

168 Theorem 3.5.5. and

Assume that there exists a function

liminf f(t) = -oo t-oo

f such that F(t)

and lim sup f(t) t--+oo

= oo.

= f'(t),

(3.5.15)

Then every bounded solution of (3.5.2) is oscillatory. Proof: Let x(t) be an eventually positive and bounded solution of (3.5.2) and set z(t) = x(t)- px(t- r). Then

(z(t)- f(t))'

= q(t)x(g(t))

~ 0.

If z(t)- f(t) ~ 0 eventually, then z(t) ~ f(t) eventually, which contradicts (3.5.15) and the boundedness of z. Hence z(t)- f(t) > 0 eventually, which is impossible since z(t) is bounded. 0 Example 3.5.2. The equation

(x(t) + x(t -11")) 1 - tx(t- 211") = -tsin t

(3.5.16)

satisfies the assumptions of Theorem 3.5.5. Hence every bounded solution of (3.5.16) is oscillatory. In fact, x(t) = sint is such a solution.

3.6. Sublinear Equations Consider the nonlinear neutral differential equations of the form

(x( t) + p(t)x( r(t))) 1 + f(t, x(gi (t)), ... , x(gN( t)))

= 0.

(3.6.1)

Definition 3.6.1. f(t, Yb ... , YN) is said to be strongly sublinear if there exist constants (3 E (0, 1) and m > 0 such that izi-Pif(t, z, ... , z)i is nonincreasing in

lzl for 0 < lzl

~

m.

The following notations will be employed:

= 19~N max g;(t) R(g.,r) = {t E [t 0 ,oo): g*(t)

to S g*(t) < r(t)}.

169

Oscillation of First Order Neutral Equations

Theorem 3.6.1.

Assume that

(i) p E C([t0 ,oo),(O,oo)),

1 < p.

~

p(t)

~ 11

where p. and

11

are constants;

(ii) r E C([t 0 ,oo),R) is strictly increasing, and satisfies that r(t) limt_. 00 r(t) = oo; (iii) g; E C([to,oo),R), limt- 00 g;(t) = oo, (iv)

(v)

< t, and

i = 1,2, ... ,N,

f

E C([t 0 ,oo) x RN,R) is nondecreasing in each y; and strongly sublinear, andy;J(tt,yt, ... ,yN)?.O, tOforyty;>O, 1~i~N;

fn(g•,r) lf(t, a, ... , a)idt = oo for every constant

a=/= 0.

(3.6.2)

Then every solution of Eq. (3.6.1) is oscillatory. Proof: Assume the contrary and let x( t) be a nonoscillatory solution of Eq. (3.6.1 ). Without loss of generality we may assume that x(t) is eventually positive. Set

z(t) = x(t) + p(t)x(r(t)). Then z(t) > 0 and z'(t)

~

(3.6.3)

0 for all large t. From (3.6.3) and using (i), we have

(3.6.4) for all large t, where r- 1 is the inverse function of rand r- 2 (t) = r- 1 (r- 1 (t)). Setting y(t) = ~z(t) and noting that f is sublinear, from (3.6.1) and (3.6.4) we have

(3.6.5) provided T 2:: to is sufficiently large. From (3.6.2) and (3.6.5), it is easy to show that ~~~z(t) = 0. Thus there exists T1 ;::: T such that y(r- 1 (g*(t))) ~ m for

Chapter 9

170

t

~

T1 • In view of (iv), we have

f (t, y( ,.-I (gi (t))), ... , y( ,.-I (gN( t)))) ~ f(t,y(T-I(g*(t))), ... ,y(T-l(g*(t))))

~ m-.8(y(T- 1 (g*(t)))).B f(t,m, ... ,m),

t ~ T1.

{3.6.6)

From (3.6.5) and (3.6.6) we have

(3.6.7) Dividing {3.6.7) by (y(t)).B and integrating it over R(g*,,.) n [T1 ,oo), we have

1

R(g• ,r)n{T1 ,oo)

f(s,m, ... ,m)ds~

pvm.B(y(TI)) 1 -P ( _ 1)( 1 -.8)
which contradicts {3.6.2).

Assume that (i) - (iv) in Theorem 3.6.1 hold excepting Theorem 3.6.2. the assumption of sublinearity of function f. Then Eq. (3.6.1) has a bounded nonoscillatory solution which is bounded away from zero if and only if ["" lf(t, b, ... , b)ldt < oo

lto

for some constant

b f. 0.

(3.6.8)

Proof: Necessity. Let z(t) be a bounded positive solution satisfying z(g;(t)) ~ c > 0 for t ~ T, 1 ~ i ~ N. Then z(t) defined by (3.6.3) is bounded and nonincreasing. Integrating (3.6.1) we have

£""

f(t, c, ... ,c)dt

~ z(T)- z(oo) < z(T) < oo

which contradicts (3.6.2). Sufficiency. Let c > 0 and T > to be such that pc ~ b and To=min {T(T), inf {g 1 (t)}, ... ,inf {gN(t)}} ~to l~T

l~T

Oscillation of First Order Neutral Equations

171

and

Loo f(t,p.c, ... ,p.c)dt ~ (p. -1)c.

(3.6.9)

Define a set XC C[To,oo) as follows:

e C[T0 ,oo):

c ~ x(t) ~ J-LC fort 2: T, x(t) = x(T) for To~ t ~ T}. (3.6.10) For every x E X we define

X= {x

t 2: T (3.6.11)

H0 (t)

= 1,

i-1

H;(t) = flp(ri(t)),

i = 1,2, ....

(3.6.12)

j=O

Then 0 < x(t)

~

t 2: T. Define a mapping Ton X as follows:

p.c,

c+ ]', f(s,x(gi(s)), ... ,x(gN(s)))ds, t;::: T 00

(Tx)(t) = {

Tx(t),

To~

(3.6.13)

t

~

T.

In view of (3.6.9), T X C X. It is not difficult to prove that T is continuous and T(X) is relatively compact in the topology of the Frechet space C[T0 ,oo). Therefore, by the Schauder Tychonoff fixed point theorem, there exists an x E X such that x = T x. That is

(3.6.14) From (3.6.11) we have

x(t) + p(t)x(r(t)) = x(t), and thus

x(t)+p(t)x (r(t))=c+

1

00

f(s,x(gi(s)) , ... ,x(gN(s)))ds,

t?_T. (3.6.15)

Chapter 3

172

Differentiating (3.6.15) we have that the function x(t) defined by (3.6.11) satisfies Eq. (3.6.1) fort;::: T and (3.6.16) Therefore, x(t) is a bounded nonoscillatory solution of Eq. bounded away from zero.

(3.6.1) which is 0

Combining Theorems 3.6.1 and 3.6.2, we obtain the following result. Corollary 3.6.1. Let g•(t) < -r(t) and (i)-(iv) in Theorem 3.6.1 bold. Tben, (3.6.2) is a necessary and sufficient condition for oscillation of all solutions of Eq. (3.6.1).

Example 3.6.1. Consider the neutral equation

(x(t) where Jl

+ JlX(t- 211'))' + (1 + Jl) cosi txl (t- ~ 11') =

0

(3.6.17)

> 1 is a constant.

It is easy to see that all the assumptions of Theorem 3.6.1 are satisfied. Therefore every solution of (3.6.17) oscillates. In fact, x(t) = sint is such a solution. The following example shows that condition (3.6.2) is not sufficient for oscillation of linear equations.

Example 3.6.2. Consider the linear equation

(x(t)

2 + 2x(t- 1')) + 3ft(t- _4f2)+2 4 x(t) = I

0.

(3.6.18)

It is easy to see that all assumptions of Theorem 3.6.1 are satisfied. But (3.6.18) has a positive solution x(t)

=t .

We now consider the nonlinear neutral differential equation described by n

(x(t)- px(t- -r))' + Lqi(t)/;(x(gi(t))) = 0 i=l

(3.6.19)

173

Oscillation of First Order Neutral Equations

where p;:::: 0, r > 0, q; E C([to,oo),~), g; E C([to,oo),R), oo, !; E C(R,R), xf;(x) > 0 for x =J 0, i = 1,2, ... ,n. f is said to be locally sublinear near x = 0 if there exists a > 0 such that f(x) is increasing in (O,a) and (-a,O) and

r

dx Jo f(x) < oo and

0 /_ -at

dx f(x) < oo.

(3.6.20)

Theorem 3.6.3. Assume that p E [0, 1), there exists i 0 E In= {1, ... , n} such that j; 0 (x) is locally sublinear near x = 0, and %(t)dt = oo. Then (3.6.19) •o is oscillatory, where

JA·

A; 0 = {t E [to,oo): to :5 g; 0 (t) :5 t}.

(3.6.21)

Proof: Let x(t) be an eventually positive solution of (3.6.19), say, x(t) > 0, x(g;(t)) > 0 for t 2:: t1 2:: to, i = 1, ... , n. Set z(t) = x(t) - px(t - r), then z'(t) :50. If z(t) > 0, t 2:: t2 2:: t1, we have limt-+oo z(t) = l1 2:: 0. If l 1 > 0, since l1 :5 z(t) :5 z(t2) fort~ t 2 and

N(t)-1

x(t) =

L

pi-lz(t- (i -1)r) + pN(t)- 1x(t- (N(t) -1)r)

i=l where N(t) is a positive integer such that T- r < t- (N(t) -1)r:::; T, we have

and

From (3.6.19) we see

-z'(t) 2:: %(t)!; 0 (x(g; 0 (t))) 2:: q; 0 (i)L.

Chapter 9

174

Integrating it from t 2 to t we have

z(t2)- z(t)

~L

lt

%(s)ds.

t,

Letting t -+ oo we obtain

JA;o

q; 0 (t)dt = oo. which contradicts that If z(t)-+ 0 as t-+ oo, then x(t)-+ 0 as t-+ oo. Then there exists t3 ~ t2 such that 0 ~ x (gio ( t)) ~ a and z( t) ~ a for t ~ ta. Since f is nondecreasing for t E A; 0 n [t3, oo)

From (3.6.19)

Then we have rz(ts)

Jz(t)

it

du fio(u) =

t3

~

1

-z'(s) J;.(z(s)) ds

A; 0 n(ts,t]

-z'(s) ds J;.(z(s))

~

1

q;.(t)dt.

A; 0 n(t 3 ,t]

Letting t -+ oo we have

JA·

q; 0 (t)dt = oo again. which contradicts •o lim z(t) = The last possibility is that z(t) < 0, z'(t) ~ 0, t ~ t 2 • Then t-oo exists a there so unbounded, is x(t) then oo, = £ -£2, £2 > 0 or £2 = oo. If 2 sequence {tn} satisfying lim tn = oo, x(tn) = max x(t), and lim x(tn) = n-+oo

t2
n--+oo

oo. Thus z(tn) = x(tn)- px(tn- r) ~ (1- p)~(t~) > 0, which contradicts lim x(t) = -...!lthe negativity of z. If z(t)-+ -£2, £2 E (O,oo), then t--+cx:> 1~p < 0 0 contradicting the positivity of x.

175

Oscillation of First Order Neutral Equations

Remark 3.6.1. The local sublinear condition for J; is important here. In fact, Theorem 3.6.3 is not true for linear equations. For example, consider the equation

(x(t)- ~x(t -1)) where q(t) = !:2(~~1~ solution x(t) =

t.

,

+

- 4t + 4 4t 2 (t _ 1 ) x(t -1) = 0

t2

(3.6.22)

. Clearly, J:"; q(t)dt = oo. But (3.6.22) has a nonoscillatory

Using Tychonoff-Schauder fixed point theorem we can prove the following result for the existence of nonoscillatory solutions. We omit the proof here. Theorem 3.6.4.

n

< 1 and L: ftoo q;(t)dt <

Let 0 ~ p

i=l

oo. Then (3.6.19) has a

0

bounded nonoscillatory solution x which does not tend to zero as t

-+

oo.

Corollary 3.6.2. Assume that 0 ~ p < 1, g;(t) ~ t, /;(x) is locally sublinear near x = 0, ' = 1, 2, ... , n. Then every solution of Eq. (3.6.19) is oscillatory if and only if n

['>O

E lt i=l

q;(t)dt

= oo.

to

3. 7. Equations with Mixed Coefficients In this section we consider neutral delay differential equations with positive and negative coefficients of the form

(x(t)- c(t)x(t- r))'

+ p(t)x(t- r)- q(t)x(t- a)= 0.

The following is an oscillation criterion for ( 3. 7.1). Theorem 3.7.1.

Assume that

(i) r E (O,oo), and r, a E (O,oo), and c,p,q E C((t 0 ,oo),R+); (ii) r ~ a; (iii) p(t) ~ q(t- T +a) and p(t) ¢. q(t- r +a) fort~ t 0 + r- a;

(3.7.1)

Chapter 9

176 (iv) 1- c(t)- J11_r+tt q(s)ds ~ 0 for all sufficiently large t; (v) every solution of the delay differential equation

q(s- T)ds) y(t- T) = 0 y'(t) + (p(t)- q(t- T + u)) (1 + c(t- T) + 1' t-r+a (3.7.2) is oscillatory. Then every solution of (3. 7.1) is oscillatory.

Proof: Assume the contrary, and let x(t) be an eventually positive solution of Eq. (3.7.1). Denote 1

q(s)x(s- u)ds. y(t) = x(t)- c(t)x(t- r) - 1 t-r+a Then it is easy to see that y'(t) :5 0 and (3.7.3) and (3.7.1),

y'(t)

(3.7.3)

y(t) > 0, eventually. In view of

= -(p(t)- q(t- T + u))x(t- T).

(3.7.4)

From (3.7.3) we have

x(t)

~ y(t) + c(t)y(t- r) + i~r+a q(s)y(s- u)ds ~

(1+c(t)+ ('

lt-r+tt

q(s)ds)y(t).

(3.7.5)

Substituting (3.7.5) into (3.7.4) and noting condition (iii), we have 1 q(s-T)ds) y(t-T) :50. (3.7.6) y'(t)+(p(t)-q(t-T+ u)) (1+c(t-T)+1 t-r+a

Hence, (3.7.2) has a positive solution, which contradicts (v).

Assume that > u;:::: 0, c,p,q E C([t 0 ,oo),R+)i

Theorem 3. 7.2.

i) r

> 0,

T

0

177

Oscillation of First Order Neutral Equations ii) there exists t1

~ t0

such that fort

0::; c(t)

p(t)

~ t1

+ 1~~" q(s + u)ds::; 1,

= p(t)- q(t- r + u) ~ 0;

J,11_ r p( s )ds > 0; there exists T;?: t 1 + max{r, T}

(3.7.7)

iii) lim inf t-oo

iv)

inf

t~T,A>O

such that

1 {~exp(.X1t-r p(s)ds)+c(t-r)exp(.xjt p(s)ds) t-r A

(3. 7.8)

Then every solution of (3. 7.1) is oscillatory. To prove this theorem we need the following lemmas.

Lemma 3.7.1. Assume that conditions i)-iii) in Theorem 3.7.2 hold. Let x(t) be an eventually positive solution of (3. 7.1). Set u(t) = x(t)- c(t)x(t- r)

-lt-
(3.7.9)

t-r

Then u'(t) $ 0,

u(t) > 0 eventually.

Proof: From (3.7.1), (3.7.9), and condition ii) in Theorem 3.7.2, we see that

u'(t) = -p(t)x(t- r) $ 0,

(3.7.10)

t;?: t2;?: t1.

Denote lim u(t) =f. We show that f;?: 0. t-+oo

First we consider the case that x(t) is unbounded, i.e., limsup x(t) = oo. t-+oo

Then there exists a sequence {sn} such that lim sn = oo, n~oo

lim x(sn) = oo

n-oo

178

Chapter 9

and x(sn) = max{x(t): t2:::; t:::; sn},

u(sn) = x(sn)- c(sn)x(sn- r)

2:: x(sn) [1- c(sn)Therefore

e=

n = 1,2, .... Hence from ii)

-1:~~" q(s + <1)x(s)ds

1:~~" q(s + <1)ds]

2:: 0,

n

= 1, 2, ....

lim u(t) = lim u(sn) 2:: 0.

t~oo

n--+oo

Next we consider the case that x(t) is bounded. Set lim sup x(t) = f < oo. t-+oo

Then there exists a sequence {sn} such that lim Sn = oo and lim x(sn) = n--+oo n-oo Denote r 1 =min {r,<1}, r 2 =max {r,r}, and

x(en) = max{x(s): sn-

r2 :::;

s:::; sn- r1 },

f.

n = 1, 2, ....

Clearly lim en= oo and lim sup x(en):::; f. From ii) n-+oo

n-+oo

x(sn)- u(sn) = c(sn)x(sn- r) +

!.~~~" q(s + <1)x(s)ds

:::; x(en)[c(sn) + !.~~~" q(s + <1)ds] :::; x(en). Taking superior limits on both sides as n --. oo we obtain that f.- f. S

f., and so

e;::: 0. Then u(t) > e;::: 0 fort ;::: t 2 •

0

Lemma 3.7.2. Assume that i)-iii) in Theorem 3.7.2 hold. Let x(t) be an eventually positive solution and let u be defined by (3. 7.9). Define a set A= {A> 0: u'(t) + ..\p(t)u(t):::; 0

holds eventually}.

Then A =f:. 0, and has an unifonn upper bound which is independent of x(t).

Proof: From Lemma 3.7.1 and its proof, there exists t 2 2:: t 1 such that u'(t):::; 0, u(t) > 0, fort ;::: t2, and

u'(t)

= -p(t)x(t- r):::; -p(t)u(t- r):::; -p(t)u(t),

t;::: t 2

+ r,

(3.7.11)

179

Oscillation of First Order Neutral Equations

i.e., 1 EA. Now we show that A has an upper bound which does not depend on x(t). Set

liminfj t-oo

1

t-r

p(s)ds

= 3k.

From condition iii) in Theorem 3. 7.2 we see k > 0. Hence there exists t3 such that

~

t2

+T

j~rp(s)ds>2k, t~t3. By Lemma 2.1.3 this together with (3.7.11) implies that u~~)) ::; p, t ~ t 3 . From x(t) = 0. (3.7.1) and condition (iii) in Theorem 3.7.2, it is easy to see that liminf t-+oo lim sn = oo, and Hence there exists a sequence {sn} such that sn ~ t 3 + 2r, n-+oo x(sn- r) = min{x(s- r): t3 ::; s::; sn}, n = 1, 2, .... Integrating (3.7.10) from Sn - T to sn we have

u(sn)- u(sn- r) = -1:-r p(s)x(s- r)ds

1

4n

::; -x(sn- r) an-r p(s)ds::; -2kx(sn- r),

n = 1,2, ... ,

and hence u(sn- r) > 2kx(sn- r). Substituting it into (3.7.10) we obtain

u'(sn) = -p(sn)x(sn- r) 1 1 >- 2k p(sn)u(sn- r) >- 2k3 p(sn)u(sn)

which means that ~xw.

2h- EA, i.e., 2h- is an upper bound of A which does not depend

o

Proof of Theorem 3.7.2: Let x(t) be an eventually positive solution. By Lemmas 3.7.1 and 3.7.2 u'(t) ::; 0, u(t) > 0, t ~ t 2 • We claim that there exists an a > 1 such that ..\ 0 E A implies that a..\ 0 E A.

