UNIVERSITY OF KANSAS DEPARTMENT OF MATHEMATICS
STUDIES IN EIGENVALUE PROBLEMS
Technical Report 21 CHARACTERISTICS AND CAUCHY PROBLEM FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER by
Jacek Szarski
Paper written under Contract Nonr 58304 with Office of Naval Research Lawrence, Kansas 1959
TABLE OF CONTENTS Introduction ..................................................
1
Chapter I. 1.
Partial differential inequalities of first order............
2.
Uniqueness and continuous dependence on initial
values of the solution of the Cauchy problem .............
2
9
Chapter II. 3.
Characteristic strips of first order .....................
4.
Local existence of the solution of the Cauchy problem for a non-linear equation. . ...... 6 .... 6 ........... .....
5.
13
20
General properties of characteristics and Plis's theorems ......................................
37
Chapter III. 6.
Characteristic strips of second order ...................
54
7.
Existence theorem and estimation of the existence domain
64
8.
Generalized Cauchy problem for a system of partial
differential equations of first order .....................
70
Chapter IV.
9.
Systems of equations with total differentials .............
78
10.
Involutory systems of equations ........................
88
Bibliography .................................................
99
CHARACTERISTICS AND CAUCHY PROBLEM FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS OF FIRST ORDER. 1. by
J. Szarski
Introduction.
The present paper deals with the theory of Cauchy
characteristics for the equation of the form zx = f(x, y1, ..., Yn, z, zyl.... , zyn)
(1. 0)
and with the Cauchy) problem for equation (1. 0) and for systems of equa-
tions of the form (2.0)
zi = fi(x, yI, .... Yn, zl! .... zm,izl ,..., zl
(3.0)
zi
(4. 0)
V
yn yl = piv(xI, .... xp, z1, ..., zm ) 0=1120...'M; V=1,2# .... P)
zx = flv (xI, ..., xP, yl, ...)i/n, zl
v-
zm zl ,..., zl yl
yn
)
(i = 1$ 2..., m; v= 1,..., P)
The paper is divided into four chapters. Chapter I deals with partial differential inequalities of first order and with the uniqueness of the solution of the Cauchy problem for equation (1. 0) and for systems (2. 0). The classi-
cal theory of characteristic strips of first order for equation (1. 0) together
with recent results obtained by A. Plis is treated in Chapter II. Characteristic strips of second order for equation (1. 0) and existence theorems for equation (1. 0) and for systems (2. 0) are dealt with in Chapter III. Chapter IV is 1.
Paper written under Contract Nonr 58304 with the Office of Naval Research
concerned with uniqueness and existence theorems for systems (3. 0) and (4. 0).
Chapter I contains published results obtained by T. Wad ewski and
by the author. The first part of Chapter II consists of classical material
and results published by A. Plis constitute the bulk of this chapter. Chapter III contains published results obtained by A. Plie. The first part of Chapter IV is classical the geometrical interpretation of compatibility conditions for systems (3. 0), given by T. Wazewski and the author, is to appear soon in Annales Polonici Mathematici{ the existence theorem for systems (4. 0) has not yet been published.
Numbers in square brackets
refer to bibliography.
CHAPTER I F31.
Partial differential inequalities of first order. We consider a system of partial differential inequalities of first order
is < ft(x, Y, Z, 4Y) where Y =
Yn),
Z=
zm),
(i = 1, 2,..., m)
zY = (zi
,...,
Yl
zl ) yn
inequality contains only derivatives of the i-th function
and the i-th z(x, Y)
We introduce the following Assumptions A.
fi(x, Y, Z, Q), where Q = (ql, ..., qn) are defined for (x, Y)
belonging to the set (2.1)
0<x
and for arbitrary Z, Q,
lyjl
(a
min(--))
satisfy the Lipschitz condition n
(3.1)
lfi(x,Y,Z,Q)-fi(x,Y,Z,d)I
M gj - gj( j =1 I
The assumption that ft are defined for arbitrary Z, Q is not essential. We make this assumption for simplicity's sake.
1.
-2-
and the following monotonicity condition:
For any fixed index 1, (1 < f
zi
m),
(4.1)
m), if we have
zf = zl
i
then
f f (x, Y. Z, Q) < f f (x, Y. Z, Q)
We shall prove now ([10], [11])
Theorem 1.1.
Suppose Assumptions A to hold true. Let ui(x, Y)
m) be continuous in (2.1) and satisfy the initial
and v (x, Y) (i
inequalities u1(0, Y) < v (0, Y)
(5.1)
(i =1, 2,.... M)
.
Suppose furthermore that if for some point (x*, Y*) i, n (2. 1) where x* > 0
and for some index io we have
u°(x*, Y*) = v°(x*, Y*) i
i
then u° and v° have first derivatives at (x*, Y*) and ppossess Stolz's differentials in case of (x''`, Y*) being a point on the side surface of the p
ry amid (2.1), and satisfy the differential inequalities uX(x*, Y*) < f °(x*, Y*, U(x*, Y*), u 0 (x*, Y*))
(6.1)
0 vX(x*. Y*) > fi°(x*. Y*. V(x*, Y*), vY (x*, Y*))
1.
The last hypothesis of our theorem is obviously satisfied if we assume that for all indices i and for all points (x*, Y*), where x* > 0, belonging to i° (2.1) u and v ° have first derivatives and possess StolzIs differential in case of (x*, Y*) being a point on the side surface of the pyramid (2.1), and satisfy inequalities (6.1). 1.
-3-
Under these assumptions the inequalities ui(x, Y) < v (x, Y)
(7.1)
(i = 1, 2...., m)
hold in (2.1), Proof.
By (5.1) and by the continuity of ul and v the set of x,
0 < x < a, such that (7.1) holds true in (2. 1) for 0 < x < x is not empty.
Let us denote by x* the least upper bound of such
We have to prove
that x* = a. Suppose it is not true and hence (8.1)
Then there exists an index io'and a point Y* such that (x*, Y*) belongs to (2.1) and (i = 1, 2,..., m), for 0 < x < x*
ui(x, Y) < v (x, Y),
(9.1)
u°(x*, 1`) = v°(x*, Y*)
(10.1)
Now, there are two cases to be distinguished. Case I.
Suppose (x*, Y*) Is an interior point of (2.1). i
i
Then by (10.1) and
by the hypothesis of the theorem u° and v° have first derivatives at (x*, Y*).
Consider the function u °(x*, Y) - v °(x*, Y) depending on the Y.