ChapterS

180

In fact, from condition (iv), there exists a> 1 such that inf inf { ~exp (.xjt p( s )ds) + c( t - r )exp (.xjt p( s )ds)

A>O t~T

A

+

1-T

l~~u q( s -

1-T

r +

q

1

)exp ( >.1 p( tt )du) }

~ a > 1.

(3.7.12)

By the definition of A we see that eventually

u'(t) + >.0 p(t)u(t) :::; 0. Set z(t) = u(t)exp(>.o

It: p(s)ds).

(3.7.13)

Then

z'(t) = [u'(t) + >.op(t)u(t)] exp (>. 0 it p(s)ds) :::; 0. I,

From (3.7.10) and (3.7.13) we have x(t- r) (3.7.12)

~

>. 0 u(t), and then from (3.7.9) and

u'(t) = -p(t)x(t- r)

= -p(t) [u(t- r) + c(t- r)x(t- T - r) + 1t-r-u q(s + q)x(s)ds] t-2r

:5 -p(t) [u(t- r) + >.oc(t- r)u(t- r) + >. 0 1t-r-u q(s + q)u(s + r)ds] l-2r

= -p(t) [u(t- r) + >. 0 c(t- r)u(t- r) + >. 0

l~~u q(s + q- r)u(s)ds]

= -p(t) [ z(t- r)exp(- >. 0 1:-r p(s)ds)

+ >.oc(t- r)z(t- r)exp(- >. 0 1:-r p(s)ds) + >.o

1~~u q( s + q -

:5 -p(t)[exp(>.o

T

)z( s )exp ( - >. 0 1: p( u )du) ds]

1~rp(s)ds) +>. c(t-r)exp(>.o 1~/(s)ds) 0

181

03cillation of Fir3t Order Neutral Eqv.ation3

i~~(f q(s + u- r)exp( >.

+ Ao

~-

(>-o

inf exp

t~T

+ >.o ~

i'

t-r

0

1'

p( s )ds) + >.o c(t -

i~~(f q(s + u- r)exp( >.o

T

p(u)du )ds] u(t)

)exp

1'

(>-o it-r t p( s )ds)

p(u)du )ds] p(t)u(t)

-a>.op(t)u(t).

This implies that a>. 0 E A. Repeating this we obtain that am >.o E A for any positive integer m, which contradicts the boundedness of A. D Theorem 3.7.3.

i)

r

> 0,

T

Assume that

>u

~

0,

c,p,q E C([to, oo),R+);

ii) p(t) ~ 0, fort~ t 1 ~ t 0 and ftc;'p(t)dt = oo, where pis defined in Theorem 3.7.1; iii) For some T

~ t1

+max {r, T }, there exists,\* > 0 such that

sup { ;. exp (,\ * t2:T

+

i'

t-r

p( u )du) + c( t -

i~~(f q(s- r + u)exp( >.*

T )exp (,\•it

1'

p(u)du) ds}

Then Eq. (3.7.1) has a positive solution x satisfying lim x(t) t-+oo

x(t)

~ exp(- >.•i:+r p(u)du)

p( s )ds)

t-r

for all large

~ 1.

(3.7.14)

= 0 and

t.

(3.7.15)

To prove this theorem we first show the following lemma. Lemma 3. 7 .3.

Let conditions i) and ii) in Theorem 3. 7.2 hold. FUrther assume

Chapter 9

182

that the integral inequality

+ l~~w q(s + u)z(s)ds

c(t)z(t- r)

1:

+

[p(s

+ r)- q(s + u)]z(s)ds :5 z(t), (3.7.16)

lim z( t) = 0 where has a positive solution z : [t 1 - m, oo) -+ (0, oo) such that t-oo m = max { r, r }. Then there exists a positive function x : [t 1 - m, oo) -+ (0, oo) satisfying the integral equation

c(t)x(t-r)+

1~~" q(s+a)x(s)ds

+ 1:[p(s + r)- q(s + u)]x(s)ds

= x(t), (3.7.17)

and 0 < x(t) :5 z(t),

t ~ t 1•

Proof: Choose T ~ t 1 such that z(t) > z(T) for t 1 functions as follows:

n ={wE C([t 1 -

m,oo),R+): 0 :5 w(t) :5 z(t),

and define a mapping T on

{

~ t 1 - m}

t

+ Jt"~; q(s + a)w(s)ds t1

-

0 :5 Xn(t) :5 Xn-l(t) :5 · · · :5 x1(t) :5 z(t),

= x(t) exists fort ~ t 1 -

m.

(3.7.19)

m :5 t < T.

By (3.7.16) Tn c n. Define a sequence {xn} on T Xn-1! n = 1, 2, .... It is easy to see that

n-+oo

t ~T

+ ft~r (p(s + r)- q(s + u))w(s)ds, (Tw)(T) + z(t)- z(T),

Then lim xn(t)

(3.7.18)

n by

c(t)w(t- r) (Tw)(t) =

m :5 t < T. Define a set of

-

t

= z,

n

by Xo

~

t1- m.

Oscillation of First Order Neutral Equations

183

It follows from the Lebesgue's dominated convergence theorem that x(t) satisfies (3.7.17). Clearly, x(t) is continuous fort~ t1- m. Since x(t) > 0 on [t 1 - m, T), it follows that x(t) > 0 for all t ~ t 1 - m.

Set

Proof of Theorem 3.7.3:

= exp(- A• £+r p(s)ds ).

z(t)

t

~ T- r.

(3.7.20)

It is obvious that z(t) is continuous fort ~ T- r and lim z(t) = 0. We shall t.-oo show that z(t) satisfies the integral inequality (3.7.16) fort~ T. From (3.7.14), we have

1 ( .\•exp .\•

lt(+r p(u)du ) rt+r-u

+ },

(

it+r

(

f.t+r

+c(t)exp A• t+r-/(s)ds

q(s- r + a)exp A• •

)

)

p(u)du ds $ 1,

t

~

T- T. (3.7.21)

We observe that roo p( S +

lt-r

T

)exp (A •

f.t+ r p( U )du) ds •+r

(3.7.22) and

j t+r-a q(st

=

T

(

+ a)exp A•

lt-rt-u

f.t+r 8

(

)

p(u)du ds

q(s + a)exp A•

jt+r p(u)du ) ds. s+r

(3. 7.23)

ChapterS

184 Substituting (3.7.22) and (3.7.23) into (3.7.21) we have c(t)exp ( A* 1

+

1""

) t+r p(u)du t+r-r

P(s

t-r

+ ~t-a q(s + u)exp ( A* 1t+r p(u)du ) t-r

ds

~+r

+ r)exp(A* {t+r p(u)du)ds ~ 1, la+r

t;::: T- r.

(3.7.24)

From (3.7.24) and (3.7.20) we obtain c(t)z(t- r)

+~~~a q(s + u)z(s)ds +

1:

p(s

+ r)z(s)ds

~ z(t),

t;::: T- r,

i.e., (3.7.16) has a continuous positive solution z(t) defined by (3.7.20). By Lemma 3.7.3, (3.7.17) has a positive solution x(t) on [T, oo) satisfying x(t) ~ z(t). Thus xis a positive solution of (3.7.1) satisfying lim x(t) = 0. 0 t-oo

Remark 3.7.1. Condition (3.7.14) is sharp in the following sense: if c,p,q are constants with c > 0 and p > q, condition (3.7.14) is not only a sufficient condition but also a necessary condition for the existence of a positive solution which tends to zero as t-+ oo. This is based on the fact that for the autonomous case, Eq. (3.7.1) has a positive solution which tends to zero if and only if its characteristic equation (3.7.25) has a negative real root. In the same sense, condition (3.7.8) is sharp.

Corollary 3. 7 .1.

Assume that conditions i) and ii) in Theorem 3. 7.2 bold,

and

li~inf{e t

00

1'

t-r

p(s)ds

+ c(t- r) + ~t-a q(s- r + u)ds} > 1.

(3.7.26)

1-r

Tben every solution of (3.7.1) is oscillatory.

Example 3. 7 .1. Consider (x(t)-

t x(t- 2))' + (1 + ~ + sint)x(t -1r)- ~ x(t- f)= 0

(3.7.27)

185

Oscillation of First Order Neutral Equations and

(x(t)-

e x(t -1)- ~ x(t- t) = 0. k x(t -1)) , + 2(13+e)

(3.7.28)

It is easy to see that for (3.7.27) and (3.7.28) all assumptions of Corollary 3.7.1 are satisfied. Therefore, these equations are oscillatory.

3.8. Linearized Oscillation The following linearized oscillation result reveals that, under some assumptions, certain nonlinear differential equations have the same oscillatory behavior as an associated linear equation with constant coefficients. Consider the neutral differential equation with positive and negative coefficients

with the following assumptions G,H~,H2 E C(R,R);

(1) P,Q1,Q2 E C([to,oo),R+), (2)

T

E (0, oo ),

(3) lim sup P(t) t~oo

u~, u2 E [0, oo ),

= P2 E (0, 1);

(5) 0 ~ G~u) ~ lfor u (6) ~~~=~

;:

1 and 0

<

:f: 0,

O"J ;:::

u2;

= Pl E (0, 1);

liminf P(t) t-+oo

lim

G(u)

U--+00

U

Ju) ~ M

H2

for u

= 1· '

:f: 0,

lim

H2(u)

u-+0

u

= 1;

(7) 1- P2- Mq2(u1- u2) > 0. We associate Eq. (3.8.1) with the linear equation with constant coefficients (3.8.2)

The following is the linearized oscillation result for Eq. (3.8.1). Theorem 3.8.1. Assume that evezy solution of Eq. (3.8.2) is oscillatozy. Then every solution of Eq. (3.8.1) is also oscillatozy.

Chapter 9

186

To prove this theorem we first show some related lemmas. Lemma 3.8.1.

Consider the neutral delay differential equation (3.8.3)

where p E ( 0, 1], T, q1, q2 E (0, oo ), u1, u2 E [0, oo). Suppose that every solution of Eq. (3.8.3) is oscillatory. Then there exists an eo > 0 such that for any e, e1. e 2 E [0, eo) every solution of the differential equation

is also oscillatory.

Proof: By Theorem 2.1.1 it suffices to show that the characteristic equation of (3.8.4) has no real roots. The assumption that every solution of Eq. (3.8.3) is oscillatory implies that the characteristic equation (3.8.5)

of Eq. (3.8.3) has no real roots. In view of f(oo) = oo we see that f(>.) > 0 for all .A E R. This means that f(O) = q1- q2 > 0 and either u1 ;::: u2 or u1 < u2 ::; T. Then f( -oo) = oo and so m := min f( .A) > 0 exists. Therefore, AER

Set

Observing that

lim (f(.A)- g(>.)) = oo. In particular, there exists a .\ 0 > 0 such IAI-+oo that f(.\)- g(.\) > T for J.AJ ;::: .\ 0 • Let TJ = J.A 0 Je.Xor + e>-out + e>. 0 u 2 and set eo= min{.5,m/2TJ}.

we have

Oscillation of Firat Order Neutral Equations

187

To complete the proof, it suffices to show that for any e 1 C:t,C:2 E [O,c:o) the characteristic equation

of Eq. (3.8.4) has no real roots. In fact, for IAI ~ Ao A- A(p- c:)e-~r + (qt - et)e-~"' 1 - (q2

+ c:2)e-~....

= j().)- [-c:Ae-~r + C:te-~"' 1 + e2e-~"'•] ~/(A)-

g(A) >

T > 0.

Also, for IAI :5 Ao A- A(p- c:)e-~r

+ (qt

- et)e-~"' 1

-

(q2

~/(A)- c:o[IAole~or ~ m-

+ e2)e-~ ....

+ e~o"'1 + e>.o"'•J

!f >0.

0

Lemma 3.8.2. Assume that FE C([T,oo),( O,oo)), HE C(R+,~), r E (O,oo), u 11 u 2 E [O,oo), c 11 c 2 E [O,oo); H(u) is nondecreasing in a neighborhood of the origin and u 1 ~ 0'2. Let m = max{ r, ut} and suppose that the integral inequality

F(t)z(t-r)+c1 1~~~· H(z(s))d s+c21:

1

H(z(s))ds :5 z(t),

t

~T

(3.8.6)

has a continuous positive solution z: [T- m, oo)-+ (0, oo) such that lim z(t) 1-ooo

0.

=

Then there exists a positive solution x : [T- m, oo) -+ (0, oo) of the corresponding integral equation

F(t)x(t-r )+ct

1~~~· H(x(s))d s+c21:

1

H(x(s))ds = x(t),

t

~ T.

(3.8.7)

Proof: Choose a Tt ~ T and a 6 > 0 such that z(t) > z(T1 ) forT- m :5 t < Tt, 0 < z(t) < 6 fort~ T1 - m and H(u) is nondecreasing in [0, 6]. Define a

Chapter 9

188

set of functions

n ={wE C([T- m,oo),R+): and define a mapping T on

0:5 w(t) :5 z(t),

t ~ T- m}

n as follows:

F(t)w(t- r) + c1 Jt~:,_2 H(w(s))ds (Tw)(t)

={

+ c2 ft~rr, H(w(s))ds (Tw)(Tl)

(3.8.8)

+ z(t)- z(T1),

T-m:5t
It is obvious that T is continuous. Also w 1, w 2 E n with w1 :5 w2 implies Tw 1 :5 Tw 2 • From (3.8.6), Tz :5 z and sow En implies 0 :5 Tw :5 Tz :5 z. Thus T : n -+ n. Define a sequence {Xn} on n as follows: Xo = z,

and

Xn = Txn-lo

n = 1,2, ....

(3.8.9)

By induction, we have that 0 :5 Xn(t) :5 Xn-l(t) :5 z(t) for

t?. T- m,

n = 1, 2, ....

(3.8.10)

Set x(t) = lim Xn(t) fort~ T- m. It follows from the Lebesgue's dominated n-+oo convergence theorem that x(t) satisfies (3.8.6). From (3.8.8) x(t) > 0 for T-m :5 t < T1. Consequently, x(t) > 0 fort ~ t - m. 0 Proof of Theorem 3.8.1: Without loss of generality we assume that x(t) is an eventually positive solution of Eq. (3.8.1) that every solution of Eq. (3.8.2) is oscillatory implies that its characteristic equation

has no real roots. Since /(0) = q1- q2 and f(oo) = oo, we see that q1 > q2. Choose 71 > 0 so small that (3.8.11) and (3.8.12)

Oscillation of First Order Neutral Equations

189

Then by assumption (5) and (6) and (3.8.1), for sufficiently large t1, when t;::: tt, we have (x(t)- p(t)G(x(t- -r)))'

+ (q1- 7J)Ht (x(t- ut))- (q2 + 7J)H2(x(t- u2))

:50. (3.8.13)

Set z(t) = x(t)- p(t)G(x(t- -r))- (q2

+ TJ) 1~~~ 2 H2(x(s))ds.

(3.8.14)

From (3.8.11) and (3.8.13) and assumption (6) z'(t) = (x(t)- p(t)G(x(t- -r)))'- (q2

:5 -(ql- 7J)Ht (x(t- u1))

+ 11)(H2(x(t- u2))- H2(x(t- Ut)))

+ (q2 + 7J)H2 (x(t- u1)) (3.8.15)

Now we claim that x(t) is bounded. Otherwise there exists a sequence {tn} such that lim tn = oo, x(tn) =max x(s), and lim x(tn) = oo. Then by n--~>oo •
+ TJ) 1~~~~ 2 H2(x(s))ds

G(x(tn- -r)) ( ) x(tn- -r) X tn-T

;::: x(tn)[1- P2- M(q2

+ TJ)(ut

- u2)]-+ oo,

as n-+ oo,

which contradicts (3.8.15). Therefore, x(t) is bounded, and so z(t) is bounded and lim z(t) = l E R. Integrating (3.8.15) we have t ..... oo

Chapter 9

190

which together with assumption (6) and the boundedness of x(t), implies that liminf x(t) = 0. We claim that lim x(t) = 0. Otherwise, let p, = limsup x(t) t-+oo

t-+oo

t-+CX>

and let {tn} and {t~} be two sequences in the interval [t1,oo) such that lim tn = oo,

n-+oo

lim t~ = oo,

n-+oo

lim x(tn) = 0

t-+oo

lim x(t~) = p,.

and

t-+oo

Since z(tn) :::; x(tn), then£:::; 0. On the other hand, by choosing sufficiently small > 0 we have

E: 1

z(t~);::: x(t~)- p(t~)x(t~- r)- M(q2 + 77) 1~~~~ 2 x(s)ds 2:: Letting n

--+

x(t~)- p(t~)(p,

+ c')- M(q2 + 77)(u1- u2)(p, + c').

oo we obtain

£ 2:: p,- P2(!-L + c')- M(q2 + 77)(u1- u2)(p, + c'). Since

E: 1

can be arbitrarily small, we have

£ 2:': 1-L- P2/-L- M(q2

+ 77)(u1- u2)!-L = [1- P2- M(q2 + 77)(u1- u2)]p,

which, together with£:::; 0 and (3.8.12), implies£= p, = 0. Hence lim z(t) = 0 and lim x(t) = 0. Then t-+oo

t-+c:x:>

(3.8.16) Now by using (3.8.14) and (3.8.16) we obtain

x(t) 2:: p(t)G(x(t- r))

+ (q2 + 77) 1~~~ 2 H2(x(s))ds

+ (qt- q2- 277) =p(t)

1:.

H2(x(s))ds

r-cr•

G(x(t- r)) x(t-r) x(t-r)+(q2+77) lt-cr,

+ (qt- q2- 277)

1

00

t-u1

H 2(x(s)) x(s) x(s)ds

H2(x(s)) x(s) x(s)ds.

191

Oscillation of Fir.st Order Neutral Equations Choose 0

< e < -21 min {P2,

}. Then by assumptions (3), (5), (6) __!L_+ 92 'I

2

x(t) 2: (P2- e)x(t- r) + (q2 + 77)(1- e) ~~~~ x(s)ds

+ (q1 -

q2- 277)(1- e)

1:

1

x(s)ds,

t;::: t2

where t 2 is sufficiently large. By Lemma 3.8.2, the equation

2 v(t) = (P2- e)v(t- r) + (q2 + 77)(1- e) ~~~~ v(s)ds

+ (q1- q2- 277)(1- e)

1:

1

v(s)ds

has a continuous positive solution v : [T - m, oo) ---. (0, oo ), where T 2: t2 is sufficiently large. Clearly, v(t) is a positive solution of the neutral equation

where e1 = 77 + eq1 - 71e and e2 = 77 - q2e - 71e are sufficiently small positive numbers. Hence, by Lemma 3.8.1., Eq. (3.8.2) has a positive solution. This 0 contradiction proves the theorem.

3.9. Equations with a Nonlinear Neutral Term Consider the nonlinear neutral differential equations of the form

(x(t)- g(x(t- r)))' + h(t, x(t- u)) = 0,

t;::: t 0

(3.9.1)

where r > 0, u 2:0, g E C(R, R), hE C([to, oo) x R, R). Theorem 3.9.1. Assume that (i) g is nondecreasing, xg(x);::: 0 and there exists a E (0, 1] such that either

lim sup lzi--ooo

IYI (xl )I =MER+, X"'

Chapter :J

192

whenever o: E (0, 1), or lim sup g(x) = c E [0, 1), lxl-+oo X

(ii) his nondecreasing in x,

h(t,x)x

~

roo jh(t, c)jdt =

ito

if o:

= 1;

0 for x =/: 0, and

00

for any c =/: 0;

(iii) There exists a positive integer n such that the differential inequality

y 1(t)

+ h(t,y(t- u) + g(y(t- r)

+ g(y(t- u- 2r) + ··· + g(y(t- 0'- nr)) ... )))

::; 0 (3.9.2)

bas no eventually positive solutions and

y'(t) + h(t, y(t- u)

+ g(y(t- u- r)

+ g(y(t- u- 2r) + ··· + g(y(t- u- nr)) ... )))

~ 0

(3.9.3) has no eventually negative solutions. Then every solution of Eq. (3.9.1) is oscillatory.

Proof: Assume the contrary, and let x(t) be an eventually positive solution of Eq. (3.9.1), say x(t) > 0, t;:::: T;:::: t 0 • Set

z(t)

= x(t)- g(x(t- r)).

(3.9.4)

Then z'(t) ::; 0, t ~ T + m, m = max {r,u}. If z(t) < 0 eventually, then there exists a t 1 ~ T + m such that z(t) ::; -f.< 0 fort 2: t 1 • Thus, from (3.9.4), g(x(t- r)) ~ £ > 0 fort;:::: t 1 • Integrating Eq. (3.9.1) and noting condition (ii) we have lim z(t) = -oo. Consequently, lim g(x(t- r)) = oo, which implies that t-+oo

t-..oo

lim x(t) = oo. Thus there exists a sequence {en} such that lim

t-+oo

n-+co

en =

oo and

193

Oscillation of First Order Neutral Equations x(~n) =

max {x(t)}-+ oo,

t,9::;;en

n-+ oo. Then

z(~n) = x(~n)- g(x(~n- T));::: x(~n)- g(x(~n))

= x(~n)[1- g(x(~n))/x(~n)]-+ oo,

as

n-+ oo,

which contradicts the fact that z(t) is eventually negative. Therefore z(t) > 0 for t ;::: T + m. Hence

+ g(x(t- T)) + g(x(t- 2T))) ;::: z(t) + g(z(t- T) + g(z(t- 2T) + · · · + g(z(t- nT)) ... ))

x(t) = z(t)

= z(t) + g(z(t- T)

(3.9.5) fort;::: T

z'(t)

+ nT.

Substituting (3.9.5) into (3.9.1) we have

+ h(t, z(t- u) + g(z(t- u- T)

+ g(z(t- u- 2T) + · · · + g(z(t- u- nT)) ... ))) :50 (3.9.6) for t ;::: u + T + nT, which implies that (3.9.2) has an eventually positive solution. This is a contradiction. A parallel argument is true if we assume that Eq. (3.9.1) has an eventually negative solution. 0 Theorem 3.9.2. Assume that (i) g is nondecreasing, xg(x);::: 0 and there exists ad> 0 such that g(d) < d, (ii) his nondecreasing in x, h(t,x)x;::: 0 and

r= lh(t, c)idt <

lto

00

for any c =I= 0.

(3.9.7)

Then Eq. (3.9.1) has an eventually positive solution.

Proof: From (i) there exists a

/3

> 0 such that

/3 + g(d)


194

Chapter 9

exists a T

~ t0

such that

£

00

h(t, d)dt

~ d- g( d) -

{j.

(3.9.8)

Define a sequence {wA:(t)} on [T- m,oo) as follows: w1(t) = {j, and fork= 1,2, ... {j + g(wk(t-

WA:+l(t) = {

+ J1

00

t

~

T- m,

r))

h(s, wk(s- u))ds, t

WA:+l(T),

~T

(3.9.9)

T-m~t~T.

We see that {j

1 +1

~ w2(t) ~ {j + g({j) + ~ {j + g(d)

00

00

h(s,{j)ds

~ d,

t

~ T.

.. · ~ d,

t

~ T.

h(s,d)ds

By induction we get {j = w1(t) < w2(t)

~

.. · ~ WA:(t)

Thus }im WA:(t) = w(t) exists and {j o;-00 theorem, from (3.9.9) we have w(t)

~

= {j + g( w(t- r)) +

w(t)

1

00

~

~d.

By the Lebesgue's convergence

h(s, w(s- u))ds,

which implies that w(t) is a positive solution of Eq. (3.9.1).

t

~ T, D

The linear equation (x(t)- px(t-

r))' + q(t)x(t- u) = 0

(3.9.10)

is a special case ofEq. (3.9.1). From the above we have the following conclusions.

195

Oscillation of First Order Neutral Equations

Corollary 3.9.1. Ifp E (0, 1), u ~ 0, and q E C([to,oo),.R.r) and there exists a positive integer n such that every solution of n

y1(t)

+ q(t) LPiy(t- u- jr) =

0

(3.9.11)

j=O

is oscillatory. Then every solution of Eq. (3.9.10) is oscillatory. Corollary 3.9.2. In Eq. (3.9.10), p E (0, 1), u ~ 0 and q E C([to, oo ), R+ ), q(t)dt < oo. Then Eq. (3.9.10) has an eventually positive solution.

It";

We now consider the nonlinear neutral differential equation

(x(t)- px"'(t- r)) 1 + q(t)xP(t- u) = 0, where p,

T

E (0, oo ),

u E [0, oo ), a and

t ~to,

(3.9.12)

f3 are quotients of odd positive integers,

q E C([to,oo),R+)·

Theorem 3.9.3. if and only if

If a E (0, 1), then every solution of Eq. (3.9.12) is oscillatory

1

00

q(t)dt =

00.

(3.9.13)

to

It";

Proof: Necessity. Assume q(t)dt < oo. Then it is not difficult to find that all conditions of Theorem 3.9.2 hold and hence Eq. (3.9.12) has an eventually positive solution. Sufficiency. Assume (3.9.13) holds. We shall show that all assumptions of Theorem 3.9.1 hold. In fact, it is easy to see that (i) and (ii) in Theorem 3.9.1 are satisfied for Eq. (3.9.12). Now check (iii) in Theorem 3.9.1. For a E (0, 1) there exists a positive integer n such that an f3 < 1. We show that the delay differential inequality

y1(t)+q(t)[y(t-u)+p(y(t-u-r)+ p(· · +py"'(t-u-nr))"' ... )"'].8 ~ 0, (3.9.14) has no eventually positive solutions. Otherwise, let y(t) be an eventually positive

Chapter 3

196

solution of (3.9.14). By the decreasing nature of y we have

Thus

Integrating the above inequality from t to oo we have

contradicting the positivity of y. Similarly, we can show that

y'(t)

+ q(t)[y(t- a)+ p(y(t- a- r) + p(· · · + py"'(t- a- nr))"' ... )"'J.B ;: :_ 0 0

has no eventually negative solutions.

Remark 3.9.1. If a= 1 and p = 1, then (3.9.13) is sufficient but not necessary for every solution of Eq. (3.9.12) to be oscillatory. Thus condition a E (0, 1) is important for the validity of Theorem 3.9.3.

In a similar way, we can prove the following result. Theorem 3.9.4. If a= 1 and p, /3 E (0, 1), then every solution of Eq. (3.9.12) is oscillatory if and only if (3.9.13) holds. Remark 3.9.1. /3 E (0, 1) is essential for the validity of Theorem 3.9.4. In fact, Theorem 3.9.4 may not be true if /3 = 1. We see this fact from the linear equation with the constant parameters of the form

(x(t)- p(t-

r))' + qx(t- a)= 0

(3.9.15)

where p E (0,1), q,r E (O,oo), a E (O,oo). (3.9.13) holds for (3.9.15) but Eq. (3.9.15) may have positive solutions if p, q, r, a satisfy some conditions, see Section 3.1.

197

Oscillation of First Order Neutral Equations Example 3.9.1. Consider

(x(t)- x 1 13 (t

-1r))' + (1 + 3sin2 t)x 113 (t- f)= 0, t 2::

?r.

(3.9.16)

It is easy to see that the conditions of Theorem 3.9.3 are satisfied. According to Theorem 3.9.3 every solution of Eq. (3.9.16) oscillates. In fact, x(t) = sin 3 t is such a solution. Example 3.9.2. Consider

( x(t)- 3xlf3(t- 1))'

+ (2t- 1)(2t2- 2t + 1)

x(t) = 0.

t(t-1) 5

(3.9.17)

For this equation, (3.9.13) is not satisfied. According to Theorem 3.9.3, Eq. (3.9.17) has nonoscillatory solutions. In fact, x( t) = (1- 3 is such a solution of (3.9.17).

t)

3.10. Forced Equations We consider the forced nonlinear neutral equation (

m

x(t)- 8p;(t)x(h;(t))

)'

n + ~/j(t,x(gj(t)))

= Q(t),

(3.10.1)

where p;, h;, gj, Q E C([to, oo ), R), lim h;(t) = oo, lim gi(t) = oo t-oo t-oo i E Im = {1,2, ... ,m}, j E In= {1,2, ... ,n}. In the following we will present some results for the existence of nonoscillatory solutions of (3.10.1). Firstly we give a result for the case that p;, i = 1, ... , m, do not change signs. Theorem 3.10.1.

Assume that

(i) 1/i(t,x)l $ 1/i(t,y)l as lxl < where 0

IYI, and

for each closed interval I= [c,c],

< c < c, we have

lfi(t,x)- fi(t,y)l $ Lj(t)lx-

Yl when x,y E I,

where Lj(t) E C([to,oo),R+) are dependent on the interval I and j E In;

f 1"; Li(t)dt < oo,

Chapter 9

198

(ii)

I:: IQ(t)ldt <

OOj

(iii) there exists an r E (0, 1) such that either m

(iiih p;(t) ~ 0,

i E Im, and L:p;(t) $ 1- r,

t

~to or

i=l m

i E Im, and- L:P;(t) $ 1- r,

(iii)2 p;(t) $ 0,

t

~to;

i=l n

(iv)

L: f 1

00

j=l

1/j(t, d)ldt < oo for some d "I 0.

0

Then Eq. (3.10.1) has a bounded nonoscillatory solution x(t) with liminflx(t)l t-+oo

> 0.

Proof: We denote by BC the space of all bounded continuous functions defined on [to, oo). Define a distance in BC by l!x- Yl! = suplx(t)- y(t)l, for x,y E t>to

BC. Let U:::: {x E BC: c $ x(t) $ ldl, t ~to} .;here 0 < c
(Tx)(t)

=

+ L:~ 1 p;(t)x(h;(t)) + Ej= 1 j,

{

/j(s,x(gj(s)))ds-

00

(Tx)(T),

t0 $ t

j,

00

Q(s)ds,

t ~ T,

(3.10.2)

< T,

where c1 and T ~ t 0 satisfy the following conditions: when (iiih holds, c < c1 < rldl and Tis sufficiently large such that h;(t) ~ to, i E Im, gj(t) ~ to, j E Im, fort~T,

loo

n

loo

"2:,},

ifi(s,d)lds+ j,

T

T

j=l

IQ(s)jds$min{ct- c,rldl-ct},

(3.10.3)

and

L hoo Lj(s)ds $ ~; n

j=l

when (iiih holds, c + (1- r)ldl < n

roo

roo

(3.10.4)

T Ct

<

He+ (2- r)jdl),

(3.10.4) holds, and

~Jr lfi(s,d)lds+ Jr IQ(s)lds$ct-c-(1 -r)ldl.

(3.10.5)

199

Oscillation of First Order Neutral Equations

It is easy to show that c :5 (Tx)(t) :5 ldl fort~ to, i.e., Tf>. c f!. We shall show that Tis a contraction on n. In fact, for x, y E f2 and t ~ T

I('Tx)(t)- (Ty)(t)l

=I ~p;(t)(x(h;(t))-

y(h;(t)))

+~!roo (fi(s,x(gj(s)))) -fi(s,y(gj(s)))dsl :5 (

~ IP;(t)i) ~~f.lx(t)- y(t)i +

t1

00

j=l

:5 (

Lj(s)ix(gi(s))- y(gi(s))ids

T

t !roo

~ lp;(t)l +

Lj(s)ds) llx- Yll

:5 ((1- r) + ~)llx- Yll

= (1- ¥)

llx- Yll·

Thus II'Tx- Tyll :5 (1- ¥) llx- Yll· By the contraction mapping theorem T has a fixed point x* inn. Obviously, x* is a bounded nonoscillatory solution of (3.10.1) and liminf lx*(t)l ~ c > 0. 0

t-oo

The following result is for the case where p;, i functions.

= 1, ... , m,

are oscillatory

Theorem 3.10.2. Assume that (i), (ii), (iii) 2 in Theorem 3.10.1 hold and xfi(t,x) ~ 0 for x =/= 0, j E In. Then condition (iv) in Theorem 3.10.1 is a necessary and sufllcient condition for Eq. (3.10.1) to have a bounded nonoscillatory solution x(t) satisfying liminflx(t)l > 0.

t-oo

Proof: Sufficiency follows from Theorem 3.10.1. We now prove the necessity.

Chapter 3

200

Without loss of generality, assume that x(t) 2: d > 0,

t 2: to. Set

m

z(t) = x(t)- 2:Pi(t)x(hi(t)). i=l If (iv) is not true, we have

z(t)- z(T)

~

£ t

n

(

Q(s)ds- ~ Jr fi(s,d)ds ..--. -oo 0

as t ..--. oo, contradicting the boundedness of x.

Example 3.10.1. Consider the equation

(x(t)

.

,

cost

+ t sm x(h(t))) + 1 + tZ x (g( t))

= Q( t)

(3.10.6)

where h, g, Q E C([O, oo ), R) satisfying t___..oo lim h(t) = oo, t-oo lim g(t) = oo and

ft"; IQ(t)idt

< oo. By Theorem 3.10.2, (3.10.6) has a bounded nonoscillatory solution x(t) satisfying liminflx(t)l > 0. t-+oo

3.11.

Notes

Theorem 3.1.1 can be seen from Arino and Gyori [5], also from Gyori and Ladas [76]. The rest of Section 3.1 is from Erbe and Kong [40]. Theorem 3.2.1 is based on Jaros [87]. Theorem 3.2.2 is taken from Chuanxi, Ladas, Zhang and Zhao [25]. Theorem 3.2.3 is adopted from Zhang and Yu [211]. Theorem 3.2.4 is based on the results of Zhang and Gopalsamy (204]. Theorems 3.2.5 and 3.2.6 are taken from Zhang and Yu [214]. Theorem 3.2.9 is from Zhang, Yu and Liu. Theorem 3.2.10 is adopted from Gopalsamy, Lalli and Zhang [55], also see Yu [177]. Theorem 3.2.11 is taken from Yu, Wang and Qian [181], also see Wang [166], Kitamura and Kusano [94]. Theorem 3.2.12 is based on Ladas and Sticas [76]. Lemma 3.3.1 is from [25]. Lemma 3.3.2 is obtained by Gyori [75]. Theorem 3.3.1 to Theorem 3.3.3 are taken from Zhang and Yu [212]. Theorem 3.3.4 to 3.3.6 are adopted from Erbe and Kong [38]. Condition (3.3.54) is new. Theorem 3.3.7 and Theorem 3.3.8 are taken from Zhang and Gopalsamy [204]. The content of Section 3.4 is based on Lalli and Zhang [117] and Gopalsamy

Oscillation of First Order Neutral Equations

201

and Zhang [56]. Lemma 3.5.2 is from Gyori and Ladas [76]. Theorem 3.5.1 is adopted from Lalli and Zhang [116]. Theorem 3.5.4 is new. Theorem 3.5.5 is based on Erbe and Zhang [44]. Theorem 3.6.1 and 3.6.2 are from Jaros and Kusano [89]. Theorem 3.6.3 is adopted from Wang [165]. Theorem 3.7.1 is from Yu and Wang [179]. Theorem 3.7.2 is a modification ofYu [176]. Theorem 3.7.3 is by Gopalsamy and Zhang [203]. Theorem 3.8.1 is taken from Chuanxi and Ladas [22]. The content of Section 3.9 is taken from Zhang and Yu [211]. Theorem 3.10.1 is adopted from Lu Wudu [132].

4 Oscillation and Nonoscillation of Second Order Differential Equations with Deviating Arguments

4.0.

Introduction

In Section 4.1 we introduce the linearized oscillation results for a certain class of second order nonlinear delay differential equations. In Section 4.2 we present results on the existence of oscillatory solutions of the delay differential equation. In Section 4.3 we introduce Sturm comparison theorems which are very useful for the oscillation theory as well as the boundary value problems. In Section 4.4 we establish some oscillation criteria for second order neutral differential equations. By way of these criteria the oscillation problem for neutral differential equations can be reduced to the same problem for the corresponding delay equations and sometimes to the corresponding ordinary differential equation. In Section 4.5 we introduce the classification of nonoscillatory solutions for second order nonlinear neutral differential equations, various existence results of nonoscillatory solutions of different types are given. In Section 4.6, we present several bounded oscillation 202

Second Order

Equation~

with Deviating

Argument~

203

criteria for the second order neutral differential equation of unstable type. Section 4. 7 deals with the forced oscillation. In Section 4.8 equations with nonlinear neutral terms are investigated. In the last section, advanced type equations are investigated, also a relation between the oscillation of equation of the advanced type and the oscillation of the corresponding ordinary differential equation is established.

4.1. Linearized Oscillation We consider the second order nonlinear delay differential equations

x"(t) + A(t)x'(t) + B(t)f(x(t- r)) = 0,

t~0

(4.1.1)

x"(t)- A(t)x'(t) + B(t)f(x(t- r)) = 0,

t~0

(4.1.2)

A,B E C(R+,(O,oo)), f E C(R,R) B(t) =be (0, oo). lim A(t) =a E (0, oo), tlim ...... oo t-+oo

(4.1.3)

and

where

r > 0,

The sunflowing equation

. y ( t - r) = 0, y "( t ) + -a y '( t ) + -b sm r

r

t~O

( 4.1.4)

is a special case of (4.1.1). Under some assumptions the following equations are called the linearized limiting equations of (4.1.1) and (4.1.2), respectively:

x"(t) + ax'(t) + bx(t- r) = 0,

t~0

(4.1.5)

= 0,

t~0

(4.1.6)

and

x"(t)- ax'(t) + bx(t- r) respectively.

We shall establish the relations between the oscillation of (4.1.1), (4.1.2) and that of their linearized limiting equations, ( 4.1.5), ( 4.1.6), respectively.

204

Chapter 4

Theorem 4.1.1. Assume that (i) uf(u) > 0 for u -:f. 0, lui~ H, where HE (0, oo) and lim u~o (ii) The characteristic equation of Eq. (4.1.5)

J(>.)