By (9.1) and (10.1) it attains maximum at Y* and hence, Y* being an in-
terior point, we have (11.1)
uYjo (x*, Y*) = v Yj°(x*, Y*)
(j = 1,..., n)
Similarly the function u°(x, Y*) - v°(x, Y*) depending on x attains its maximum in the interval 0 < x < x* at the point x*. Therefore (12.1)
to
to
ux (x*, Y*) - vX (x*, Y*) > 0
On the other hand, by (4.1), (6.1), (9.1), (10.1), and (11.1) we have successively -4-
fi°(x*, Y# U(x Y*). uY (x*, Y*))
uX (x*, Y*) - vX O(X*, Y#) <
-fi°(x*, Y*, V(x*, Y*), Y (x*, Y*)) = fi°(x*, Y*, U(x*, Y*), .Y (x*, Y*)) i i uY (x Y*)) < 0 -f °(x*, Y*, V(x*, Y*)0 which contradicts (12.1). Case II. Suppose (x*, Y*) is a point on the side surface of (2.1). We can
assume (rearanging if necessary the indices) that we have
yP =b (13.1)
p-
Mx* (p = 1,..., s){ y*q = -b q +Mx* (q =s +1,..., s +r) IYj I
< bj-Mx* (,) = s+r+l,...,n) i
i
By (10.1) and by the hypothesis of the theorem u 0 and v 0 have Stolls I s
differentials at (x*, Y*). Consider the function u °(x* yl'...' Yes Y+1""' yn) 1°(x*' v
yl"Ys' y5
depending on the point (y1,..., ye)
,...' yn)
-bp# Mx* < yp < b
in the set
(p = 1,..., s).
It attains maximum at (Y*--' Ya ) by (9.1) and (10.1), and hence from (13.1) it follows that (14.1)
Mx*,
y (x*, Y*) - y (x*, Y*) >= 0 p
(p = 1, 2,..., s)
p
By a similar argument we get (15.1)
(16.1)
u°
Yq
(xY*) - vYq° (x*, Y*) <= 0
°(x*, Y* ) -
Yj
Now, for 0 <x< x*
° (x*, Y*) = 0 yj
(q = e +1,..., s + r)
() = s+r+l,...,n)
put
Y(x) _ (bp- Mx, - bq + Mx, yj ) -5-
-
and consider the composite function u°(x, Y(x)) - v°(x, Y(x)).
It attains
maximum at x* by (9.1) and (10.1) and hence Td [u°(x, Y(x)) - v°(x, Y(x))]x,# > u°
0
and v ° having Stolz's differentials at (x*, Y*) and Y(x*) being equal
to Y* we can apply to the left-hand member of the last inequality the formula for the derivative of a composite function and get the inequality io
io
#
ux (X*, Y*) - vx (x
7.1)
io
io
, Y*) > M L (uy (x # , Y#) - vy p
(X*, Y*))
p
p
yq u
q
On the other hand, by (3.1), (4.1), (6.1), (9.1), (10.1),
yq (14.1), (15.1) and (16.1)
we get ux (x*, Y*) - vi (x*, Y*)
-f iO(x*, Y*, V(x*, Y*)r
fiO(x
<
i Y*, U(x*, Y*Luo(x* ,
Y)) *
Y(x*. Y*)) = [fi°(x*Y*. U(x*, Y*),u,(x*, Y*))
-fi°(x*, Y*r V(x*, Y*). Y(x*, Y*))] + [fiO(x*, Y 1V(bc*, Y*), io
io # -fio (x* , Y* , V(x*, Y*), vy(x Y#)] < M[ (u ,
p
(° q
i,
yq
Y(x*,
io
Y*))
#
(X*. Y*) - v (x , Y N
yP
yp
(x*rY*)),
y9
which contradicts (17.1). Since in both cases I and II we got a contradiction,
our assumption (8.1) was false and thus the theorem is proved. Remark. In case of m = 1 the condition (4. 1) is automatically satisfied. Corollary 1.1.
Replace the set (Z.1) y
-6-
0 < x < a,
(18.1)
Y arbitrary.
Suppose all assumptions of theorem 1.1 to be satisfied with (2.1) replaced by (19.1) and assume that for each point (x*, Y*)j where x* > 0, belonging to (18.1), such that
u°(x*, Y*) = v°(x*, Y*), the functions
u° and v° have
Stolz's differentials at (x*, Y*). Then inequalities (7.1) hold, true in (18.1). Proof.
AA
Let (x, Y) be an arbitrary point in (18.1). If the b3 in (2.1) are belongs to (2.1). Then applying
chosen sufficiently large, the point
theorem 1, 1 we get (7, 1) at ( .-,Y), which completes the proof.
The next theorem we are going to prove concerns differential inequalities involving absolute values of functions and of their derivatives
[11].
Theorem 2.1. Suppose f1(x,Y,Z,Q), to be defined and non-negative for (19.1)
Ixl < a, IYj) <_ bj-Mlxl ,
(j = 1, 2,...,n)
and for arbitrary Z, Q and to satisfy (3. 1) and (4. 1). Let v (x, Y) (i =1, 2,..., m) be positive and continuous in (2.1). Let- ui(x, Y)
(i=1, 2,..., m) be continu-
ous in (19.1) and satisfy the inequalities lu1(0. Y)I< vi(0, Y)
(i=1, 2,..., m)
Suppose furthermore that if for some point (x*, Y*) in (19.1), where
Ix* I > 0
and for some index i° we have (20.1)
lui°(x*,Y*)I = v°(Ix*I,Y*)
then u° and v° have first derivatives at (x*, Y*) and (Ix* I, Y*) respectively and possess Stolz's differentials in case of (x*, Y*) being a point on the side surface of (19.1), and satisfy the differential inequalities
-7-
luX (x*,Y*)I < f1O(Ix*I,Y*, IU(x*,Y*)I, IuY (x*,Y*)1) (21.1)
Vx(1x*I,Y*) > f
(Ix*I,Y'`v(I-*1,Y*),
where luI ° = (1uoyl I ., Iu°ynI) and similarly for Y
IvY(Ix*I,Y*)I) Iv Y°I
Under these assumptions the inequalities
Iul(x,Y)I < vi(IxI,Y) (i=1,2,...,m)
(22.1)
hold true in (19.1).
Proof. First we shall show that inequalities (22.1) are satisfied in (2. 1). Put ui(x, Y) = (ui(x, Y)l
in (2.1)
and
fl(x, ' Z, Q) = f i(x, Y, Z.
(23.1)
I Q 1)
for (x,Y) in (2.1) and Z,Q arbitrary, where IQI = (Igll,..., Ignl). It is obvious that fi(x, Y, Z, Q) satisfy assumptions A. Suppose now that for some i
i
(x*, Y*) in (2.1) and some index io we have u°(x*, Y*) = v°(x*,Y*).
Then
(20.1) is satisfied and hence, by the hypotheses of our theorem, inequalities (21.1) hold true. But, since v°(x*, Y*) > 0, we have
ui°(x*, Y*) 4 0
and
hence ( 24.1)
i°(X*,
ux
Y*) G ux
i°(x*,
Y*)
l°u Y(x*, Y*) I = lui°(x Y )I Y
By (23.1) and (24.1) inequalities (21.1) imply (6.1). Thus we see that fi, ui,
and v' satisfy all the hypotheses of theorem 1.1 and therefore we have (7.1) in (2. 1). But, (7.1) in (2.1) is equivalent to (22.1) in (2. 1).