= >.. 2 +a>.+ be-).r = 0

f(u) u

= 1, (4.1.7)

has no negative roots. Then every solution of Eq. (4.1.1) whose graph lies eventually in the strip R+ x [- H, H] is oscillatory. Theorem 4.1.2. Assume that (i) uf(u) > 0 as u "I 0, lim fsu) = 1,

JuJ--oo

(ii) The characteristic equation of ( 4.1.6)

(4.1.8)

has no positive roots. Then every solution of Eq. (4.1.2) is oscillatory. The methods of the proofs of Theorems 4.1.1 and 4.1.2 are similar. So we only show the proof of Theorem 4.1.1 in the following. We first prove some Lemmas which will be used in the proofs of Theorems 4.1.1 and 4.1.2. Lemma 4.1.1. Assume that a, b, r E (0, oo) and every solution of Eq. (4.1.5) is oscillatory. Then there exists an e E (0, b) sucl1 that every solution of the equation

z"(t) +(a+ e)z'(t) + (b- e)z(t- r) = 0

(4.1.9)

is oscillatory also.

Proof: From Theorem 6.0.1 we will see that ( 4.1.5) is oscillatory is equivalent to the statement (4.1.7) has no real roots. It is easy to see that>. ;:::: 0 does not satisfy (4.1. 7). Therefore, is oscillatory is equivalent to ( 4.1. 7) having no negative roots. Since f(O) = b > 0, f(-oo) = oo, and (4.1.5) is oscillatory, we have m =min{!(>.):>. ~ 0}

> 0,

205

Second Order Equationa with Deviating Argumenta

and hence J(>.) = ).. 2

+a>.+ be->.r 2:: m for >. ~ 0.

From the fact that

we know that there exists a

>.o < 0 such that for >. ~ >.o

Set

Then for

>.

~

>. 0

For>. E (>.o,O] >. 2 +(a+ e)>.+ (b- e)e->.r

= J(>.)- e(e->.r- >.)

Therefore, >. 2 +(a+ e)>.+ (b- e)e->.r = 0 has no negative roots. Consequently every solution of (4.1.9) is oscillatory. Lemma 4.1.2. Assume that A, BE C(~,(O,oo)), f E C(R,R), uf(u) > 0 as u -:f:. 0 and lui~ H where HE (O,oo), and f is nondecreasing in [-H,H]. If y(t)2::

1 1: 00

B(u)f(y(u-r))exp(

-1

8

A(v)dv)duds,

t2::T (4.1.10)

206

Chapter

bas a positive solution y(t) : [T- r, oo)

z(t) =

lX> lr"

--+

4

(0, H], then

B(u)f(z(u -r))exp(

-1·

(4.1.11)

A(v)dv)duds

bas a positive solution z(t) on [T- r,oo) and 0 < z(t) ~ y(t).

(4.1.12)

Proof: Let C([T - r, oo ), R+) be the space of all continuous and nonnegative functions on [T - r, oo ). Define a set X by

X={wEC([T-r,oo),R+):

O~w(t)~1,

t~T-r}.

(4.1.13)

Define an operator T on X by ylt)

(Tw)(t) =

{

fo00 J;B(u)f(y(u- r)w(u- r)) X

exp(-

~ (Tw)(T)

J: A(v)dv)duds,

t

+ (1- ~),

It is easy to see that TX C X, (Twt)(t) ~ (Tw 2 )(t).

~

(4.1.14)

T

T- r

~

t

~

T.

Tis continuous, and w 1 (t) ~ w 2 (t) implies that

Define a sequence of functions {wk(t)}, k

=1, wk(t) = (Twk-d(t),

=0, 1, 2, ...

as follows:

w 0 (t)

k = 1,2, ....

(4.1.15)

By induction 0 ~ Wk(t) ~ Wk-t(t) ~ · · · ~ wo(t) = 1, Then the limit w(t)

= k-+oo lim wk(t),

t ~ T- r.

t ~ T- r exists and 0 ~ w(t) ~ 1,

t ~

207

Second Order Equations with Deviating Arguments T- r. By the Lebesgue's monotone convergence theorem

yfu f:X' J;B(u)f(y(u- r)w(u- r)) w(t) =

{

t ~T

exp(- J: A(v)dv)duds, ~w(T)+(1-~),

(4.1.16)

T-r5:t5:T.

w(t) is continuous on [T- r, oo) and w(t) > 0 fort E [T- r, T). Then w(t) > 0 for t ~ T - r. Set (4.1.17)

z(t) = y(t)w(t). Then z(t) is a positive solution of (4.1.11) and satisfies (4.1.12).

0

Proof of Theorem 4.1.1: Assume the contrary. Let x(t) be an eventually positive solution of (4.1.1) with 0 < x(t) 5: H, t ~ T. Set

u(t) = x'(t)exp(1t A(s)ds). Then

u'(t) = -B(t)f(x(t- r))exp(1t A(s)ds) < 0,

t

~ T1 = T + r.

(4.1.18)

There are two possibilities for u :

(i) u(t) > 0,

t

~

Tb (ii) there exists T2 ~ T1 such that u(t)

< 0 fort~ T2.

For (i), x'(t) > 0 fort~ T1 . Hence

x"(t) < -B(t)f(x(t- r)) 5: -CB(t),

t ~ T1 + r

where 0 < C =min {f(x): x(TI) 5: x 5: H}, which implies that lim x(t) = -oo, t-oo a contradiction. For the case (ii), x'(t) < 0 fort~ T2. Then lim x(t) = i exists. We show t-oo that i = 0. In fact, if f > 0

B(t)f(x(t-

r))-+ bf(i) > 0,

as

t-+ oo,

Chapter 4

208 and hence there exists a Ta ;::: T2 such that

B(t)f(x(t- r)) >

bf~£)

and

A(t):::; a+ 1,

t;::: Ta.

for

From ( 4.1.1) we obtain

x"(t) +(a+ 1)x'(t) + ~ f(£):::; 0,

t;::: Ta

which implies that

x'(t) +(a+ 1)x(t)--> -oo,

as

t--> oo,

and hence x'(t) --> -oo, as t --> oo. This is impossible. So lim x(t) = 0. It is t .... oo

easy to see that lim x'(t) = 0 also. t .... oo

From condition (ii) and Lemma 4.1.1 there exists an c E (0, b) such that

>. 2 +(a+ c)>.+ (b- c)e-.\r > 0 for

>. < 0.

(4.1.19)

From (4.1.3) and condition (i) there exists a T4 such that

A(t) < a+t:,

and

B(t)

f(x(t- r)) ( ) > b-t:, xt-r

for

t;::: T4.

Substituting them into (4.1.1) we obtain

x"(t) +(a+ t:)x'(t) + (b- t:)x(t- r):::; 0,

t;::: T4 ,

and hence

which leads to the integral inequality

By Lemma 4.1.2, the integral equation

z( t)

= (b-

c)/,""£"

z( u- r )e(a+~)(u-s) duds,

t ;::: T4

Second Order Equationil with Deviating Argumentil

has a positive solution and 0 < z(t)

~

209

x(t). It is easy to see that

z"(t) +(a+ c)z'(t) + (b- c)z(t- r) = 0 which contradicts (4.1.9). Similarly we can prove that (4.1.1) has no eventually D negative solution with 0 > x(t);:::: -H. The following results are about the existence of nonoscillatory solutions, where condition (4.1.3) is no longer required. Theorem 4.1.3. Assume that (i) there exist a> 0, b > 0 such that A(t);:::: a, B(t) ~ b, t;:::: 0; (ii) there exists an H > 0 such that uf(u) > 0, as u E (O,H], f(u) ~ u for u E [0, H], and f is nondecreasing on [0, H]; (iii) The characteristic equation (4.1. 7) has a real root. Then Eq. (4.1.1) has an eventually positive solution x(t) lying in the strip R+ x (0, H] eventually. Theorem 4.1.4. Assume that (i) uf(u) > 0 as u :f: 0; (ii) there exists an M > 0 such that f(u) ~ u for u;:::: M (iii) there exist a > 0, b > 0 such that A(t) ;:::: a, B(t) ~ b, non decreasing on [0, oo ); (iv) (4.1.8) has a real root. Then (4.1.2) has an eventually positive solution.

t ;:::: 0; and f

is

Proof of Theorem 4.1.3: Let Ao be a real root of (4.1. 7). Clearly, Ao < 0. Then y(t) = e>-ot is a solution of (4.1.5). Choose T so large that y(t) ~ H for t;:::: T- r. From (4.1.5)

Integrating it twice we obtain

Chapter 4

210

Then

y(t);:::

1

00

B(u)f(y(u-r))exp(

;;

-1"

A(v)dv)duds,

t;:::T.

-1"

A(v)dv)duds,

t;:::T

By Lemma 4.1.2, the equation

z(t)=

1

00 ;;

B(u)f(z(u-r))exp(

has a positive solution z(t) and 0 satisfies

z"(t)

< z(t) :5 y(t) :5 H, t;::: T-

+ A(t)z'(t) + B(t)f(z(t- r))

= 0,

r. Clearly z(t)

t;::: T,

i.e., z(t) is a positive solution o£(4.1.1) on [T,oo) and 0 < z(t)

:5 H, t;::: T.

0

The proof of Theorem 4.1.4 is similar to the above, we omit it here.

4.2. Existence of Oscillatory Solutions In the oscillation theory of the differential equations with deviating arguments the problem of the existence of oscillatory solutions is a difficult problem.

4.2.1.

We consider the second order delay differential equations of the form

x"(t)

+ a(t)x(t) + h(t,x(gi(t)), ... ,x(gn(t))) = 0

(4.2.1)

compared with the equation

y"(t) + a(t)y(t)

=0

where a E C 1 (R+, R), g; E C(R+, R), t - r 1,2, ... ,n, r > 0, and hE C(R+ X Rn,R).

(4.2.2)

< g;(t)

~

t,

t ;::: 0,

t

=

Assume that (i) there exist two constants A and B sucll tllat 0 < A :5 a(t) :5 B, t E R+, a' (t) is of a constant sign; (ii) lh(t, u1, u2, ... , un)l :5 b(t)F(Iull, !u2!, ... , lunl)

Theorem 4.2.1.

211

Second Order Equations with Deviating Arguments where bE C(R+,R+), FE C(R+,R+), and F (iii) ifO < u; :5 v;, i = 1,2, ... ,n, then

and

1

oo

o

b(t)dt < oo,

1

> 0 when u; > 0,

ds = F(s,s, ... ,s)

00

0

1,2, ... ,n;

00.

Then Eq. (4.2.1) has an oscillatory solution.

Proof: First we show some details of the behavior of the solutions of Eq. ( 4.2.2). It is well known that condition (i) implies that every solution of (4.2.2) is oscillatory. Let Y1(t),y2(t) be solutions of (4.2.2) which satisfy the initial conditions Yl(O) = 1, yHO) = 0, Y2(0) = 0, y~(O) = 1. Then

I

Yl(t) Y2(t) y~(t) yHf)

1-=

1,

t

~

0.

(4.2.3)

Suppose y(t) is a solution of (4.2.2) and define

w1(t,y,y')

=y

12

+a(t)y2

2 1 ,, ') ( =a(t)y +y. w2t,y,y

Then along any solution of Eq. ( 4.2.2)

dw~t(t)

= a'(t)y2(t),

(4.2.4)

dw2(t) = _ a'(t) [ '( )] 2 a2(t) y t · dt If a'(t)

< 0' < 0' then .!!!!!1. dt -

-

.!!!!!2. dt

0 and so > ' -

+ a(t)y~(t) :5 [y~(oW + a(O)y~(O) = a(O) [y~(t)] 2 + a(t)y~(t) :5 [y~(OW + a(O)y~(O) = 1,

[y~(tW

(4.2.5)

Chapter 4

212 and

a~t) [y~(tW + y~(t) ~ a(10) [y~(oW + y~(o) = 1, (4.2.6)

a~t) [y~(tW + y~(t) ~ a!o) [y~(0)]2 + y~(O) = If a'(t) > 0 ' then we have .!!!!!1. > 0' dt -

[YWW + a(t)y~(t) ~

.!!!!!.2. < 0 ' and so dt -

[y~(oW

+ a(O)y~(O) =

+ a(t)y~(t) ~ [y~(oW + a(O)y~(o) =

[y~(tW

a!o).

a(O) (4.2.7) 1;

and

att) [y~ (t)j2 + y~(t) att) [y~(tW

~ a(~) [y~ (0)]2 + y~(O) =

+ y~(t) ~

a(10)

1,

[y~(oW + y~(O) = a!o) .

From (4.2.5) and ( 4.2.8) we have y2(t)< 1

-

a(O)

<

'iift) -

a(O) A

[yHf)] 2 ~ a(O)

y~(t)~

-dtr ~ 1

[y~(t)] 2 ~ 1

or

yHf)~ 1,

or

[yHt)] 2 ~ a(t) ~ B,

or

olo> ' [YWW~ ;fijt ~ ~.

or

y~(t)~

Set M 1 =max {

/a(O) {1 VA ,1,VA.'

fi} , y-;;w

and

M~ = max { Va(O), ,JB, 1,

{of;}.

(4.2.8)

213

Second Order Equations with Deviating Arguments

Then

s Mb IY~ (t)l s M~

IYl(t)l

I

IY2(t)l

s Ml,

(4.2.9)

IY~(t)l

s M~.

(4.2.10)

Since y1 is oscillatory, there exists a sequence {tn} such that limn-co tn = oo yHt;) = 0, i = 1,2, ... , and Yl(t2m-1) < 0, Yl(t2m) > 0, m = 1,2, .... From (4.2.6) and (4.2.7) we have y~(t;) ~ 1 or yHf;) ~ ;f~~ ~ ~, i = 1,2, .... Let M2 =min {1,

~ }. Then (4.2.11)

Now we turn to Eq. (4.2.1). Let p = 3MM21 ' ~

Choose to {

k = (1 + p)Ml,

},

( 1

F s, ... ,s

) ds.

(4.2.12)

r so large that

00

ito

iJ?(u) = {ku

•

b(t)dt < mm

{

iJ?( k + 1) 1 } 2M? '2M1F(k + 1, ... k + 1) .

(4.2.13)

Define x(t)

= PY1(t) + Y2(t)

-it

[Yl (s )y2(t)- Y2(s )Yl (t)]h(s, x(g1 (s )), ... , x(gn(s )))ds,

to

t ~ to (4.2.14)

and x(t) = PYl(to)

+ Y2(to),

for

to-rS t S to.

(4.2.15)

It is easy to show that x( t) is a solution of Eq. (4.2.1) with the initial condition (4.2.15). Assume x(t) exists on some interval [to, T). Then we show that x(t) and x'(t) are bounded on [to, T). In fact, in view of condition (ii) and (4.2.9) we have that fortE [t0 , T)

lx(t)l S PIY1(t)l

+ IY2(t)l

Chapter

214

4

+ [' IYl(s)yz(t)- Yz(s)yl(t)ilh(s,x(gl(s)), ... ,x(gn(s)))lds lto

$

pM1 + M1 +2M?

f' b(s)F(Ix(gl(s))l, ... , lx(gn(s))l)ds

lto

= k +2M{ f' b(s)H(s)ds

(4.2.16)

}to

where H(t) = F(lx(g 1 (t))l, ... , lx(gn(t))l). By condition (iii) we have

H(t):::; F(k + 2Mf

1:

b(s)H(s)ds, ... , k + 2Mf

1:

b(s)H(s)ds).

(4.2.17)

From (4.2.17) and (4.2.12) we have

!

(l>(k+2Mf

1:

b(s)H(s)ds)

< - F(k + 2Mr $

1,:

2Mib(t)H(t) b(s)H(s)ds, ... , k +2M? J,: b(s)H(s)ds)

2Mib(t).

Integrating the above inequality from t 0 to t, and noting that (1>( k) = 0 and (4.2.13) holds, we have

q>

(k +2M{

1>(

s )H( s )ds) :::; 2Mi

1>(

s )ds

$2M? {'X) b(s)ds:::; (t>(k + 1).

lto

Since (1>( u) is increasing we have

k + 2Mi

1' to

b(s)H(s)ds $ k + 1.

(4.2.18)

From (4.2.16) and (4.2.18) we obtain lx(t)l $ k + 1 for t 0 $ t < T. On the other hand, for to-r$ t $to, lx(t)l = IPYl(to) + Y2(to)l :::; M1p + M1 = k $ k + 1.

215

Second Order Equations with Deviating Arguments

Then

lx(t)i ~ k + 1,

t0

ix(g;(t))i ~ k + 1,

-

to~

r ~

t < T,

and

t < T.

(4.2.19)

In view of (4.2.10), (4.2.13), (4.2.14) and (4.2.19) we have

lx'(t)i ~PlY~ (t)i

+

+ IY;(t)i

Jt iYI(s)y;(t)- Y2(s)y'(t)iih(s, x(gi(s)), ... ,x(gn(s)))ids to

~PM{+ M{ + 2MIM{

J'

b(s)F(ix(gi(s))i, ... , ix(gn(s))i)ds

to

~PM{+ M{ + 2M M{F(k + 1, ... , k + 1) 1

~PM{

= (P

1'

b(s)ds

to

+M'+M{

+ 2)M{,

t E [t 0 , T).

(4.2.20)

Now we claim that for the maximal existence interval of x(t), [to, T), we have T = oo. Otherwise, T < oo. Then rewrite ( 4.2.1) in the form

x'(t)

= z(t)

z'(t) = -a(t)x( t) - h (t, x(gi (t)), ... , x(gn(t))). Define Q = C([-r,O],R 2 ) and

f(t,r):=f(t,cp,'I/J)= (

llrll =

sup -r$8$0

ir(B)i, for

r = (cp,¢)

(4.2.21)

En.

Denote

¢(0) ) , -a(t)cp(O)- h(t,cp(gi(t)- t)), ... ,cp(gn(t)- t)

and u(t) = (

x(t)) z(t)

,

Ut

= u( t + 8).

Then ( 4.2.21) becomes u'(t)

= f(t,u 1 ).

(4.2.22)

Chapter

216

4

By (4.2.19) and (4.2.20) we have

llutll

$ k + 1 + (P + 2)M~,

(4.2.23)

t E [to,T)

and

I -a( t)
+ b(t)f(l
Since a(t) and b(t) are bounded on a.ny finite interval, and 11JI(O)I, l
w=

[to,T]

X

{r: llrll $ k+ 1 +(P+2)M~, r En} c [to,oo)

X

n

there is a tw, such that for tw $ t < T, (t, Ut) ~ W, which contradicts ( 4.2.23). It follows that T = oo. Finally, we conclude that x( t) is an oscillatory solution. By ( 4.2.9), ( 4.2.11 ), (4.2.12), ( 4.2.13) and condition (ii) we have

$ -PM2

+ M1 + 2Mf

t•m-\ b(s)f(lx(gl(s))i, ... ,(x(gn(s))l)ds

ito

$ -3Mt + M1 + 2Mf F(k + 1, ... , k + 1)

loo

b(s)ds

to

1

2

$-3M1 +MI+2M1F(k+l, ... ,k+1) 2MF(k 1 + 1, ... ' k+) 1

x(t2m) 2:: PM2- M1- 2Mf

i

t 2m

to

b(s)F(Ix(gt(s))i, ... , lx(gn(s))i)ds

217

Second Order Equations with Deviating Arguments

D

Hence x(t) is oscillatory. We consider the second order linear equations of the form n

x"(t) + a(t)x(t)

+L

b;(t)x(t- r;) = f(t).

(4.2.24)

i=l

From Theorem 4.2.1 we have the following result. Assume that 0 < A ~ a(t) ~ B, a'(t) is of a constant Corollary 4.2.1. lf(t)ldt < oo. Then Eq. (4.2.24) has an sign, J000 lb;(t)ldt < oo, i = 1, 2, and oscillatory solution.

Jt

n

= Z::: u; + 1, b(t) = max{lb1 (t)l, ... , Ibn( t)l, lf(t)l}. i=l Then F(uh ... ,un) > 0 for u; > 0, i = 1,2, ... ,n, and

Proof: Let F( u1, ... , un)

lh(t, UI, ••. , Un)l =

I~ b;(t)u;- f(t)l

~ b(t) ( ~ lu;l + 1) =

lb(t)l F(lu1l, ... , luml).

Therefore, all assumptions of Theorem 4.2.1 are satisfied and hence Eq. (4.2.24) has an oscillatory solution. Example 4.2.1. Consider

X

t X (t "( t ) + ax (t ) + -2sin -t +1

where a > 0, T > 0. Since has an oscillatory solution.

f0

00

T

)= 0

(4.2.25)

I ;J~i ldt < oo, by Corollary 3.2.1. Eq. (4.2.25)

Chapter

218

4.2.2.

4

We consider the initial value problem of the equation (4.2.26)

y"(t) = p(t)y(g(t)) where p,g E with

C([t 0 ,oo),~),

y(s)

g(t) :5 t,

= <,o(s)

for

lim g(t) = oo, and g is nondecreasing; t-+00

s E Et 0 ,

y(to) =yo, (4.2.27)

y'(to) = where E 10 ={to} U {g(t): g(t) and (4.2.27) by y(t,u,y~).

y~,

<,o E C(E,.,R)

< t 0 , t > t 0 }. We denote the solution of (4.2.26)

For a given <,o E C( E 10 , R), the solution y( t, <,o, y~) is a one-parameter family. Define

soo s-oo

S0

lim y(t) = oo} = {y: t-+oo lim y(t) = -oo} = {y: t-+oo lim y(t) = 0 = {y: t-+oo

s- = {y: y(t)

monotonically,

y(t) ¢. 0}

is oscillatory}.

and

K"" = {y~ E R: y(t,<,o(t),y~) E S 00 } K-oo =

{y~ E

R: y(t, <,o(t), y~) E

s-oo}

K 0 = {y~ E R: y(t, <,o(t), y~) E S 0 } K'"" = {y~ E R: y(t,<,o(t),y~) E S'""}

We state the following results without proof. Theorem 4.2.2. For any initial function <,o E C( E 10 , R), the sets K 00 , K-oo are nonempty and given by nonoverlapping halflines ( -oo, a) and ((3, oo) (a :5 (3), respectively. Denote F = R- (K"" U K- 00 ). If a< (3, then for every y~ E F = [a, (3] the corresponding solution is unbounded and oscillatory.

219

Second Order Equations with Deviating Arguments Let v(t) = sup {s: g(s)

Theorem 4.2.3.

lim sup t-oo

i

< t}, t;;::: to. Assume

...,(t)

(s- t)p(s)ds

< oo,

(4.2.28)

t

and lim sup [' (s- g(t))p(s)ds > 1. t-oo

(4.2.29)

}g(t)

Then the initial value problem for Eq. (4.2.26) with y(t) = cp(t) fort E Et 0 has a unique oscillatory solution. In fact, (4.2.28) implies that a = {3. (4.2.29) guarantees that the corresponding solution is oscillatory. Finding the conditions so that a < {3 is an open problem.

4.3. Sturm Comparison Theorems We consider the second order linear delay equations of the form (p(t)x'(t))'

+

t

q;(t)x(r;(t))

+

lt

k(s, t)x(s)ds = 0,

t E [a, b)

(4.3.1)

r(a)

i=l

and (p(t)y'(t))'

+ tQ;(t)y(r;(t)) +

1;a) K(s,t)y(s)ds

= 0,

t E [a, b)

(4.3.2)

where a < b :5 oo, p E C 1 ([a, b), (0, oo)), q;, Q;, K(s, t), k(s, t) are continuous functions over [a, b) and {(s,t): s :5 t, a :5 t < b}, respectively. Also, r;(t) :5 t, where r; is continuous, i = 1, ... , n, and r(t) = min{r;(s), s;;::: t, i = 1, ... ,n}.

(4.3.3)

For a given initial function cp E C[r(a), a], there exists a unique solution x(t) to Eq. (4.3.1) in [a, b) with x(t)

= cp(t)

t E [r(a),a],

and

x'(a+)

= cp'(a-).

(4.3.4)

Chapter 4

220

Let .,P(t) E C[r(a),a] be an initial function for Eq. (4.3.2), and y(t) be the corresponding solution to Eq. ( 4.3.2) with the initial condition given by .,P( t). For Eqs. (4.3.1) and (4.3.2) assume the following comparison conditions hold. (At) Q;(t) ~ lq;(t)l, ~

(A2) K(s,t)

= l, ... ,n,

i

s,t E [a, b),

lk(s,t)l, ~

(A3) .,P(t)f.,p(a)

t E [r(a),a].

l
If we assume that all of the q;(t) and k(s, t) are nonnegative, then we can relax (A 3 ) to get the following conditions:

(BI) Q;(t)

~

(B2) K(s,t)

q;(t) ~

~

~

k(s,t)

(B3) .,P(t)f.,p(a)

~

i = 1, ... , n,

0,

0,

s, t E [a, b),

.,P(t)j.,p(a)

0,

Likewise, if we assume that
(C 1 )Q;(t)~O,

(C2) K(s,t)

~ 0,

K(s,t)

~

~ ~


t E [r(a),a].

0 we can relax (At) and (A 2 ) to get the

i=l, ... ,n,

k(s,t),

s,t E [a, b),

(C 3 ) .,P(t)f.,p(a) ~
(DI) Q;(t)

~

(D2) K(s,t)

q;(t) ~

~

0,

k(s,t)

~

i = 1, ... , n,

0,

s,t E [a, b),

(D3) .,P(a) =f 0 and .,P(t) does not change sign in [r(a),a],
= 0,

1

(4.3.5) Conditions (DI)-(D 3 ) imply conditions (B 1 )-(B 3 ). In fact, from (D 3 ), we see that x'(t)/x(t)-+ oo as t-+ a+. A new initial point a- can be chosen so that

221

Second Order Equations with Deviating Arguments

with the shifting of the initial interval to [r(a-),a-], the conditions (BI)-(B3) now hold.

Assume that one of the sets of comparison conditions (AI/BdCI/D 1 ) - (A3/B3/C3/D3) holds, and that the solution y(t) ofEq. (4.3.2) does not vanish in [a, b). Then, for all t E (a, b)

Theorem 4.3.1.

y'(t)

x'(t)

y(t) y(a)

x(t)

-<-y(t) - x(t)

(4.3.6)

and

-. -< - x(a)

(4.3.7)

As a consequence, x( t) does not vanish in (a, b). Proof: Let r(t) = -p(t)x'(t)/x(t), R(t) = -p(t)y'(t)/y(t), then, from (4.3.1) and (4.3.2), we obtain the following Riccati equations:

~

r 2 (t) 1 r (t) = - ( ) pt

+ L.Jq;(t) i=l

x(r;(t)) () X t

+

1' r(a)

x(s) k(s,t) -() ds X t

(4.3.8)

and

R'(t) = R 2 (t) p(t)

+

t

i=I

Q;(t) y( r;(t)) y(t)

+ [' }r(a)

K(s, t) y(s) ds y(t)

(4.3.9)

According to the first two comparison conditions, R(t) satisfies the differential inequality

R'(t);::: R 2 (t) p(t)

+ :tlq;(t)ly(r;(t)) + [' i=I

y(t)

Jr(a)

ik(s,t)i y(s)ds. y(t)

(4.3.10)

From (4.3.5) (4.3.11) To prove (4.3.6) it suffices to show that R( t) ;::: r(t), for all t E (a, b). To this end, we first assume that (4.3.10) and (4.3.11) hold strictly. Otherwise, we can first

Chapter

222

4

establish the required result with y(t) replaced by Ye(t) (c > 0), which satisfies the modified delay equation

(p(t)y~(t))' + tQ;(t)ye(r;(t)) + i=l

1'

K(s,t)ye(s)ds +e = 0

(4.3.12)

r(a)

and the initial condition (4.3.13) From {4.3.12), we get the modified Riccati inequality

R~(t) > R~(t) + p(t)

t

i=l

lq;(t)l Ye(r;(t))

Ye(t)

+

1'

r(a)

lk(s,t)IYe(s) ds. Ye(t)

(4.3.14)

The general result then follows from the specialized result by letting e-+ 0. We shall show that under the hypothesis of strict inequalities, the conclusion (4.3.6) actually holds with a strict inequality sign. Suppose the contrary, let q >a be the first point at which R(q) = r(q). Then for all t E [a,q), R(t) > r(t). It follows that

x'(t) y'(t) y(t) < x(t)'

forall

tE[a,q).

(4.3.15)

Integrating it over the interval [s,t] C [a,q], we obtain

y(s)>lx(s)l x(t) y(t)

(4.3.16)

for all s < t E [a, o"]. Combining this with the third comparison condition, we see that (4.3.16) actually holds for all s < t E [r(a),q]. Hence the right hand side of (4.3.10) is not smaller than that of (4.3.8) for all t E [a, (7]. The various sign requirements in the comparison conditions are imposed to ensure that the product of the coefficient and the fraction in each of the terms of ( 4.3.8) and (4.3.9) do inherit the desirable comparison property. We have now concluded that R'(t) > r'(t) for all t E (a,q). In particular, R'(q) > r'(u). This contradicts the assumption that R(u) = r(u) and R(t) > r(t) fort < u and completes the 0 proof of the theorem.

223

Second Order Equations with Deviating Arguments Associate ( 4.3.2) with the delay equation n

(p(t)z'(t))'

+L

['

Q;(t)z(r;(t))

+ }.

K(s, t)z(s)ds = 0

(4.3.17)

T(a)

i=l

where

'f;(t)

~

r;(t)

~

t,

i

= 1, ... ,n.

(4.3.18)

We assume that the initial condition ( 4.3.19)

holds. Furthermore, assume that

Q;(t),

~

K(s, t)

(4.3.20)

0

and

.,P'(t)
or

~0

m

[r(a),a].

( 4.3.21)

Theorem 4.3.2. Let y(t) and z(t) be, respectively, positive solutions of (4.3.2) and (4.3.17) in [a, b), with the same initial value given by .,P(t). Suppose that (4.3.18) and (4.3.20) hold and that .,P'(t) ~ 0 in [r(a), a]. Then for all t E [a, b)

z'(t) < y'(t) z(t) - y(t)

(4.3.22)

~

(4.3.23)

and

z(t)

y(t).

On the other hand, if.,P'(t) ~ 0 in [r(a),a] and both z'(t) and y'(t) are nonnegative in [a, b], then the reverse inequalities hold in ( 4.3.22) and ( 4.3.23). Proof: The function S(t)

S'(t) = S 2 (t) p(t)

+

t

= -p(t)z'(t)/ z(t) satisfies the Riccati equation

i=l

Q;(t) z('f;(t)) z(t)

+ [' }T(a)

K(s, t) z(s) ds. z(t)

(4.3.24)

224

Chapter ~

Since z(t) is decreasing (increasing), z(r;(t)) the Riccati inequality

S'(t)

($)z(r;(t)). Hence S(t) satisfies

~ ($)S2(t) + :tQ;(t)z(r;(t)) + p(t)

z(t)

i=l

4

1t

r(a)

K(s,t)z(s) ds. z(t)

The comparison between S(t) and R(t) can then be carried out as is done in the proof of Theorem 4.3.1 for R(t) and r(t). Theorem 4.3.2 asserts that for a decreasing solution, a "shorter" memory slows down oscillation, whereas for an increasing solution, it speeds up oscillation (in the sense that the solution reaches its maximum or rebounces faster, and not that the solution becomes zero faster). We now apply Theorem 4.3.2 to a oscillation problem. We consider delay equations of the form

(p(t)x'(t))' +

t

q;(t)x(r;(t))

+

i=l

lt

k(s, t)x(s)ds = 0,

t

~a

(4.3.25)

r(t)

with the assumptions that q;(t)~O,

k(s,t)~O

(4.3.26)

and

r(t)= min {r;(t): i=O,l, ... ,n}-+oo, t-+oo

t-+oo.

(4.3.27)

We compare ( 4.3.25) with another delay equation

(p(t)y'(t))'

+

t~1

Q;(t)y(r;(t))

+

t

hoo

K(s, t)ds

= 0,

t

~a.

(4.3.28)

The following results can be easily deduced from Theorems 4.3.1 and 4.3.2. Theorem 4.3.3.

Suppose that for sufficiently large s, t

Q;(t)

~

q;(t)

~

0,

K(s,t)~k(s,t)~O,

r;(t)

~

r;(t),

i = l, ... ,n, s
i = 0, ... , n.

(4.3.29) (4.3.30) ( 4.3.31)

225

Second Order Equations with Deviating Arguments

H Eq. (4.3.25) is oscillatory, so is Eq. (4.3.28). Theorem 4.3.4. Suppose that p(t) ordinary differential equation

= 1 and (4.3.26),

y"(t) + 8(t)y(t)

(4.3.27) hold, and the

=0

(4.3.32)

is oscillatory. Then so is the delay equation (4.3.25), where

8(t)=![tqi(t)ri(t)+ t i=l

lt

sk(s,t)dt].

(4.3.33)

r(t)

Proof: Suppose the contrary, and ( 4.3.25) has an eventually positive solution x(t). It is easy to see that x'(t) > 0 eventually. Suppose that lim x'(t) = p. > 0 and hence x(t) "'p.t, t-+ oo. From ( 4.3.25) t-+oo

x"(t) "'p.t8(t). The convergence of x'(t) implies the integrability of x"(t). Hence

1

00

t8(t)dt <

(4.3.34)

00.

But is is well known that ( 4.3.34) implies the nonoscillation of (4.3.32) which contradicts our hypothesis. Therefore, lim x'(t) = 0. Let us take a point on the t-+oo

solution curve, ( t1, x( t1)). Denote by L the straight line joining this point and the origin (0, 0). Concavity implies that L may intersect the solution curve in at most two points. In the case that there are two points of intersection, t 1 and t2, without loss of generality we may assume t 1 < h· The part of the straight line between these two points lies below the curve. Let t 3 ~ t 2 be so large that r(t) > t1 for all t > ta. For any t > ta, the line joining (0,0) and (t,x(t)) lies below L. Hence the part of this line between r(t) and t lies below the solution curve. This implies that

x(r(t))

~ r~t)

x(t),

for all

t > ta.

(4.3.35)

The case in which Lis tangent to the solution curve at t 1 can be treated as the degenerate case with t 1 = t 2 • Now suppose that (ft. x( t 1 )) is the only point of intersection. If L lies below the solution curve in the interval [a, t 1 ], then ( 4.3.35)

Chapter

226

4

actually holds for all t > a. Finally, let us note that the remaining case is void since concavity and lim x'(t) = 0 dictate that the curve must meet L again. t-oo

We rewrite ( 4.3.25) in the form

(p(t)x'(t))'

+

(t q;(t) x(r((;)) + [' t i=I

lro(t)

X

k(s, t) x((s)) ds)x(t) X t

=0

(4.3.36)

and regard it as a linear equation without delay. By ( 4.3.35) the coefficient is larger than B(t). Therefore, from the classical Sturmian theory ( 4.3.36) or equivalently ( 4.3.25) oscillates faster than ( 4.3.32), so we have a contradiction. 0

4.4. Oscillation Criteria 4.4.3.

Nonlinear case

In this section we deal with the oscillatory behavior of solutions of the neutral differential equation

(x(t)- px(t-

r))" + q(t)f(x(t- a(t))

= 0

( 4.4.1)

under the assumption p and

T

are positive numbers;

q, a E C(R+,R+), lim

t-oo

f E C(R,R),

f

(t- a(t)) = oo, a(t) > r;

is increasing and f(-x) = -f(x);

f(xy) 2: f(x)f(y) as xy > 0,

f(oo) = oo, and

f(y) --. oo or 1 as y--. 0. y The purpose of this section is to establish a relation between the oscillation problems of ( 4.4.1) and a corresponding ordinary differential equation. The following lemmas will be used to prove the main results. Lemma 4.4.1. yE

C 2 ([T,

Assume that g E C(R+, R+), g(t) 5 t and lim g(t) = oo, t-oo

oo ), R) and y(t) > 0,

y'(t) > 0,

y"(t) 50 on [T, oo).

227

Second Order Equations with Deviating Arguments Then for each k E (0, 1) there is a Tk 2: T such that

y(g(t));::: k

g~t)

t 2: Tk.

y(t),

( 4.4.2)

Proof: It suffices to consider only those t for which g(t) t > g(t) 2: T,

< t. Then we have for

y(t)- y(g(t))::; y'(y(t))(y- g(t)), by the mean value theorem and the monotone properties of y'. Hence

~ < 1 + y'(g(t)) y(g(t))

y(g(t)) -

(t- g(t))

'

t > g(t);::: T.

Also,

y(g(t)) 2: y(T) so that for any 0

+ y'(g(t)) (g(t)- T)

< k < 1 there is a Tk ;::: T

with

t _> Tk.

y(g(t)) > k (t) y'(g(t)) - g ' Hence, ~

< t + (k -1)g(t) <

y(g(t)) -

kg(t)

_t_

- kg(t) '

The proof is complete. We discuss Eq. (4.4.1) for the cases that 0 respectively. Lemma 4.4.2.

D

<

p

< 1,

p = 1, and p

> 1,

Assume that p E (0, 1) and (H) holds. If the equation

( 4.4.3)

228

Chapter

4

is oscillatory for some 0 < >. < 1, then the nonosciilatory solutions of Eq. (4.4.1) tend to zero as t -+ oo. Proof: Without loss of generality let x(t) be an eventually positive solution of ( 4.4.1) and define

z(t) = x(t)- px(t- r). From Eq. (4.4.1) we have z"(t) :50 for t;::: T. If z'(t) < 0 eventually, then lim z(t) = -oo. But z(t) < 0 eventually implies that lim x(t) = 0 which t ...... oo

t--..tx>

contradicts the fact that lim z(t) = -oo. Therefore, z'(t) > 0 fort ;::: T. There 1--+oo

are two possibilities for z(t) :

(a) z(t)>Ofort;:::T1 ;:::T, (b) z(t) < 0 fort;::: T1. For case (a), by Lemma 4.4.1, for each k E (0, 1) there is a Tk ;::: T such that

z(t- u(t));::: Since 0

k(t-u(t)) t z(t),

t;::: Tk.