Now, the case
-a < x < 0 is reduced to the case just settled by the transformation x = -i;. Corollary 2.1. (25.1)
Replace the set (19.1) l x I < a,
Y arbitrary. -8-
Suppose all hypotheses of theorem 2.1 to be satisfied with (19.1) replaced by (25.1) and assume that for each point (x*, Y*) in (25.1), where
Ix* I
> 0, such
that (20.1) holds true, the functions u-1O and v ° have Stolz's differentials at
(x*, Y*) and (Ix* I, Y*) respectively. Then inequalities (22.1) hold Uxue: in (25.1).
Proof. Corollary 2.1 follows from theorem 2.1 just as corollary 1. 1 did from theorem 1.1.
§2. Uniqueness and continuous dependence on initial values of the solution of the Cauchy problem. We will give now an application of theorem 2. 1. Theorem 1. 2. (1.2)
Let ui(x, Y) (i = 1, 2,..., m) be continuous in the set
Ixl < a,
Iyjl
Assume that for IxI > 0, ui(x, Y) have first derivatives in the interior of (1. 2)
and possess Stolz's differentials on the side surface of (i'; 2). Suppose finally
that ui satisfy the initial inequalities (2. 2)
(i = 1,..., m)
I ui(0, Y) I < c
and differential inequalities for Ixl > 0 in (1. 2) n
m
(3.2)
IuXI
I +C
Iukl
j=1
(i=1,2,....m)
k
where c, M, C are non-negative constants and L > 0. Under these assumptions the inequalities (4. 2)
Iui(x,Y)I < c emLlxl +
L
are satisfied in (1. 2). -9-
(emLlxl
- 1)
m)
Proof. Put \\_m
n
fl(x,Y,Z,Q)= Lzk+Mqj+C+E (i=1,Z,...,m), k=1
and
vl(x, Y) = (c + E) em
j=1
Lx + C+E (mLx- 1) riTL
(i = 1, 2,..., m)
.
Then, from (2. 2) and (3. 2) it follows that Iul(0, Y) I < v'(0, Y) and
IuXI < f 1(Ix1, Y, IUI, 141) in (1. 2) for IxI > 0. On the other hand, we have
vx = f1(Ix1,Y, Ivi, Iv4I) in (1. 2) for x > 0. Thus we see that
f i,
ul, and vi satisfy all the assump-
tions of theorem 2.1 and hence
< vi(Ix1, Y) = (c+E) emLIxl
(emLIxl-
(5.2)
Iul(x, Y) I
in (1. 2).
But, since (5. 2) is true for arbitrary E > 0, we get (4. 2) making E
+ C+E
1)
in (5. 2) tend to zero. Corollary 1. 2.
Theorem 1. 2 holds true if we replace the set (1. 2) y
Y arbitrary
IxI < a,
(6. 2)
and assume that ui(x, Y) possess Stolz's differentials in (6. 2) for IxI > 0. Theorem 2. 2. Consider two systems of equations (7. 2)
zX
= f(x, Y, Z, zY)
m)
(8. 2)
zi = gl(x, Y, Z, zY)
m)
where fl and gl are defined for (x, Y) in (1. 2) and Z, Q arbitrary. Suppose -10-
that
n
m (9. 2)
i if (X, Y, Z, Q) - fi(x, Y, Z, Q) I < L
I zk- zk I + M
(10.2)
I I qJ - qJ I
J=1
k=1
Ift(x, Y, Z, Q) - gl(x, Y, Z, Q) I
,
Let z i(x, Y) be continuous in (1. 2), have first derivatives in the interior of (1. 2) for IxI > 0, possess Stolz's differentials on the side surface of (1.2)
and satisfy system (7. 2) for
Let zt(x, Y) be a solution of (8.2) having similar properties, and assume that IxI
> 0.
I z t(0. Y) - zt(0, Y) I
(11. 2)
< c (i =1, 2,..., m) .
Under these assumptions the inequalities (12.2)
I zt(x, y) -
zi(x, y) I
< c emLI xI
+
i. (emLl xI - 1)
(i =1, 2,..., m)
hold true in (1. 2).
Proof.
Put ut(x, Y) = a (x, Y) - zi(x, Y)
.
Then by (9. 2), (10.2), and (11.2) all the hypotheses of theorem 1. 2 are satis-
fied for ui(x, Y) and hence we get (12.2). Corollary 2. 2.
Theorem 2. 2 holds true if we replace the set (1. 2) by (6. 2)
and assume that 1z' and zt possess Stolz's differentials in (6. 2) for IxI > 0. Theorem 3. 2. Suppose fi(x, Y,Z. Q)I to satisfy all the hypotheses of theorem 2. 2.
Then the solution zi(x, Y) of the Cauchy problem z1(0, Y) = W1(Y)
(i = 1, 2,..., m)
for system (7. 2), which is continuous in (1. 2), has first derivatives in the
Al-
interior of (1. 2) and possesses Stolz's differential on the side surface of (1. 2), is unique and depends continuously on cpt(Y) and on the right-hand members of (7. 2).
Proof. The conclusion follows from theorem 2.2 if in (12.2) we let
c
and C tend to zero.
Corollary 3. Z. Theorem 3. 2 holds true if we replace the set (1. 2) by (6. Z)
and assume that the solution zi(x, Y) possesses Stolz's differential in (6. 2). We will state now without proof a more general uniqueness criterion for one equation, which was obtained by T. Wazewski L131 by means of the theory
of ordinary differential inequalities. Theorem 4. Z. Consider the equation
zx = f(x, Y, z, zl,) .
(13.2)
Assume that f(x. Y, z, Q) is defined for (x, Y) in (2. 1) and z, Q arbitrary
and satisfies the inequality If(x, Y, Z, Q) - f(x, Y, z, Q) I < Q(x, J . - 2 l) + M , jqj- qj I
j=l for x > 0 in (Z. 1), where o(x, Y) is continuous and non-negative for y > 0, 0 < x < a. Suppose that the unique integral y(x) of the equation d Tx-
= U(x, y)
lim y(x) = 0,
xi0 is identically zero.
y'(0) = 0
Then the solution z(x, Y) of the Cauchy problem
-12-
z(0, Y) = p(Y) for the equation (13.2), which is of class C1 in (Z.1), is unique.
CHAPTER II Characteristic strips of first order.
g 3.
Consider a general first order partial differential equation (1. 3)
where X = (x1,..., xn),
F(X, z, zX) = 0, zX = (z x ,..., zx ) n
1
and F is supposed to be of class
CI in some domain SZ. Denote the projection of D.
For a fixed point (X, z) E D
(Z. 3)
Sl
on the space (X, z) by
consider the equation for P = (pl""' pn)
F(X, Z, P) = 0.