< z(t) < x(t), from Eq. (4.4.1) we have z"(t) + p(t)f( k(t -tu(t)) z(t)) :50.

Using Theorem 7.4 in [86] we see that ( 4.4.3) has an eventually positive solution. This contradicts the assumption. For the case (b), as mentioned before, we lead to that lim x(t) = 0. 0 t--+oo

Theorem 4.4.1. that

1 t

limsup t--+oo

In addition to the conditions of Lemma 4.4.2, assume further

if lim J(yl = 1 (u-(t-u(t)+r))q(u)du >

{

t-cr(t)+r

p,

O,

y--+0

(4.4.4)

if lim !JJtl y--.0

Then every solution of Eq. (4.4.1) is oscillatory.

y

y

= oo.

Second Order Equation3 with Deviating Arguments

229

Proof: As in the proof of Lemma 4.4.2, it is sufficient to show that z(t) < 0 for t :;:: Tis impossible under the assumptions. Suppose that x(t) > 0, z"(t) ::; 0, z'(t) > 0, and z(t) < 0 eventually. Then

z(t- u(t)

+-r) > -px(t- u(t)).

Substituting this into Eq. (4.4.1) we have

z"(t)- q(t) f(z(t- u(t) + r)) ::; 0. p

Integrating (4.4.5) from s to t for t

11'

z'(t)- z'(s)--

p •

(4.4.5)

> s we have q(u)f(z(u- u(u) + r))du::; 0.

(4.4.6)

Integrating (4.4.6) ins from t- u(t) + r tot, we have

z'(t)(u(t)- r)- ['

dz(s)

lt-tT(t)+r

11'

(u- (t- u(t) + r)]q(u)f(z(u- u(u) + r))du::; 0.

--

t-tT(t)+r

P

Hence for t sufficiently large

z(t- u(t) + r)- z(t) 11t --

P t-17(t)+r

[u- (t- u(t) + r)]q(u)f(z(u- u(u) + r))du::; 0.

Dividing the above inequality by z(t- u( t) + r) and noting the negativity of this term, we have

1_

z(t) z(t-u(t)+r)

(4.4.7)

- pz (t - 0'1(t) + T ) We note that z(t)

lt

<

t-17(t)+r

[u- (t- u(t)

+ r)]q(u)f(z(u- u(u) + r))du:;:: 0.

0 eventually implies that lim z(t) = 0. From (4.4.7) we t-+oo

Chapter

230

have

11'

1>P

t-a(t)+r

4

f(z(u- u(u) + r)) ) du () {

[u-(t-u(t)+r)]q(u)

Z

t- U t

+T

0

which contradicts (4.4.4).

Lemma 4.4.3. Assume (H) holds and p = 1. Then the nonoseillatory solutions x(t) of Eq. (4.4.1) are bounded provided every solution of the equation

z"(t) is oscillatory, where Q(t) =

+ q(t)f(Q(t)z(t))

3;, (

(4.4.8)

= 0

t - u( t)) 2 •

Proof: Let x(t) be an eventually positive solution of Eq. (4.4.1) and z(t) = x(t)- x(t- r). Then z"(t) $ 0 fort~ to. If z'(t) < 0 fort~ to, then we have lim z(t) = -oo. Thus , ..... 00

x(t) $ x(t- r)

for all large

t

which implies that x(t) is bounded, a contradiction. Therefore z'(t) t1

(4.4.9)

> 0, t 2:

2: to.

Assume z(t) > 0, t ~ t2 2: ft. By Lemma 4.4.1, for any k E (0, 1) and i = 0, 1, 2, ... there exits T; 2: to = To such that

z(t- u(t)- ir) 2:

k(t- u(t)- ir) z(t), t

t-

u(t) 2: T;.

Since n-1

x(t- u(t))

=L

z(t- u(t)- ir)

+ x(t- u(t)- nr)

i=O

n

2: L:z(t- u(t)- ir) i=O -1

(here

L: = 0), from Eq. (4.4.1) we have i=O

z"(t) + q(t)f( ~ z(t- u(t)-

ir)) 50.

(4.4.10)

231

Second Order Equations with Deviating Arguments Using (4.4.10) we obtain

z"(t)

+ q(t)f( ~ ~(t- u(t)- ir)z(t)) s; 0,

i.e.,

z"(t) Since nr

+ q(t)f(~(n + 1)(t- u(t)- ~r)z(t)) s; 0.

s; t- u(t)- T0 < (n + 1)r, z"(t)

we have

+ q(t)J( 2~t[(t- u(t)?- Tg]z(t)) s; 0.

Choose T ;::: T0 large enough, then it follows that

Noting that z( t), z(T) are upper and lower solutions of Eq. ( 4.4.8) respectively, and using a known result in [86], we see there is a solution x(t) of Eq. (4.4.8) satisfying z(T) s; x(t) s; z(t), contradicting the fact that Eq. ( 4.4.8) is oscillatory. Assume z(t) < 0 for t ;::: t 2 ;::: t 1 • Then x(t) < x(t- r), t ;::: t 2 which D implies that x(t) is bounded. Definition 4.4.1. Let E be a subset of R+· Define

Pt(E)

= J.L{E n [0, t]} t

and

p(E) =lim sup Pt(E)

(4.4.11)

t-oo

where J.L is the Lebesgue measure. Lemma 4.4.4. Assume (H) holds and p > 1. Then the nonoscilla.tory solutions x(t) of Eq. (4.4.1) satisfy x(t) < px(t- r) eventually provided the following conditions bold:

i)

+ q(t)f(R(t, >-.)z(t))

= 0

< ), < 1, where R( t, >-.)

=i

z"(t)

is oscillatory for all 0

(4.4.12) p ·-~<<);

Chapter 4

232

lim sup p-;tfr J:(t- u)q(u)f(u- u(u)

ii)

+ r)du > 0

(4.4.13)

t-+oo

tf/.E

holds for some p 1

> p and any set E with p(E) = 0.

Proof: First we claim that if a set E C R+ and p(E) = p > 0, then for any t 0 E R+ and integer n, there exists aTE [t 0 , t 0 +r) such that the set {T+ ir }~ 1 intersects E at least n times. If not, there exists a to E R+ and an integer N, such that {T + ir }i: 1 intersects Eat most N times for any T E [to, to+ r). This implies that p{E} < oo. But p(E) = p > 0 means there exists tn-+ oo such that Ptn(E) ~ ~ > 0. Thus

and this is impossible. Let x(t) be an eventually positive solution of Eq. (4.3.1) and set z(t) = x(t)- px(t- r). Then z"(t) $0 eventually. There are three possibilities: (a) z'(t)

> 0, z(t) > 0,

(b) z'(t) < 0, z(t) < 0,

(c) z'(t)

> 0, z(t) < 0

eventually. (a) Assume z'(t) > 0, z(t) > 0, t ~ t 0 ~ 0. Then (4.4.10) holds and for any t E Rr0 = {t; t + u(t) ~To}, there exists a positive integer n such that T0 $ t- u(t)- nr
x (t- u(t))

=L

piz(t- u(t)- ir)

i=O n

~

.l:>iz(t- u(t)- ir), i=O

from Eq. (4.4.1) we have

+ pnx(t- u(t)- nr)

233

Second Order Equations with Deviating Arguments

In view of (4.4.10) we have

z"(t) + q(t)f (

tk ~p'(t- u(t)- ir)z(t) n

.

)

50,

i.e., ) kr n n+l k z"(t) + q(t)f [ ( -(t- u(t)) p _ - - L pi z(t) 50. t i=l p 1 t ]

(4.4.14)

Since

We have k

p"+l - 1

-(t- u(t)) p -1 t

kr

n

.

--Lip' t i=l

= (p-k1)2t [(t- u(t))(p"+2- pn+l- p + 1)- r(npn+2- (n + 1)pn+l + p)] k

= (p _ 1)2 t [(t- u(t)- nr)p"+2- (t- u(t)- (n + 1)r)p"+ 1

- (t-u(t)+r)p+ (t-u(t))] k

~ (p _ 1)2t[Topn+2- Topn+l- (t- u(t)

+ r)p + (t- u(t))] (4.4.15)

for some A E (0, 1) if T0 and t are sufficiently large. Substituting (4.4.15) into (4.4.14) we have

z"(t) + q(t)f ( tA p!=..!..ill z(t) ) 50. T

Noting that z(t), z(To) are upper and lower solutions of (4.4.12) and by a known result in [86] we see there is a solution y(t) of (4.4.12) satisfying z(T0 ) 5 y(t) 5 z(t), contradicting the fact that Eq. (4.4.12) is oscillatory for all 0
234

Chapter

< 0, z(t) < 0, t

(b) Assume z'(t) some f.> 0.

~ t0 ~

4

0. Then z(t) $ -f.t, t ~to for

We claim z(t) ~ -p~fr essentially, where p1 {t: z(t) < -p~fr}. Then p(E) = 0.

> p is arbitrary, i.e., if E =

Otherwise, p(E) = p > 0. As in the beginning of the proof, for any n, there exists a T 1 E (t 1 ,t 0 + r) such that the set {T1 + ir}~ 1 intersects Eat least n times. Assume M = max x(t). Then if n is sufficiently large, ta~t~ta+r

contradicting that x(t) > 0 eventually. It is easy to see that condition ii) implies that

[XJ q(u)f(u- a(u) + r)du = oo.

(4.4.16)

Condition (ii) also implies that z'(t) < -fl. for all fl. > 0 eventually. For otherwise, there exists a fl. > 0 such that z'(t) ~ -fJ,, t ~ T2 • On the other hand, x(t- r) ~ -~ z(t), thus

z"(t)

+ q(t)f(- ~z(t- a(t) + r))

$0.

Integrating it from T2 to t we get

z'(t) + Noting that z(t- a(t)

£.

q(u)f(-

+ r)

~z(u- a(u) + r))du $0.

$ -f.(t- a(t)

£. q(u)f(~(u!(~) £. z'(t) +

+ r)

a(u)

q(u)f(u- a(u)

we have

+ r))du

+ r)du

$ 0,

$ -z'(t) $ fl,

(4.4.17)

235

Second Order Equations with Deviating Arguments which contradicts ( 4.4.16). Therefore, z' ( t) ( 4.4.17) there is a T,. such that

On Ec

< - J.l for all

J.l

> 0 eventually. From

n [T,., oo),

Thus

J(!!:_)p;tfr [' (t- J.L)q(u)f(u- u(u) + r)du::::; 1, P

lr~

and Ptfr

~~ (t- u)q(u)f(u- u(u) + r)du::::; !(\)'

which contradicts ( 4.4.13) since J.l (c) Assume z'(t) > 0, z(t) obvious.

( 4.4.18)

> 0 is arbitrary and f( oo) = oo. < 0, t ~ t 0 ~ 0. Then x(t) < px(t- r) is 0

Corollary 4.4.1. In addition to the assumptions of Lemma 4.4.3, further assume that <1 is a positive constant, and oo

rT+ir+or

'2:: } i=O

7

(u- T)q(u)du = oo

.

(4.4.19)

T+•r

holds for any T E J4, and 0 < a ::; r, then all nonoscillatory solutions of Eq. (4.4.1) tend to zero as t ~ oo.

Proof: If not, there exists an eventually positive solution x( t) satisfying limsup x(t) > 0, and this can only occur when z"(t)::::; 0, z'(t) > 0 and z(t) < 0, t-+oo

t ~ t 0 ~ 0, hence z'(t) ~ 0, z(t) ~ 0 as t ~ oo. If liminf x(t) > 0, then t-+oo x(t) ~a> 0, t ~ t 1 ~to. Integrating Eq. (4.4.1) twice we get z(t)+

1

00

(u-t)q(u)f(a)du
Chapter 4

236

which implies lim sup

1

00 (

u - t )q( u )du :::,; 0,

t

t-+ex>

contradicting ( 4.4.19). Thus

> 0 and liminf x(t) = 0.

lim sup x(t) t-+oo

Then we can choose tz

> t1

t-+oo

such that x(tz- u) > x(t1- u). We claim that

~to

liminf x(t 2 - u + nr) > 0.

(4.4.20)

n-+oo

In fact n

x(ti- u + nr) = L:z(tj- u + ir) + x(tj- u),

j = 1,2.

i=l

Since z(tz- u

+ ir) ~

z(t 1 - u

+ ir) fori=

liminf x(t 1 n-+oo

-

u

1, 2, ... , n, and

+ nr) ~ 0.

We have liminf x(tz- u n-+oo

Now, choose to :::,; t1

+ nr) ~ x(tz- u)- x(t1

- u) > 0.

< tz < t3 such that for any T E [tz, t 3],

x(tt- u) < x(tz- u):::,; x(T- u). From the above discussion, we see that (4.4.20) holds, i.e., there exists a p. such that x( t 2 - u + nr) ~ p. for all n. It is easy to see that for T E [t 2 , t 3 ], n

x(T-u+nr)

= Lz(T-u+ir)+x(T-u) i=l n

~ L:z(t 2 -u+ir)+x(t 2 -u) i=l

=x(tz-u+m)~p..

>0

237

Second Order Equations with Deviating Arguments From {4.4.1) we have

-z'(s) + z(t 0 )

+

J.t q(u)f(x(u- a))du ~ 0,

f\u- t 0 )q(u)f(x(u- a))du

lto

to~ s ~ t,

~ 0,

t

~to.

{4.4.21)

Hence

z(to)

+ f(J.L)

t

ita~ir (u- to)q(u)du ~ 0, t•+•r

i=O

and then

contradicting {4.4.19).

D

Corollary 4.4.2. In addition to the assumptions of Lemma 4.4.4, assume further that a is a positive constant,

1

00

(4.4.22)

(u- t)q(u)du = oo

and oo

rT+ir+or

L f(pi) } T+•r i=O

.

1

hold for any T E R+ and 0 < a (4.4.1) tend to zero as t--+ oo.

~ T.

(u- T)q(u)du

= oo

( 4.4.23)

Then all nonoscillatory solutions of Eq.

Proof: If not, similar to the proof of Corollary 4.4.1 we see that there exists an eventually positive solution x(t) satisfying lim sup x(t) t-+oo

> 0 and liminf x(t) t-+oo

= 0.

From the proof of Lemma 4.4.4 this can only occur when z"(t) ~ 0, z'(t) > 0, and z(t) < 0, t ~ t 0 . Choose t2 > t 1 ~to such that x(t2- a) > x(t 1 -a).

238

Chapter

4

Since n

x(t2- 17 + nr)

=LPn-iz(t2- 17 + ir) + p"x(t2- 17), i=l n

x(t1- 17

+ nr) =

LPn-iz(tl- 17

+ ir) + pnx(t1

- 17),

i=l

z(t2and x(t 1

-17

17

+ ir) ~ z(t1- 17 + ir),

+ nr) > 0, x(t 2 -

17

n

i

= 1,2, ... ,n,

= 0, 1, ... , we see

+ nr) ~ p"(x(t 2 - 17)- x(t 1 - 17))

= Ap".

Similar to the proof of Corollary 4.4.1, we can show that there is an interval [t2, t3] such that

x(T-

17

+ nt) ~

Apn

forTE [t2, t 3 ] and all n. From ( 4.4.21) we get

z(to)

+ f(A)

t lta+ir i=O

(u- t2)q(u)j(pn)du

:<::::;

0

t•+•r

which contradicts ( 4.4.23). We are now ready to state an oscillation criterion. Tbeorem 4.4.2. Assume (H) holds and p ~ 1. In addition to the conditions of Lemma 4.4.3 and Lemma 4.4.4 for the cases p = 1 and p > 1, respectively, we assume (4.4.4) holds. Then Eq. (4.4.1) is oscillatory. Proof: If not, let x(t) be an eventually positive solution of Eq. (4.4.1). Then we have z'(t) > 0, z(t) < 0 eventually, where z(t)

= x(t)- px(t- r).

The following is similar to the proof of Theorem 4.4.1. We omit it here.

0

239

Second Order Equation& with Deviating Arguments 4.4.4. Further discussion for the linear case Now we pay attention to the following linear equation

(x(t)

+ px(t- r))" + q(t)x(t- u) = 0.

(4.4.24)

Assume that p;::: 0 and q E C(R+,R+)· Let x(t) be an Lemma 4.4.5. of ( 4.4.24). Set solution positive eventually y(t) = x(t) + px(t- r), and

z(t)

= y(t) + py(t- r).

(4.4.25)

Then z(t) > 0, z'(t) > 0 and z"(t) $ 0 and

z"(t) + rit; z(t- u) $ 0

(4.4.26)

eventually, where q*(t) =min {q(t),q(t- r)}. Proof: Suppose x(t) > 0 fort 2:: t 0 • Then y(t) > 0 fort;::: to+ r, and y"(t) fort 2:: t1 =to +max {u, T }. Therefore y'(t) > 0 fort ;::: t 1. Then

<0

z"(t) = y"(t) + py"(t- r) $ -q*(t)[x(t- u) + px(t- r- u)] = -q*(t)y(t- u).

(4.4.27)

Similar to the above we have z(t) > 0, z'(t) > 0 and z"(t) $ 0 for t ;::: t 2 and

z"(t) $ -q*(t)y(t- u) q*(t) $ - l+p [y(t-u)+py(t-u- z)] =- q*(t) z(t- u). l+p

;:::

t1

4

Chapter

240

0

The proof is complete.

Let p > 0, q E C(R+,R+)· Assume the second order ODE

Theorem 4.4.3.

t-u

y"(t) + Aq(t) - t - y(t)

=0

(4.4.28)

is oscillatory for some A E (0, 1). Then every solution of Eq. (4.4.24) is oscillatory. Proof: Otherwise, let x(t) be an eventually positive solution of (4.4.24) and z(t) is defined by (4.4.25). Then z(t) satisfies all conditions of Lemma 4.4.1. So for every k E (0, 1), there exists a tk ~ 0 such that

t- (j

z(t- u);?: k - t - z(t),

t;?: tk.

for

(4.4.29)

By Lemma 4.4.5, ( 4.4.26) is true. Combining ( 4.4.26) and ( 4.4.29) we have z"(t)

+ k(t- u)q•(t) (1 + p)t

z(t) < 0

-

,

for

t >_ tk,

( 4.4.30)

which implies that

z"(t) + k(t- u)q•(t) z(t) (1 + p)t

=0

has a nonoscillatory solution, contradicting the assumption.

( 4.4.31)

0

From this theorem every oscillation criterion for the second order ODE ( 4.4.28) becomes an oscillation criterion for the second order NDDF ( 4.4.24). For example, we have the following Corollary. Corollary 4.4.3. Let p > 0, q E C(R+, R+)· Then every solution of Eq. (4.4.24) is oscillatory iffor some a E (0, 1)

1

00

t"'q(t)dt =co.

(4.4.32)

Second Order Equations with Deviating Arguments

241

Then we turn to the equation with variable p as follows:

(x(t)

+ p(t)x(t- r))" + q(t)x(t- u) =

Theorem 4.4.4.

0,

t 2:: to.