Equation (2. 3) possesses in general an infinite number of solutions P =
(p1pn). For a given solution P let us consider the n-dimensional plane in the space (X, z) passing through the point (X, z) n (3. 3)
z-z-
pi(xi - xi) = 0 . i =1
The family of planes (3. 3) with P = (p1,..., pn) satisfying (2. 3) admits in
general an envelope, which is a cone with the vertex at the point (X, z). Let us apply Lagrange's method for finding the equation of this cone. To this effect we put the derivatives of the function
P . (
- xi) + XF(X, a, P)
J =l
-13-
,
with respect to the variables pi, equal to zero and obtain Xi - xi = X F
(4. 3)
(i = 1, 2,..., n) .
i
Eliminating the parameters (p1...., pn) and X from (2. 3), (3. 3) and (4. 3) we obtain the equation of the Monge's cone with vertex (X, z). By (3. 3) and
(4. 3) the straight lines generating the cone are given by the parametric equations Xi - xi = X Pi (X, z, P) (5. 3)
(i = 1, 2,..., n)
n J =l
j
where P satisfies equation (2. 3). The direction of such a straight line is called characteristic direction defined by the equation (1. 3) at the point (X, z). A curve
xi =xi(t)
(i=1, 2,...,n) , z = z(t)
in the space (X, z) is called characteristic of the equation (1. 3) if at each
point it is tangent to one of the characteristic directions. Linear case.
Suppose now that (1. 3) is linear, that is to say
F(X, Z' P) = ) aj(X) pi j4=1
Equations (5. 3) assume then the form xi - xi = Xai(X)
z - z =LLpjaj(X) -14-
Lpl al(X) = 0, whence
with l
xi - ii = Xai(X), z- z = 0 . This means that in the linear case the Mange's cone degenerates to a single (6. 3)
line (6.3) parallel to the plane z = 0. Hence it follows that in the linear case
all the characteristics are parallel to the plane z = 0 and can be obtained from those situated in the plane z = 0 by a parallel shift in the direction of the taxis. Therefore, in order to have a complete picture of characteristics in the
linear case it is sufficient to consider the curves xi=xi(t) satisfying the system of ordinary differential equations, dxi
3t = ai(xl,...,xn)
(7. 3)
called characteristic equations for the linear equation n
=0
) aj(X) zx
(8. 3)
i
J=l Integrals of (7. 3) are called, likewise, characteristics of (8. 3). Now it is easy to prove the following Theorem 1. 3. Suppose ai(X) are continuous in a domain G. Then a function z(X)
of class C1 in G is a solution of (8. 3) in G if and only if z(X) is
constant along every characteristic of (8. 3), that is to say along every integral of (7. 3).
Construction of the solution of the Cauchy problem for a linear equation by
means of characteristics. Consider the linear equation in the normal form n (9. 3)
bl(x, y) zy = 0
zx +
3
J=1 -15-
with the initial condition z(x, Y) = (Y)
(10.3)
The characteristic equations assume now the form dyi
(11.3)
= bi(x, yl,.., yn)
(i = 1, 2,..., n)
Theorem 2. 3. Suppose that bi(x, Y) are of class C1 in a domain G and
that the n-dimensional parallelepiped x=x,
(R)
I yi
- yi I < ai
(i =1, 2,..., n)
is contained in G. Let w(Y) be a function of class C1 in R. Denote by Z(R) the subdomain of G composed of points situated on integrals of (11.3)
issued from the points of R. Under these assumptions there exists a unique solution of (9. 3) satisfying (10. 3) which is of class C1 in Z(R) . Proof.
Denote by yi = pi(x' ' ql""' .n) the unique integral of (11.3)
passing through the point (I;, rill..., tin). The functions fpi(x, C, H) are of
class C1 in the domain of their existence ([ 2 ]
,
p. 155, Satz 1).
Hence and by the definition of the domain Z(R) the functions cpi(x, x, Y) are
of class C1 for (x, Y) E Z(R). Consequently the function (12.3)
z(x, Y) = W(cpl(x, x, Y Y ) ,--
. Tn(x, x, Y))
'is of class C1 in Z(R). Since qi(x, z, Y) = yi, the function z(x, Y) satisfies the initial condition (10.3). Let Y = Y(x) be an arbitrary integral of (11.3) contained in Z(R) and denote by
yn) the point Y(x). Then, by
definition of functions (pi we have (pi (x, x, Y (x)) = yi and hence
-16-
(i =1, 2...., n)
z(x, Y(x)) = W(1f) = const.
We have thus proved that z(x, Y) is constant along each integral of (11.3)
contained in Z(R). By theorem 1. 3 it follows that z(x, Y) satisfies equation (9. 3) in Z(R).
To prove the uniqueness of the solution suppose zl(x,Y) and z2(x,Y) are two solutions of (9. 3) and (10.3) which are of class Cl in Z(R) and let
(x*, Y*) be an arbitrary point of Z(R). Denote yi = cpl(z, x*, Y*) or, more briefly, Y = 4, (c, x*, Y*). By theorem 1. 3 and by the initial condition (10.3) we have z1(x, 40, x*, Y*)) = z1(x, 4, (x, x*, Y*)) = W(4, (x, x*, Y*)) = W(Y)
(13.3)
z2(x, $(x, x*, Y*)) = z2(x, 4, (x, x*, Y*)) = W(4 (x, x*, Y*)) = W(I)
for all x's for which 4,(x, x*, Y*) is defined. Since by the definition of (pi we Y* = cj,(x*, x*, Y*), it follows from (13.3), if we put in particular
have x = x*,
that
zl(x*, Y*) = z2(x*, y*) = W(Y)
,
which completes the proof of the uniqueness. Remark.
The geometrical meaning of the construction of the solution (12. 3)
is very simple. For every (9, H) E R consider in the space (x, Y, z) the curve (14.3)
Yi = p1(x, 6, H),
z = W(H) .
Then the curves (14.3) generate the surface z = z(x, Y), where z(x, Y) is defined by the formula (12.3). Characteristic strips.
Let z(X) be a solution of the equation (1. 3), which is of class C2 in -17-
a domain G. and consider the system of equations dx (15. 3)
ZTF
= pi(X, z(X), a
For any integral X(t) _ (xi(t),..., xn(t))
(X))
of (15.3) the curve df = z(t)
z = z(X(t))
X = X(t),
(i = 1, 2,..., n)
is a characteristic of (1. 3) lying on the surface z = z(X). The planes tangent
to this surface along this characteristic are determined by the coefficients pi(t) = zx (X(t)) i
P(t) = zX(X(t))
(16.3)
(i = 1, 2,..., n). Put P(t) = (pl(t),..., pn(t)), then we have and by (15.3) dz t)
irt-
j[=1
pj(t) Fp (X (t), z(t), P(t)) j
.
Differentiating pi(t) we get (17.3)
dpi
_
zx x(X(t));FP (X(t), z(t), P(t)) =1
ij
j
On the other hand we have for X e G the identity F(X, z(X), zX(X)) - 0
and hence differentiating with respect to xi and then substituting X = X(t) we obtain
(18.3) (X(t),z(t),P(t)) +Fz(X(t),z(t),P(t))Pi(t)+
.
(X(t),z(t),P(t))zxj x(X(t)) =0. i
j=1
Subtracting (18.3) from (17.3) we finally get, in view of zx ixj = zx x
ji
-18-
'
dp
ti- = - F (X(t), z(t), P(t)) - Pi(t) F (X(t), z(t), P(t))
(19.3)
i
Thus, by (15.3), (16.3), (19.3) we have arrived at the conclusion that functions
X(t), z(t), P(t) just defined satisfy the following system of ordinary differential equations dxi
(i = 1, Z,..., n)
Fp (X, z, P)
i
n (20.3)
dz
=,
J=1
dpi
pi P (X' Z' P) J
F (X, z, P) - Pi Fz (X, z, P) i
(i = 1, 2,..., n)
called characteristic equations of (1. 3). Any solution X(t), z(t), P(t) of (20.3)
is called characteristic strip of first order, or briefly, characteristic
strip of (20.3) and can be regarded as either a curve in the 2n+l dimensional space (X, z, P) or as a curve in the n+l dimensional space (X, z) (21.3)
X = X(t),
z = z(t)
with n- dimensional lplanes, determined by the coefficients pl(t),..., pn(t),
assigned to each point of this curve. For a fixed t the point (X(t), z(t)) and the plane determined by the coefficients pl(t),..., pn(t) and passing through
this point will be called element of contact. An element of contact is called integral element of contact (with respect to (1. 3) ) if F(X(t), z(t), P(t)) = 0.
A characteristic strip is called integral characteristic strip if all its elements are integral elements. Now, we have the following
-19-
Theorem 3. 3. A characteristic strip is integral if and only if at least one
of its elements is integral. Proof.
The proof follows immediately from the formula F(X(t), z(t), P(t)) = 0
which is a consequence of equations (20.3). X(t), z(t), P(t) being an integral characteristic strip the curve (21.3) is a
characteristic called characteristic corresponding to the respective character-
istic strip. 4.
Local existence of the solution of the Cauchy problem for a non-linear
equation.
We consider the equation in normal form z x = f(x, Y, z, zY) ,
(1.4)
where Y = (y1,..., yn) and f is of class C1 is some domain 61, with the initial condition z(x, Y) = w(Y)
(2.4)
Since in the present case
.
the characteristic equations (20.3)
of = 1,
now take the form n
T = -if (X# Y, z, Q), i
=p
qj fq (x, Y, z, Q)
1=l
(3.4)
ax - -f yi (x, Y , z, Q) f' qifz(x, dqi
where Q
Y,
z, Q),
1
dpf Tx = x(x,
Yi
z, Q )
t P f7,%X' Y , z, Q
_ (ql,..., qn).
If we consider integral characteristic strips (Y(x), z(x), Q(x)), i. e. characteristic strips satisfying the equation -20-
p = f(x, Y. z, Q)
(4. 4)
,
it is sufficient to take into account the system of ordinary equations dyi
= -f 4i(x, Y, z, Q)
dz
-
-
n)
(i n
(5.4)
Ux = f(x, Y, z, Q)
gjfq (x, Y, Z' Q) j=1
dqi
= f y ( x, Y , z, Q )
CTx
i
+
qi f z ( x, Y . z, Q)
(i =1 , Z ,..., n )
.
called characteristic equations for the equation (1. 4). Indeed, from (3. 4)
and (4. 4) follows (5. 4).
On the other hand, if for any solution Y(x), z(x), Q(x)
of (5.4) we put p(x) = f(x, Y(x), z(x), Q(x)),
then (4. 4) holds true and equations (3. 4) are satisfied since dq.
dy
= fx+fyl
+fz dx+fqa-JRgjfz)
Definition 1. 4.
fx+pfz
A function z(x, Y) which is of class CI in some domain D is
said to be generated by characteristics of (1. 4) if for every (x*, Y*) E D there exists a solution (6.4)
(7.4)
(Y(x), z(x), Q(x)) of (5. 4) such that
Y(x*) = Y* z(x) = z(x, Y(x))
in some neighborhood of x*. We have the following -21-
Q(x) = zY(x, Y(x))
Theorem 1. 4. For a function z(x, Y) of class C1 in some domain D to be
a solutioin of (1. 4) it is sufficient that it be generated by characteristics. This condition is also a necessary one for a solution z(x, Y) of class Proof.
C2.
Sufficiency. Suppose z(x, Y) to be of class CI in some
1)
domain D and to be generated by characteristics of (1. 4). Let (x*, Y*) be
an arbitrary point in D. Then, by our assumptions there exists a solution (Y(x), z(x), Q(x)) of (5. 4) satisfying (6. 4) and (7. 4). From (5. 4) and (7. 4) it
follows that
J =l
yJ
Y(W), z(W), Q(W))
(x, Y(W)) f
q
j
and hence, since
dy
-u
fq (x, YW(), zM(), QW()) J
we get
zx(x, Y(x)) = f(x, Y(x), z(x, Y(x)), zY(x, Y(x)) .
In particular for x = x* we have by (6. 4) zx(x*, Y*) = f(x*, Y*, z(x*, Y*), z
(xY*) being an arbitrary point in D the last equality completes the proof of sufficiency. 2)
Necessity.
Let z(x, Y) be a solution of (1. 4) which is of blass
C2 in some domain D and let (x*, Y*) be an arbitrary point of D. Consider the system of ordinary differential equations (8. 4)
dyi -T
fgi(x, Y. z(x, Y), zy(x, Y))
(1 = 1, 2,..., n)
1. The necessity of the condition for a solution of class Cl will be discussed in paragraph 5. -22-
The right-hand members of (8. 4) being continuous in D, there exists a solution Y(x) of (8. 4) satisfying (6. 4) and defined in some neighborhood N of x'0`.
Now, put z(x) = z(x, Y(x)),
(9.4)
Q(x) = zY(x, Y(x)) .
Then, (6.4) and (7.4) are satisfied for x E N. From (1. 4), (8. 4), and from the first formula (9. 4) differentiated with respect to x it follows that the first n +1 equations (5.4) are satisfied by Y(x), z(x), Q(x) for xE N. It
remains to be proved that the last n equations (5.4) are satisfied too. Now, by (8. 4) and (9. 4) we have d (10.4)
n = zy x(x, Y(x)) zy (x, Y(x)) fq (x, Y(x), z(x), Q(x)) y i 1 j=1 i
But, substituting z(x, Y) into (1. 4) and differentiating the identity thus obtained
with respect to yi we get zxyi (x, Y) = f (x, Y, z(x, Y), zy(x, Y)) +
Y, z(x, Y), zY(x, Y))zyi (x, Y)
yi n
+ L fq (x, Y, z(x, Y), zY(x, Y)) zy Y j =l
(x, Y)
') 1
)
Substituting in the last formula Y = Y(x) we obtain from (10.4) dqi = MR-
fyi x, Y(x), z(x), Q(x)) + qi(x) f. (x, Y(x), z(x), Q(x))
for x e N, which completes the proof. According to theorem 1. 4, solutions of equation (1. 4) which are of class
CZ have to be looked for among th, functions generated by characteristics of (1. 4).