(4.4.33)

Assume that

(i) T > 0, 0' > 0; (ii) q E C(R+, R+) and q(t) 2:: qo > 0; (iii) p E C(R+, R) and there exist constants Pl and P2 such that Pl :5 p(t) :5 P2, and p(t) is not eventually negative. Then every solution of Eq. (4.4.33) is oscillatory. Proof: If not, let x(t) be an eventually positive solution. Set

z(t) = x(t) + p(t)x(t- r). It is not difficult to show that z(t)

( 4.4.34)

< 0 eventually, which contradicts (iii).

0

Example 4.4.1. Consider the equation

(x(t)+

(~+sint)x(t-211'))" + (~+sint)x(t-411')=0,

t;:=:O. (4.4.35)

It is easy to see that the assumptions of Theorem 4.4.4 are satisfied here. Therefore, every solution of Eq. ( 4.4.35) oscillates. For example, x( t) = l!~:i! 1 is such 2 a solution.

The following example shows that, if the hypothesis (ii) of Theorem 4.4.4 is violated, the result may be wrong.

Example 4.4.2. Consider the equation (x(t) + (t -1)_ 1 , 2 x(t -1))" +

~c

3 12 (t-

2)_ 1 , 2 x(t- 2) = 0,

t;::: 2. (4.4.36)

All assumptions of Theorem 4.4.4, except (ii) are satisfied. Note, however, that x(t) = t 112 is a nonoscillatory solution.

Chapter 4

242

4.5. Classification of Nonoscillatory Solutions We consider the second order nonlinear neutral differential equations of the form

(

m x(t)- t;p;(t)x(tr;) ) " n

+ Lfi(t,x(gji(t)), ... ,x(gjt(t))) =0,

t?_t 0 •

(4.5.1)

i=l

In this section we assume that (i)

T;

> 0, p;

E C([t 0 ,oo),R+), i = 1, ... ,m and there exists 8 E (0,1) such

m

that 'L:p;(t) $ 1- 8, t?. to; i=l

(ii) 9i• E C([to, oo ), R), lim 9is(t) = oo, j t-oo

= 1, ... , n,

s = 1, ... , I!;

(iii) fiE C([ta,oo) x R 1,R), Ytfi(t,yt, ... ,yt) > 0 for YIYi > 0, i = 1, ... ,£, j = 1, ... , n. Moreover

whenever Set

IY;i

$ lx;i and x;y; > 0, i

= 1, ... ,l, j = 1, ... , n.

m

y(t) = x(t)- LP;(t)x(t- r;).

(4.5.2)

i=l

We shall give the classification of nonoscillatory solutions of Eq. ( 4.5.1). First we show some lemmas which will be useful for the main results. Lemma 4.5.1. Let x(t) be an eventually positive (or negative) solution of Eq. (4.5.1). If lim x(t) = 0, then y(t) is eventually negative (or positive) and t-oo

lim y(t) = 0. Otherwise, y(t) is eventually positive (or negative).

t-oo

Proof: Let x(t) be an eventually positive solution of Eq. (4.5.1). From (4.5.1), y"(t) < 0 eventually. Thus y'(t) is decreasing and y'(t) > 0 or y'(t) < 0 eventually. Also y(t) > 0 or y(t) < 0 eventually. If lim x(t) = 0, from (4.5.2) we have lim y(t) t-+oo

plies that y'(t)

= 0.

t-oo

Since y(t) is monotonic, so lim y'(t) t-+oo

=

0 which im-

> 0. Therefore, y(t) < 0 eventually. If lim x(t) = 0 fail, then t-oo

243

Second Order Equations with Deviating Arguments

limsup x(t) > 0. We show that y(t) > 0 eventually. If not, then y(t) < 0 t-+oo

eventually. If x(t) is unbounded, then there exists a sequence {tn} such that lim tn = oo, x(tn) = max x(t), and lim x(tn) = oo. From (4.5.2), we have n-oo

t 0 ::;_t~tn

n--..oo

y(tn) = x(tn)- tp;(tn)x(tn- r;)

~ x(tn)(1- tp;(tn)).

i=l

(4.5.3)

•=1

Thus lim y(tn) = oo, a contradiction. If x(t) is bounded, then there exists a n-+oo sequence {tn} such that lim tn = oo, and lim x(tn) = limsup x(t). Since n-oo

n--+oo

t-oo

sequences {p;(tn)} and {x(tn- r;)} are bounded, there exist convergent subsequences. Without loss of generality we may assume that lim x(tn - r;) and n-+oo lim p;(tn), i = 1, ... , m, exist. Hence n-+oo

0

~ n-oo lim y(tn) =

lim (x(tn)-

n-oo

~ p;(tn)x(tn- r;)) L-J i=l

a contradiction again. Therefore, y(t) > 0 eventually. A similar proof can be given if x(t) < 0 eventually. D m

Lemma 4.5.2.

Assume that lim LP;(t) = p E (0, 1), and x(t) is an event-ex:> i=l

tually positive (or negative) solution of Eq. (4.5.1). If lim y(t) = a E R, then t-+oo lim x(t) = 1 ~P. If lim y(t) = oo (or - oo), then lim x(t) = oo (or - oo). t--+oo t-+oo t-oo

Proof: Let x(t) be an eventually positive solution ofEq. (4.5.1), then x(t) ~ y(t) eventually. If lim y(t) = oo, then lim x(t) = oo. Now we consider the case that t--+oo

t--+oo

lim y(t) = a E R. Thus y(t) is bounded which implies that x(t) is bounded t-+oo (see (4.5.3)). Therefore, there exists a sequence {tn} such that lim tn = oo n-+oo and limn-+oo x(tn) =lim sup x(t). As before, without loss of generality, we may t-+oo

Chapter

244

4

assume that lim p;(tn) and lim x(tn - T;), i = 1, ... , n, exist. Hence n-+oo

n-+oo

m

lim p;(tn) lim x(tn- T;) lim x(tn)-'""' n-co = n-+oo n-+oo L..J i=l

~lim sup

x(t)(1- p)

t-oo

i.e. -1 a ~lim sup x(t). t-oo - P

(4.5.4)

lim x(t~) = liminf x(t). On the other hand, there exists {t~} such that R-t>OO t-400 lim p;(t~) and lim x(t~- T;), '= Without loss of generality, we assume that n-+oo n-+oo 1, ... , m, exist. Hence a= lim y(t~) n-oo

m

lim x(t~)-'""' = n-+oo LJ i=l

lim p;(t~) lim x(t~- t;) n-+oo

n-+oo

:$liminf x(t)(1- p) t-+oo

or -1 a :$liminf x(t). t-+oo - p

(4.5.5)

lim x(t) = 1 ~ • A similar proof can be Combining (4.5.4) and (4.5.5) we obtain t-+oo P 0 given if x(t) < 0. We are now ready to prove the following results. m

Theorem 4.5.1.

Assume that lim L:P;(t) = p E [0, 1). Let x(t) be a t-+00 i=l

nonoscillatozy solution of Eq. (4.5.1). Let S denote the set of all nonoscillatory solutions of Eq. (4.5.1), and define lim y(t) = 0, lim y'(t) = 0}, S(O,O,O) ={xES: lim x(t) = 0, t-+oo t--+oo t-+oo

245

Second Order Equation& with Deviating Arguments

S(b,a,O) ={xES: lim x(t) = b :=_a_, lim y(t) =a, lim y'(t) = 0}, 1- p

t--+oo

y--+oo

t-+oo

lim y(t) = oo, lim y'(t) = 0}, S(oo,oo,O) ={xES: lim x(t) = oo, t-oo t-+oo t-+oo

lim y(t) = oo, lim y'(t) = d S(oo,oo,d) ={xES: lim x(t) = oo, t-+oo t-+oo t-+oo

:f. 0}.

Then S = S(O, 0, 0)

u S( b, a, 0) U S( oo, oo, 0) U S( oo, oo, d).

Proof: Without loss of generality, let x(t) be an eventually positive solution. lim y'(t) = 0, i.e., x E If lim x(t) = 0, by Lemma 4.5.1, lim y(t) = 0 and t--+oo t__,.oo t_..oo S(O,O,O). If limx(t) = 0, fails, by Lemma 4.5.1, y(t) > 0 eventually, it is easy t-+00

to see that y'(t) > 0, y 11 (t) < 0 eventually. If lim y(t) = a > 0 exists. Then t-+oo limx(t) = 1 ~ P = b, i.e., x E S(b,a,O). limy'(t) = 0, by Lemma 4.5.2, we have t-+oo t--+oo If lim y(t) = oo, by Lemma 4.5.2, lim x(t) = oo. Since y"(t) < 0 and y'(t) > 0, t-+oo

t-+oo

we have limy'(t) = d, where d = 0 or d t--+oo x E S(oo, oo, d).

> 0. Then either s E S(oo,oo,O), or 0

In the following we shall show some existence results for each kind of nonoscillatory solutions of Eq. (4.5.1).

Assume there exist two constants k 1 > k2 > 0 such that

Theorem 4.5.2.

m

m

LP;(t)exp(k1r;) > 1 ~ LP;(t)exp(k 2 r;), ~1

(4.5.6)

~1

and

... , exp( -k2gjt(u)))du eventually. Then Eq. (4.5.1) has a solution x E S(O,O,O).

(4.5.7)

Chapter

246

4

Proof: Let BC be the space of all bounded continuous functions on [to, oo) with the sup norm 1\xl\ = suplx(t)l. Define t~to

n=

{x E BC: exp(-k 1 t) $ x(t) $ exp(-k2t) and

lx(t2)-x(ti)I$Lit2-tii,

tht22::to},

where L 2:: k1 • Then f! is a nonempty, closed convex bounded subset of BC. For the sake of convenience, denote n

f(u,x(g(u))) = Lfi(u,x(gji(u)), ... ,x(gjt(u))), j=l (4.5.8) n

f( u,exp( -k2g(u))) =

L

fi( u, exp( -k29jl (u)), ... , exp( -k29jt( u))).

j=l Define a mapping T on

n as follows:

EP;(t)x(t- r;)- j 100 (u- t)f(u, x(g(u)))du, (Tx)(t) = {

( 4.5.9)

i=l

exp(- K(x)t), where K(x) = to,

ln(T;)(T) ,

t 0 $ t < T,

T is a sufficiently large number so that t - r; 2::

9i•(t)2::to, i=1, ... ,m, j=1, ... ,n, s=1, ... ,1!,fort2::T.

Condition (4.5.7) implies that

£

00

f(u,exp(-k2g(u)))du < oo.

Assumption (i) at the beginning of this section implies that for given a E (1-h, 1)

(4.5.10) Therefore, T can be chosen so large that for t 2:: T

(4.5.11)

247

Second Order Equations with Deviating Arguments

and m

a+ l::exp(- k2 (t- r;)) ~ 1. i=l

Hence by (4.5.6) and (4.5.7) m

('Tx)(t) ~ LP;(t)x(t- r;) i=l

m

~ LP;(t)exp(- k2(t- r;)) i=l

= ( ~

~p;(t)exp(k2r1 ))exp(-k2t)

exp( -k2t),

for

t 2:: T,

and

('T x)(t) 2::

L p;(t)exp(- k (t- r;)) - loo (u- t)f(u,exp( -k g(u)))du m

1

2

~1

t

= exp( -k1 t) + ( ~p;(t)exp(k 1 r;) -1 )exp( -ktt)

-1

00

(u- t)/(u,exp( -k2 g(u)))du

2:: exp( -k 1 t) for t 2:: T, i.e., exp(-k 1 t)

~

('Tx)(t)

~

By the definition of K(x), and exp(-k 1 T) that k2 ~ K(x) ~ k1. Hence

exp( -k1 t)

~

(Tx)(t)

~

t 2:: T.

exp(-k2t), ~

('Tx)(T)

exp( -k2 t),

t0

~

~

exp(-k 2 T), we know

t < T.

Chapter

248

4

Now we show that (4.5.12)

Without loss of generality assume t2 and (4.5.12), we have

~

tt

~to.

~

For t2

tt

~

T, using (4.5.11)

I(Tx)(tt)- (Tx)(t2)l m

:5 L lp;(tt)x(tt - r;)- p;(t2)x(t2- r;)l i=t

m

:5 L (p;(t2)lx(t2- r;)- x(t1- r;)l + IP;(t2)- p;(tl)lx(tl- r;)) i=l

+

11:

2

(u- tt)f(u,x(g(u)))du +

1~ (t

2 -

tt)f(u,x(g(u)))dul

:5 [ ~ (p;(t2) + exp( -k2(t- r;)))L

+ 1.<>0 /(u,exp(-k2g(u)))du]lt2 -td :5 { [ tp;(t2) +

t

exp(- k2(t1 - r;)}]

+ (a-~ p;(t2)) }Lit2 -

= [~ exp(- k2(h- r;)) +a] Llt2- td :5 Llt2 -ttl· For to :5 tt :5 t2 :5 T, we have I(Tx)(t2)- (Tx)(tt)l = jexp(- K(x)(t2))- exp(- K(x)(tt))j

:5 IK(x)(t2)- K(x)(tt)l :5 Llt2 -ttl·

ttl

Second Order

Equation~

249

with Deviating Argv.menu

For to < t1 :::; T :::; t2, we have I(Tx)(t2)- (Tx)(ti)I :5 I(Tx)(t2)- (Tx)(T)I + I(Tx)(T)- (Tx)(t2)l :::; Llt2 - Tl +LIT - til = Llt2- t1l·

We have proved that (4.5.12) holds for all t 0 :::; t 1 :5 t2. Therefore, Tfl ~fl. Clearly, T is continuous. Since Tfl ~ fl, Tfl is uniformly bounded. Set X En, we see that I(Tx)(t)l :5 exp( -k2t) I(Tx)(t2)-(Tx)(t1)l :5Lit2-tll,

to :5tl :::;t2.

For given e: > 0 there exists aT'> to such that exp(-k2t) <~fort 2:: T'. Then (4.5.13)

for t 2 2:: t 1 2:: T'. For to:::; t1 :::; t2 :::; T', we have whenever

lt2 - t11 <

e:

L.

(4.5.14)

(4.5.13) and (4.5.14) implies that Tfl is equicontinuous. Therefore, Tfl is relatively compact. By Schauder's fixed point theorem there is an x* E fl such that x* = T x*. Then x* is a positive solution of Eq. (4.5.1 ), and x* E S(O, 0, 0). 0 m

Theorem 4.5.3.

Assume that lim EP;(t) = p E [0, 1). Then Eq. (4.5.1) has t-+oo i=l a nonoscillatory solution x E S(b, a, 0) (b =f:. 0, a =f:. 0) if and only if (4.5.15)

Proof: Necessity. Without loss of generality, let x(t) E S(b, a, 0) be an eventually positive solution of Eq. (4.5.1). By Theorem 4.5.1 we know that b > 0, a> 0.

250

Chapter

4

Using the notation (4.5.8), from (4.5.1) and (4.5.2) we have

y"(t)

= -f(t,x(g(t)).

Integrating it from s to oo for s 2: to we have

1

00

y'(s) =

f(u,x(g(u)))du.

Integrating it from T to t for T sufficiently large, we have

y(t) = y(T) + J;cu- T)f(u,x(g(u)))du

+ Since lim x(gjh(u)) u-oo

= b > 0,

j

ioo

(t- T)f(u,x(g(u)))du.

= 1, ... , n,

h

( 4.5.16)

= 1, ... ,£,there exists aT 2: t 0

such that x(gjh(u)) 2:! for u 2: T. Hence from (4.5.16) we have

which implies that (4.5.15) holds. Sufficiency. Set b1 > 0 and A > 0 so that A < (1- p )b 1. From ( 4.5.15) there exists a sufficiently large T so that fort 2: T we have t- r; 2: t 0 , i = 1, ... , m, and gjh(t) 2: t 0 , j = 1, ... , n, h = 1, ... , £, and

( 4.5.17) Define n to be the set of all continuous functions x( t) on [to, oo) such that 0 :5 x(t):::; b1 , t 2: t 0 , and define a mapping Tin as follows:

n

m

A+ ;;p;(t)x(t- r;)

t

+ JTuf(u,x(g(u)))du

{

(Tx)(t)

=

+ f1

00

(Tx)(T),

tf(u,x(g(u)))du, t 0 $t
t 2: T

( 4.5.18)

251

Second Order Equations with Deviating Argument.! Set

x 0 (t)=0,

t~to

xk(t)=(Txk-t)(t), It is easy to see that x 0 (t)

t~to,

< x 1 (t) =A$ b1 , t

(4.5.19)

k=1,2, .... ~ t0 •

By induction, we have

Thus limk ..... ooXk(t) = x(t) exists and A$ x(t) $ bt, Lebesgue's convergence theorem, from ( 4.5.19) we obtain

t $ [t 0 , oo ). By the

A+ L;~ 1 p;(t)x(t- r;) + J;uf(u,x(g(u)))du x(t) =

{

+ J,

00

t

tf( u, x(g( u )) ) du,

x(T),

~

T

to$

Hence, x(t) is a positive solution of Eq. (4.5.1). Since 0 Theorem 4.5.1, x E S(b,a,O).

t < T.

< A$ x(t) $ bt, from 0

Similarly, we can prove the following results. m

Theorem 4.5.4.

Assume that lim L:;p;(t) = p E [0, 1). Then Eq. (4.5.1) has t-(X) i=l

a nonoscillatory solution x E S( oo, oo, d)

( d f. 0) if and only if

i~ltfi(u,dtgit(u), ... ,dtgit(u))ldu
m

Theorem 4.5.5.

Assume that lim L:;p;(t) t-0() i=l

forsome

d1 f.O.

(4.5.20)

= p E [0, 1). Further assume that

Chapter

252

4

and ( 4.5.22)

where b1 d 1 > 0. Then Eq. (4.5.1) bas a nonoscillatory solution x E S(oo, oo,O). Example 4.5.1. Consider

(4.5.23) where q(t) = 2(t(~~r>;'" . It is easy to see that (4.5.15) holds. Therefore, (4.5.23) has a nonoscillatory solution x E S(b, a, 0), b =f. 0, a =f. 0. In fact, x(t) = 1- ~ is b = 1. such a solution, where a=

t,

Example 4.5.2. Consider

(x(t)- tx(t- t))"

+ q(t)x 113 (t) =

(4.5.24)

0

where

For large t, q(t) "' kc 5 13 • It is easy to see that ( 4.5.21) and ( 4.5.22) are satisfied. According to Theorem 4.5.5, Eq. (4.5.28) has a solution x E S( oo, oo, 0). In fact, x(t) =-/tis such a solution of (4.5.24). Remark 4.5.1. The above arguments can be applied to the equation

(x(t) - ~

~ p;( t)x(t- r;)

)" = ~

~.f; (t, x(gj 1 (t) ), ... , x(gjt(t))),

t ;:: t 0 • (4.5.25)

For instance, under the assumptions of Theorem 4.5.1, we have S = S(O, 0, 0) U S(b, a, 0) US( oo, oo, d) US( oo, oo, oo ). Theorems 4.5.3 and 4.5.4 hold also for Eq. (4.5.25). Furthermore, Eq. (4.5.25)

253

Second Order Equations with Deviating Arguments has a nonoscillatory solution x(t) E S( oo, oo, oo) if

1~ I~ fi(t, d1gj1(t), ... , dlgjt(t)) ldt < oo

for some

d1

f:. 0.