Now, we are going to prove a local existence theorem concerning the
solution of the Cauchy problem (1. 4) and (Z. 4). -23-
We will construct the solution
by means of characteristics. Theorem Z. 4.
Let f(x, Y, z, Q) be of class C2 in a neighborhood of the
point (z, Y, z, Q). Suppose that w(Y) is of class C2 in a neighborhood of
the point Y and that w(Y)
,
wyi (Y) = qi
(1 = 1, 2,..., n)
Under these assumptions there exists a unique solution of (1. 4) and (2. 4)
which is of class C2 in a neighborhood of the point (x, Y). Proof.
The uniqueness of the solution follows immediately from the
theorem 3. 2.. Now, to establish the existence of the solution, denote (11.4)
yi = yi(x, 9. H, to A),
z = z(x, to H, to A),
qi = qi(x, t, H, t , A)
,
where H = (nl""' nn)' A = (X1""' fin)' the unique solution of the system (5.4) passing through the point (f, H, ,, A), i.e. such that (12.4)
yi(f.+, to H, to A) = ni,
z(t, to H, to A) _ t ,
t, H, to A) = ki
The functions (11.4) are of class Cl in some neighborhood of the point
x = t = z , H=Y,
t=z,
A =d ([2], p. 155, Satz1).
Now, put Yi(x. H) = yi(x, x, H, w(H), (jy(H)) (13.4)
z(x, H) = z(x, x, H, OH), wY(H)) qi(x, H) = qi(x, x, H, w(H), wY(H))
The functions (13.4) are of class Cl in some neighborhood of the point x =x, H= Y (14.4)
and by (12.4) they satisfy Yi(x, H) = ni,
z(x, H) = w(H), qi(x, H) = wYi (H) -24-
By (5.4) we have
r __ fqi '
(15.4)
where, for example,
f
_ 8x = f Oz
,
-
qj fq
q
8x = fyi+ qi f z
denotes f(x, Y, z, Q) with the substitution
Y = Y(x, H) _ (Yl(x, H)
H),..., qn(x, H))
H)
Consider now the system of equations Y = Y(x, H)
(16.4)
Since, by (14.4)
D(Yl,..., Yn)
Y(x,Y) =Yr
D(n1,..., Tin)
the system (16.4) admits a unique solution (17.4)
H = H(x, Y)
l(x, Y),..., nn(x, Y))
which is of class C1 in some neighborhood V of the point (z, Y) and satisfies H(x, Y) = Y
(18.4)
Put
z(x, Y) = z(x, H(x, Y))
(19.4)
for (x, Y) a V. We shall prove that z(x, Y) is a solution of (1. 4) and (2. 4) which
is of class C2 in V. Now, z(x, Y) is obviously of class Cl in V and by (14.4) and (18.4) satisfies (2. 4). To prove that it satisfies equation (1. 4) we will show first that (20.4)
zyi(x. Y) = qi(x, H(x. Y)) (i = 1, 2,...oil) -25-
Now, by (19.4) the last formula is equivalent to 8yi = qi(x, H(x, Y))
an (x,
which in turn is equivalent (by the substitution Y = Y(x, H)
and in view of
H(x,Y(x,H)) =H) to 8
H) .t
(21.4)
= qi(x, H)
Y=Y(x,H)
i Differentiating the identity
ni = j(x, Y(x, H))
with respect to ilk we get 8yi
ajk
Y=Y(x,H) a "k
8yi
i
ayayi
Hence multiplying (21. 4) by
and summing with respect to i we obtain a-
k
Y
equivalent equations (since in some neighborhood we have Det(8) $ 0 ). (22.4)
nl(x,H) =
8
^^
gi(x,H) enk
(k=1,2,...,n)
Thus, in order to prove (20.4) it is sufficient to show that (22.4) holds true.
To this effect put for fixed H _
q
(x) -
8yi qi 8r7k
nk
Differentiating with respect to x we get
_
dcpk
cTx =
r1kx -
2^ 8qi x
8yi
'qk -
-?6-
8
qi i
yi
Now, by (15.4) we have ey
zxn k
= L fy any + fz zn + k
j
k
i
8qi fyi+ gifz
ax
a2Y i 8xank
ank
( fqi
a2yi aqk 8x
Since z nkx '
are continuous and consequently
zr)kx = zxgk ,
a2-yi a 2yl allk ax = axarlk
the last four formulas give dyk -C-f
= fZ pk
Hence q)k(x) satisfies a linear, homogeneous, differential equation, and
since
by (14.4) q)k(x)
=
71
(
,H))-
qj(x, H)
ayi enk x=
=w
(H) - W
nk
'7k
(H) = 0
,
we finally get cpk(x) - 0, and thus (22.4) and consequently (20.4) is proved.
Now we can prove that z(x, Y) satisfies (1. 4) in V, i. e. (23.4)
zx(x, Y) = f(x, Y, z(x, Y), zY(x, Y))
in
V.
By (19.4) and (20.4) the last formula is equivalent to zx(x, H(x, Y)) +
\
8 n.
zq (x,H(x,Y)) ax = f(x, Y, z(x, H(x, Y)), Q(x, H(x, Y))) .
Substituting in the last formula Y = Y(x, H) we get an equivalent equation -27-
an
=f
(x, H) 8x (x,
zx(x, H) +
(24.4)
.
Y=Y(x,H)
Hence, instead of (23.4) it is sufficient to prove (24.4). Now, by (15.4) and (22.4) we have 8yi 8nj
an
(z5.4)
x(x,H) +
zn(x,H) 8x
Y=Y
i
j
8tjj
$x
j
Y =Y
On the other hand, differentiating with respect to x the identity yi(x, H(x, Y)) = yi we get a
yi 8x H=H(x,Y)+
whence, putting Y = Y(x, H),
8yi
8tj 0
aqj H H(x,Y) we obtain
ayi
851 ax + Z'-'iia8x j j Y=Y(x,H)
0
and finally by (15.4) ayi
an
=f
anj Y= ,F
qi
The last formula together with (25.4) gives (24.4) and consequently (23.4). By (20.4) and (Z3.4) the derivatives Remark 1. 4.
zxx , zxvj,
z yiyj
are continuous in V.
Suppose the inverse transformation (17.4) is defined and of class
Cl in some domain D of the space (x, Y). Denote by D the image of D by means of the transformation x = x, H = H(x, Y) and assume that D(yl(x, H),..., yn(x, H)) 4 0 D(n1,..., nn)
_2g_
in D .
Then, it is easy to check that the argument of the proof of theorem Z. 4
holds true in D (instead of V) and therefore the formula (19.4) defines a solution of (1. 4) and (2. 4) which is of class C2 in D.
We will take advantage of this remark in the discussion of the local existence of a solution of the Cauchy problem (1. 4) and (2. 4) "along a char-
acteristic". Local existence of a solution of the Cauchy problem "along a given charac-
teristic". Consider the equation (1, 4) with the initial condition (2. 4) and introduce the following
Assumptions B.
f(x, Y, z, Q) is of class C2 in a domain Sl and w(Y) is
of class C2 in a neighborhood of the point Y, where (x, Y, w(Y), wy(Y)) a SZ Let (26.4)
Y = Y(x), z = z(x),
Q = Q(x)
be a characteristic strip of (1. 4), i. e. a solution of the system (5. 4), defined in the interval (27.4)
al < x < a2
and satisfying the initial conditions (28.4)
Y(x) = Y,
z(x) - w(Y),
Q(x) = wy(Y)
We shall solve the following problem. Problem 1. 4. Establish the necessary and sufficient conditions for a solution
of (1. 4) and (2. 4) to exist and to be of class C2 "along the characteristic" (corresponding to the strip (26.4)) ,
-29-
Y = Y(x),
(29.4)
al < x < a2
i. e. in a domain containing the curve (29.4). We will prove first the following Lemma 1. 4.
Suppose that the curve (29.4) is continuous and that Y(x, H)
is of class Cl in a domain U containing the segment
al<x
(30.4)
D(Y1(x, H),..., yn(x, H))
(31.4)
D(r71,..., rin)
+t
0,
a1
<x < a2
H=Y
Under these assumptions the transformation Y = Y(x, H)
(32.4)
admits a unique inverse transformation H = H(x, Y)
(33.4)
which is of class C1 in a domain D containing the curve (29.4) and satisfies
H(x,Y(x)) = Y,
(34.4)
al < x < a2
.
Denoting by 15 the image of D by means of (33.4) we have furthermore D(Yl(x, H),..., Yn(x, H))
(35.4)
D(ri1,..., rln)
t+ 0
for (x, H) E D.
Let e > 0 be an aribtrary number such that a1 + e < a2 - e .
Proof.
We shall prove first that there exists an rlE> 0 such that transformation (32.4) is inversible 1.
1,
in the domain
By this we mean that (x,H) c VE, (x,H') a VE and Y(x,H)=Y(x,H*) imply H =9 -30-
aI + E < x < a2 - E,
IH-YI <
nE
.
Suppose the contrary; then, there would exist sequences {xk},
{Hk
{Hk}
such that Hk + Hk
,
Y(x,k, Hk) = Y(34k, Hk)
lk Hk = lim Hk = Y,
lim xk = x,
aI + E < x < a2- E
Hence, by the standard argument we would obtain
D(yl(x, H),..., 'n(x, H)) D(nl,..., nn)
= 0
,
IH=Y
which contradicts assumption (31.4).Now, let EV > 0 be a sequence tending
to zero and put
CO
V= U VEV V=1
V is then a domain containing the segment aI < x < a2, H = 7C and transformation (32.4) is inversible in V. In virtue of (31.4) we can assume that the were chosen so that (35.4) holds for (x, H) E V.
domains VE V
Now, in virtue of the theorem on implicit functions it follows from (30.4) and (31.4) that for each x*, aI < x* < a2 there exists an inverse transforma-
tion of (32.4), say H = H(x*; x, Y)
which is of class CI in a neighborhood D(x*) of the point (x*, Y(x*)) and
satisfies (36.4)
H(x*;x*, Y(x*)) = Y
.
We can assume D(x*) to be so small that (37.4)
for
(x, li(x*;x, Y)) E V
Denote D=
U
a1<x*
(x, Y) E D(x*) D(x*)
then D is a domain containing the curve (29.4). If we put H(x, Y) = H(x*;x, Y) for (x, Y) a D(x*)
then, by (37.4) and by the property of the domain V, H(x, Y) is a uniquely
determined inverse transformation of class C1 in D. By (36.4), (37.4) and the choice of the domain V , (34.4) and (35.4) hold true. Theorem 3. 4. [ 7 ]. Let assumptions B be satisfied, and denote
z = z(x, H),
Y = Y(x, H),
(38.4)
Q = Q(x, H)
the solution of (5. 4) satisfying initial conditions (39.4)
'fix*, H) = H,
z(c, H) = W(H),
Put
uij (x) _
Q(x, H) = wY(H) .
8yi(x, H)
n
14 =i
Under these assumptions the necessary and sufficient condition for a solution of (1. 4) and (2. 4) to exist and to be of class C2 "along the characteristic"
(29.4) is that Det(uij(x)) + 0,
(40.4)
Proof.
al < x < a2
.
Since the solutions of (5. 4) are uniquely determined by their
initial values, we have by (28. 4) and (39. 4) Y(x, Y) = Y(x),
(41.4)
Therefore,
z(x, Y) = z(x),
Q(x, Y) = Q(x) .
Y(x), z(x), Q(x) being defined in the interval (27.4), it follows
that the functions (38.4) are defined ([ 2] , p. 149, Satz 3) class Cl ([ 2 ], p. 155, Satz 1)
and of
in some domain U containing the segment
al<x
Then by (40.4), (41. 4) and
by the properties of the functions Y(x, H) stated above all the assumptions of lemma 1. 4 are satisfied. Hence the transformation (32.4) admits a unique inverse transformation (33.4) which is of class C1 in a domain D containing the curve (29.4) and satisfies (34.4). Furthermore (35.4) holds true. Hence, it follows by remark 1. 4 that formula (19.4) defines a solution of (1. 4) and (2. 4) which is of class C2 in D. 2) Necessity.
Suppose z(x, Y) to be a solution of (1. 4) and (2. 4) which
is of class C2 in some domain D containing the characteristic (29.4). Denote by Y = Y(x, H) the solution of the system (8. 4) satisfying the initial condition
Y(z,H) = H.
(42.4)
Then, by the second part of the proof of theorem 1. 4, z(x, Y) being a solution of class C2, the functions Y(x, H), z(x, Y(x, H)), zy(x, Y(x, H)) satisfy equations (5. 4). We have furthermore by (2. 4) and (42.4) (43.4)
zY(x. Y(x, H)) = WY(H) .
z(x, Y(x, H)) = W(H),
The solutions of (5. 4) being unique it follows from (39.4), (42.4) and (43.4) that Y(x, H) = Y(x, H), z(x, Y(x, H)) = z(x, H), zY(x, Y(x, H)) = Q(x, H)
(44.4) and hence
Q(x, H) = zy(x, Y(x, H)) ,
z(x, H) = z(x, Y(x, H)),
for (x, H) E U. Consequently we have dyi
fgi(x, Y(x, H) z(x, Y(x, H)), zY(x, Y(x, H)))
for (x, H) a U. Differentiating the last formula with respect to nj and putting H
we get by (41.4) and (44.4) that for every fixed j the functions u1.] ,(x)
(1 = 1, 2,..., n) satisfy in the interval (27.4) the following linear system of or-
dinary differential equations with continuous coefficients -33-
n
au (45.4)
fq
k=1
zyk(x,Y(x)) - ' fq q (Z) zy y (x,Y(x)]uk (Z) y - fqiz(Z) ik m im mk
where Z = (x, Y(x), z(x), Q(x)). Now, in view of (39.4) we have Det(uij(z))
and hence by the classical theorem we obtain (40.4). We will state now without proof a theorem by T. Wazewski [14] concerning the existence and the estimation of the existence domain of the solution of the Cauchy problem (1. 4) and (2. 4).