(4.5.26)

We now present another result for the linear equation (x(t)- px(t- r))"

+ q(t)x(g(t))

= 0,

t;::: to,

(4.5.27)

where the condition p E (0, 1) is not required. Theorem 4.5.6. Assume that (i) p, r > 0, q E C([t 0 ,oo),R+), g E C([to,oo),R), g(t+r) < t, and lim g(t) = oo; (ii) there exists a constant a > 0 such that for sufficiently large t

t-oo

1 01 r -ep

1100

+p

(s- t- r)q(s)exp[a(t- g(s))]ds:::; 1.

(4.5.28)

t+r

Then Eq. (4.5.27) has a positive solution x(t) satisfying that x(t) ---+ 0 as

t

---+ 00.

Proof: If the equality in (4.5.28) holds eventually, then we can verify that x(t) = e-"'t is an expected solution. Otherwise, we assume that there exists T >to, such that t + r ;::: 0, g(t + r) ;::: to fort ;::: T, and

fJ

1 := -e-"'r P

+-1 hoo

(s- T- r)q(s) exp[a(T- g(s))]ds

P T+r

< 1,

(4.5.29)

and (4.5.28) holds for t ;::: T. Let BC denote the Banach space of all continuously bounded functions defined on [t 0 ,oo) endowed with the sup norm. Let n be the subset of BC defined by

n

= {y E BC:

0:::; y(t):::; 1

for

Define a map T : n ---+ BC as follows:

(Ty)(t) = ('Jiy)(t)

+ (12y)(t)

t;:::

t 0 }.

Chapter

254

4

where

(1iy)(t) = {

~e-<>ry(t

+ r),

(1iy)(T)

+ exp[c:(T- t)] -

t?. T 1,

t0 $ t $ T

and (12y)(t) = {

~ ft0:.r(s- t- r)q(s) exp[a(t- g(s))]y(g(s))ds,

t?. T

(12y)(T),

to $ t $ T

where c; = ln(2- /3)/(T- to). We can show that the map T satisfies all the assumptions of Krasnoselskii's fixed point theorem, and soT has a fixed pointy inn. Clearly, y(t) > 0 fort?. t 0 • It is easy to check that x( t) = y(t)e-<>t is a solution of Eq. (4.5.27). D Corollary 4.5.1. Assume that 0 q*, r > r such that eventually

< p < 1,

0$q(t)$q*,

r > 0, and there exist constants

g(t)?.t-r.

If the "majorant" equation

(x(t)- px(t- r))"

+ q*x(t- r) =

0,

t?. t 0

has a positive solution, then Eq. (4.5.27) also has a positive solution. Proof: ( 4.5.30) has a positive solution if and only if

has a real root a. Clearly a must be negative. Let p.

or

= -a > 0, then

(4.5.30)

255

Second Order Equations with Deviating Arguments Hence

1 +-11"" -e-pr p

p

(s- t- r)q(s)exp[Jl(t- g(s))]ds

t+r

< ~e-pr + Lep(r-r) PJl2

- p

=1

'

i.e., (4.5.28) holds. Then, by Theorem 4.5.5, Eq. (4.5.27) has a positive solution.

4.6. Unstable Type Equations 4.6.5. Equations with constant p

Consider the second order linear neutral differential equation of the form (x(t)- px(t-

r))"

= q(t)x(g(t)),

t ~to,

(4.6.1)

limg(t) = oo, r > 0. where p E R,q E C([t 0 ,oo),R+), g E C([t 0 ,oo),R), t-+oo We will show later that Eq. ( 4.6.1) always has an unbounded, nonoscillatory solution. Therefore, we only need to find conditions for all bounded solutions of Eq. (4.6.1) to be oscillatory. Theorem 4.6.1. Assume that (i) 0 < p < 1, T > 0 are constants; (ii) g(t) :5 t and g is nondecreasing fort~ t 0 ; (iii)

lim sup t-+oo

f91(t) (s- g(t))q(s)ds > 1.

( 4.6.2)

Then every bounded solution of Eq. (4.6.1) is oscillatozy. Proof: Assume the contrary, and let x(t) be an eventually positive bounded solution of Eq. (4.6.1). Define z(t) = x(t)- px(t- r).

(4.6.3)

lim z(t) = oo, > 0 fort~ T1 > T, then t-+oo which contradicts the boundedness of x. Therefore, z'(t) :5 0 fort~ T.

We have z"(t)

> 0 fort~ T

~ t 0 • If z'(t)

There are two possibilities for z( t) :

256

Chapter 4

(a) z(t) > 0

fort~

T; (b) z(t) < 0 fort~ T2

~

T.

In case (a), integrating (4.6.1) from 8 tot we have

z'(t)- z 1(8)

=it

q(u)x(g(u))du.

(4.6.4)

Integrating (4.6.4) in 8 from g(t) tot, we have

z'(t)(t- g(t))- z(t) + z(g(t))

=

r 1.t q(u)x(g(u))du d8

}g(t)

=

r

(8- g(t))q(s)x(g(8))ds

}g(t)

> [' (s- g(t))q(s)z(g(8))ds }g(t)

~ z(g(t))

r

(s- g(t))q(s)ds.

}g(t)

Hence for t

~

T

z(t) + z(g(t)) (

r

}g(t)

(s- g(t))q(s)ds -1) :50

which contradicts the positivity of z(t) and (4.6.2). In case (b), we have

x(t) < px(t- r) < p2 x(t- 2r) < · · · < pnx(t- nr) fort ~ T2 + nr, which implies that lim x(t) = 0. Consequently, lim z(t) = 0, a t-oo t--+oo contradiction . 0

Remark 4.6.1. Theorem 4.6.1 is also true for p = 0. Theorem 4.6.2.

Assume that

(i) p < 0 and q(t) > 0, t ~to; (ii) g(t) = t- u, where u is a constant, u > r;

Second Order

Equation~

with Deviating

Argument~

257

(iii) There exists o: > 0 such that (4.6.5)

lim sup q(t)fq(t- r) = o: t-+oo

and (s- (t- (u- r)))q(s)ds > 1- o:p.

limsupit t-+oo

(4.6.6)

t-(u-r)

Then every bounded solution of (4.6.1) is oscillatory. Proof: Assume the contrary, and let x(t) be a bounded, eventually positive solution of Eq. (4.6.1) and z(t) be defined as in (4.6.3). As shown before, z"(t) > 0, z'(t) < 0, and z(t) > 0 eventually, where z(t) is defined by (4.6.3). From (4.6.5) and (4.6.6), there exists a constant k > 1 such that

lim sup t-+oo

it

(s- (t- (u- r)))q(s)ds > 1- ko:p,

(4.6.7)

t-(u-r)

q(t)fq(t- r) < ko:,

t

~ t1

(4.6.8)

where t1 is a sufficiently large number. We rewrite Eq. (4.6.1) in the form z"(t)- p (q(t) )z"(t- r) = q(t)z(t- u). q t- T

(4.6.9)

Substituting ( 4.6.8) into (4.6.9) we have z"(t)- ko:pz"(t- r) ~ q(t)z(t- u),

t ~ t 1•

(4.6.10)

Set w(t) = z(t)- ko:pz(t- r).

(4.6.11)

Then w"(t) ~ q(t)z(t- u) > 0,

t ~ t 1•

(4.6.12)

By the boundedness of x(t), it is easy to see that w(t) > 0, t2 ~ t 1 • Since z( t) is decreasing, w(t) = z(t)- ko:pz(t- r) ~

(1- ko:p)z(t- r),

w'(t) ~ 0,

t ~ t 2.

t ~

(4.6.13)

Chapter

258

4

Combining (4.6.12) and (4.6.13) we have 1

w"(t) ~ 1 k q(t)w(t- (u- r)). - ap Integrating (4.6.14) from

8

tot

fort~ 8 ~

t2 we have

f.'

w'(t)- w1(8) ~ 1 _ 1kap • q(u)w(u- (u- r))du. Integrating (4.6.15) in

8

(4.6.14)

(4.6.15)

from t- (u- r) tot we have

w'(t)(u- r)- w(t) + w(t- (u- r)) ~

=

1-

1-

1k

1' f.' 1' 1'

ap t-(a-T) ~

1k

(u- (t- (u- r)))q(u)w(u- (u- r))du

ap t-(a-T)

r)) k > w(t- (u1-

ap

q(u)w(u- (u- r))duds

(u- (t- (u- r)))q(u)du,

t ~ t2•

t-(a-r)

Thus

w(t) + w(t- (u- r)) [

1-

1'

1k

ap t-(a-r)

(u- (t- (u- r)))q(u)du- 1]

~0 D

which contradicts (4.6. 7). Theorem 4.6.3. Assume that (i) p = 1, r > 0; (ii) g(t) ~ t, and g is nondecreasing fort~ t 0 ; (iii) either

1

00

tq(t)dt

=

00

(4.6.16)

to

or lim t

t-+oo

1

00

t

q( s )ds = oo.

(4.6.17)

259

Second Order Equations with Deviating Arguments Then every bounded solution of Eq. (4.6.1) is oscillatory.

Proof: Assume the contrary and let x(t) be a bounded eventually positive solution of Eq. ( 4.6.1) and z(t) be defined as in ( 4.6.3). There are two possibilities: (a) z"(t) ~ 0, z'(t) ~ 0, z(t) < 0 fort~ t 1 ~ t 0 , (b) z"(t) ~ 0, z'(t) ~ 0 and z(t) > 0, t ~ t 1 ~to. In case (a), there exists a finite number a > 0 such that lim z(t) t-oo Thus there exists t2 ~ t1 such that -a < z(t) < - ~ , t ~ t2, i.e.,

-a< x(t)- x(t- T) < -

2a ,

t

= -a:.

~ t2.

Hence x(t- T) > ~. t ~ t2, then there exists t 3 ~ t2 such that x(g(t)) > ~. ~ t 3 • From Eq. (4.6.1), we have

t

z"(t) ~ ~ q(t),

t ~ t3.

x(t)>x(t-T),

t~t1

(4.6.18)

In case (b), we have

then there exists M > 0 such that x(t)

~

M,

z"(t) ~ Mq(t),

t

~ t1•

Hence

t ~ t3•

(4.6.19)

In both cases we are lead to the same inequality (4.6.19). (4.6.19) from t toT forT> t ~ t 3 we have

z'(T)- z'(t)

~MiT q(s)ds,

t3 $

t < T.

Hence -z'(t)

which implies that

~MiT q(s)ds,

t3 $ t < T,

ft'; q( s )ds < oo and so -z'(t)

~M

1""

q(s)ds.

Integrating

260

Chapter

4

Integrating the above inequality from t to T for T > t we have

z(t)

~ z(T) +MiT J.oo q(u)duds = z(T)

+M [

lT(u-

t)q(u)du + (T- t)

!roo q(u)du],

t

~ tz

which leads to a contradiction to the boundedness of z in either case of (4.6.16) and (4.6.17). 0 Example 4.5.1. Consider

211" ( x(t)- x(t- 211") ) , == --x(t11"). t -11"

(4.6.20)

It is easy to see that all assumptions of Theorem 4.6.3 are satisfied. Therefore, every bounded solution of Eq. ( 4.6.20) is oscillatory. Eq. ( 4.6.1) may have unbounded oscillatory solutions. For example, (4.6.20) has a solution x( t) = t sin t.

Theorem 4.6.4. Assume that p > 1. Then Eq. ( 4.6.1) has a bounded positive solution if and only if

roo tq(t)dt < oo.

(4.6.21)

ito

Proof: Necessity. Let x(t) be a bounded positive solution of Eq. (4.6.1) and z(t) be defined by (4.6.3). Then Eq. (4.6.1) becomes

z"(t) = q(t)x(g(t)). There are only two possibilities for z(t) :

(a) z"(t)

~

(b) z"(t)

~

0, 0,

z'(t) :::; 0, z'(t) :::; 0,

z(t) < 0, z(t) > 0,

t t

~ ~

t1 t1

~to; ~to.

As in the proof of Theorem 4.6.3, in either case we are lead to the inequality

z 11 (t) ~ Mq(t),

for

t

~

tz,

261

Second Order Equations with Deviating Arguments

where M > 0 is a constant and t 2 is a sufficiently large number. Integrating it twice we have

~MiT J.T q(u)duds

z(t)- z(T)

T >t

= M iT(u- t)q(u)du,

~ t2.

Letting T-+ oo, we see that (4.6.21) holds. The sufficiency part of Theorem 4.6.4 follows from the following more gen0 eral result. Theorem 4.6.5.

Assume p > 1 and q E C([to, oo), R) such that

1

00

sjq(s)jds < oo.

(4.6.22)

to

Then Eq. (4.6.1) bas a bounded positive solution. Proof: Let T ~ to be sufficiently large so that t t ~ T, and

1

00

p-1

sjq(s)jds $ - - , 4 t+r

t

+r ~

~to,

g(t

+ r)

T.

~ t0

for

(4.6.23)

Consider the Banach space BC of all continuous bounded functions defined on [to, oo) with the sup norm. Set

n={xeBC:

~$x(t)$2p,t~t 0 }.

Clearly, n is a bounded closed convex subset of BC. Define a map T: BC as follows:

(Tx)(t) =

{

n-+

p-1+~x(t+r)

- ~ ft";r(s- t- r)q(s)x(g(s))ds,

(Tx)(T),

t

~

T

to$ t $ T.

(4.6.24)

262

Chapter

For any

X

4

E n, from (4.6.23) we have

(Tx)(t)

~ p

1100

(s- t- r)jq(s)jjx(g(s))jds < 2p,

for

t :2: T,

12--1100

(s- t- r)jq(s)llx(g(s))jds :2: ~,

for

t :2: T.

+ 1 +p

t+.-

and (Tx)(t) :2: p-

p

t+.-

Therefore, Tn ~ n. We shall show that T is a contraction on we have

n.

In fact, for any

XJ' X2

E

n,

1 p

I(Txt)(t)- (Tx2)(t)j ~- lxt(t + r)- x2(t + r)l

11

+-

00

p t+.-

(s- t- r)jq(s)llxt (g(x))- x2(g(s))jds

~ llx1-x211~(1+P~ 1 )

=~(1+~)11xl-x211, t~T which implies that

!)

Since !(1 + < 1, it follows that Tis a contraction. Hence there exists a fixed point X En. Then x(t)

1100

1 = p -1 + -x(t + r)--

p

p t+.-

(s- t- r)q(s)x(g(s))ds,

t ~ T.

Second Order

Equation~

263

with Deviating Arguments

Differentiating it twice, we have

(x(t + r)- x(t))"

= q(t + r)x(g(t + r)),

t ~ T,

i.e., x(t) is a bounded positive solution of Eq. (4.6.1).

D

Remark 4.6.2. By a similar proof we can show that Theorem 4.6.5 is also true for p E (0, 1). The following result is about the existence of asymptotically decaying positive solutions of Eq. (4.6.1).

Theorem 4.6.6. Assume that 0 < p < 1 and there exists a constant a > 0 such that eventually

pe 01 r

+

1

00

(s- t)q(s )exp [a(t- g(s))]ds

~ 1.

(4.6.25)

Then Eq. (4.6.1) has a positive solution x(t) satisfying x(t)-+ 0 as t-+ oo. Proof: It is easy to see that if the equality in ( 4.6.25) holds eventually, then Eq. ( 4.6.17) has a positive solution x( t) = e-at. In the rest of the proof we may assume that there exists a number T > t 0 such that t- r ~ t 0 , g(t) ~ t 0 for t ~ T and (3 := pe 01 r

+ ~00 (s- T)q(s)exp[a(T- g(s))]ds < 1,

(4.6.26)

and condition ( 4.6.25) holds fort ~ T. Let BC denote the Banach space of all continuous bounded functions defined on [t 0 ,oo) with the sup norm. Let n be the subset of BC defined by

n=

{y E BC: 0

~

y(t)

~ 1,

t

~to}.

Define a map T : n -+ BC as follows:

(Ty)(t) = (1i.y)(t)

+ (12y)(t)

Chapter

264

4

where

pe"' .. y(t- r),

(1iy)(t) = {

(1iy)(T)

t?. T

+ exp[e(T- t))- 1,

(4.6.27)

to $ t $ T

and

(12y)(t) = {

j,""'(s- t)q(s)exp[x(t- g(s))]y(g(s))ds, t?. T

(4.6.28)

to $ t $ T,

('J2y)(T),

where e = ln(2- [3)/(T- t 0 ). It is easy to see that the integral in (4.6.28) is defined whenever y E il. Clearly, the set n is closed, bounded and convex in BC. We shall show that for every pair X' y E n

1ix + 'J2y E il. In fact, for any x,y

(1ix)(t)

+ (12y)(t) =

En, we have

pe"' .. x(t- r) +

$ pe"' .. $ 1,

(4.6.29)

+

for

1

00

1

00

(s- t)q(s)exp[a(t- g(s))]y(g(s))ds

(s- t)q(s)exp[a(t- g(s))]ds

t ?. T,

and

(1ix)(t)

+ (12y)(t) =

(1ix)(T)

+ (12y)(T) + exp[e(T- t)]-1

= {3 + exp[e(T- t)] - 1

$ {3 + exp[e{T- t 0 ))-1

=1

for

t 0 $ t $ T.

Obviously, (1ix)(t) + (12y)(t)?. 0 fort?. t 0 • Thus, (4.6.29) is true. From (4.6.26), we know that pe"' .. < 1, which implies that 1i is a contraction. We now shall show that 12 is completely continuous. In fact, from (4.6.25), there exists a positive constant M such that

1

00

q(s)exp[a(t- g(s))]ds $ M,

for

t?. T.

265

Second Order Equations with Deviating Arguments

Thus we have

I:t(12y)(t)i = il"' q(s)exp[a(t- g(s))]y(g(s))ds +a ~

j

00

(s- t)q(s)exp[a(t- g(s))]y(g(s))dsi

M +a,

for

t > T,

and d

dt(12y)(t) = 0,

for

t 0 ~ t < T,

which shows the equicontinuity of the family 72!1. On the other hand, it is easy to see that 72 is continuous and the family of 72 n is uniformly bounded. Therefore 72 is completely continuous. By Krasnoselskii's fixed point theorem, T has a fixed pointy E !1. That is,

pe"'Ty(t- T) y(t) =

{

+ Jt,(s- t)q(s)exp[a(t- g(s))]y(g(s))ds,

t;::: T t0

y(T) + exp[c(T- t)]- 1,

~

t

(4.6.30) ~

T.

Since y(t) ;::: exp[c(T- t)]- 1 > 0 for to ~ t < T, it follows that y(t) > 0 for t;::: to. Set x(t) = y(t)e-"'t. Then (4.6.30) becomes

x(t)=px(t-T)+

j

00

(s-t)q(s)x(g(s))ds,

t;:::T.

Thus x(t) is a positive solution of Eq. (4.6.1) and x(t)-+ 0 as t-+ oo. Remark 4.6.3. For the case p 4.6.6 still holds. Corollary 4.6.1. a > 0 such that

= 0, if g(t) < t,

Assume tl1at 0