Theorem 4. 4. Let f, f
Ix-xI < K1,
yl
,
fz , f
qi
be continuous in the domain
Iyi - yiI < K1,
Iq1-q1
Iz-zI
,
and satisfy the inequalities
Ifl, Ifyi1, IfzI, Ifgil < Ml Assume that f
yi
,
fz and fq satisfy a Lipschitz condition with respect to Y,z,Q i
with the constant Ml.
Let w(Y) be of class C I in
Ix-xI
< Kl
and satisfy inequalities wyi I
< Ml,
I qi l
< Ml,
I w(Y) - z I <
K .
I wyil'i) -4k! <
-
,
where K < K1 . Suppose finally that wyi satisfy a Lipschitz condition with the
constant Ml and set K2
b =
[(n+i)(M+K+1 where M1< M.
Under these assumptions there exists a unique solution z(x, Y) of (1. 4)
and (2. 4) which is of class CI in
-34-
Ix-xl <
IYi-
b,
Yil
4
- Mlx-xj
t
n
Moreover zy (x, Y) are Lipschitzian with respect to Y and z(x, Y) is i generatedl)y characteristics corresponding to characteristic strips Y(x,H)
z(x, H), Q(x, H) determined by initial values (46.4)
Y(ic, H) = H,
, f" f
The assumption that f
Q(x, H) = wY(H)
z(1, H) = w(H), qi,
and w
Yi
yi
be Lipschitzian is essential in
theorem 4. 4. Indeed, the following examples [14] show that if this assumption
fails to be true, then it may happen that there does not exist in any nieghborhood of (x, Y) a solution of (1. 4) and (2. 4) which belongs to class Cl and is
generated by characteristics determined by initial values (46.4). Example 1. 4. Consider the equation (47.4)
zx(x, Y) = (zy(x, Y))2
and let w(y) be a function of class C1 in a neighborhood of 0, such that w'(y) does not satisfy a Lipschitz condition in any neighborhood of 0. There
and { nv} such that
exist then two sequences nv
: 0, r1v _ 0, nv T nv,
0(n v ) -w'(nv ) wl(nv) T w,(nv),
Denote by C(ii) the characteristic strip y(x, ii), z(x, ri), q(x, n),
satisfying
initial conditions z(0, n) = w(n),
Y(0, rl) = n,
q(0, n) = w' (n)
It can be easily checked that Y(x, n) = n - 2w'(n)x
Now, we will prove that there does not exist in any neighborhood of (0, 0) a solution of (47.4) satisfying the initial condition -35-
s oo.
z(0, Y) = W(Y)
,
which is of class Cl and is generated by characteristics y = y(x, n). To do so, assume that there exists a solution z(x, y)
having the required prop-
erties in some neighborhood of (0, 0). Then the solutions of characteristic equations being unique we would have
(48.4)
q(x, n) = z
z(x, n) = z(x, YY(x, n)),
y (X, Y_(-' TO
in a neighborhood of x = n = 0. Now, for nv -
2(() W nv - WIN we have xv - p 0
V
and
(49.4)
Y(xy, nv) = Y(xv, nv)
Hence, by (48.4) (50.4)
z(xv, nv) = z(xv, iv),
q(xy, nv) = q(xv, nv) ,
for sufficiently large v. By (49.4), (50.4), and by the uniqueness.of characteristic strips it follows that C(nv) = C(Tiv) for v sufficiently large. But, the
last equality is in contradiction to Y(0, nv) = nv++ nv = y(0, Example 2. 4.
Consider now the equation zx(x, Y) = cp[zy(x, Y)]
(51.4)
,
where T(q) is of class C1 in a neighborhood of q = 0 and cp'(q) does not satisfy :a Lipschitz condition in any neighborhood of q = 0. Then, there exist
two sequences { nv} , { nv} such that _ _ nv
Let
s 0, nv 0,
Icp'(nv)-cpI w
'(nv) ++
77V++ nv, 2
W(Y) = 2 -36-
m
nv - nv
and, as in example 1. 4; denote by C(tl) the corresponding characteristic strip of equation (51.4). Then we have j(x, rl) = 77 -gp'(i7)x rl
x
V
and for
- rlV
V
cp'(iv) - (Orly)
xv-- 0 and (49.4) holds true. Assuming now that there exists a solution z(x, y) of (51.4) having the required properties in some neighborhood of (0, 0) we would have (48.4) and
by a similar argument to that used in example 1. 4. we would obtain a contradiction.
95. General properties of characteristics and Plis's theorems. Let f(x, Y, z, Q) be of class C1 in some domain 62 and let z(x, Y) be a solution of the equation (1. 4) which is of class C1 in a domain D. Definition 1. 5. [ 8 ].
A solution Y(x) of the system of equations (8. 4) is
called basic for the equation (1. 4) and for its solution z(x, Y) if the functions (1. 5)
Y(x),
z(x) = z(x, Y(x)), Q(x) = zY(x, Y(x))
satisfy the characteristic equations (5. 4).
Definition 2.5. [8] We say that the solution z(x, Y) of (1.4) has the property P if every solution of (8.4) is basic. Remark 1. 5. Property P for a solution z(x, Y) of (1. 4) implies that z(x, Y)
is generated by characteristics (see definition 1. 4). Remark 2. 5. From the second part of the proof of theorem 1. 4 it follows
that every solution of (1. 4) which is of class C2 has the property P. We will prove now a more general theorem concerning property P.
-37-
Theorem 1. 5. Let f(x, Y, z, Q) be of class C1 in some convex domain SZ,
and let z(x, Y) be a solution of (1. 4) which is of class C1 in a domain
Suppose furthermore that z
D.
(x, Y)
yi
satisfy a Lipschitz condition
with respect to Y. Under these assumptions z(x, Y) has the property P.
Proof. Let Y(x) be an aribtrary solution of (8. 4) and set qi(x, h)
[z(x, Y(x) + Hi) - z(x, Y(x))] h-1
,
where h + 0 and Hi = (0,..., 0, h, 0,..., 0) h standing on the i-th place. We have obviously by (1. 5) (2. 5)
lim ql(x, h) = z (x, Y(x)) = qi(x) yi h0
uniformly with respect to x.
Differentiating qi(x, h) with respect to x for fixed h we obtain by (1. 4) and (8. 4) dgi(x, h)
dx
=h
-1 1
zx(x,Y(x) +H1) - zx(x,Y(x)) + +
Lz
(x, Y (x) + Hi) - z Ix,Y(x)I] y] yj
x
(3.5)
= h-1{f(x,Y(x)+Hi, z(x,Y(x)+Hi),zy(x,Y(x)+Hi)) -f(x.Y(x),z(x,Y(x)),zy(x,Y(x))
(x,Y(x)+H1) - zy3(x,Y(x))] fq](x,Y(x), z(x,Y(x)), zy(x,Y(x)))l
.
Now, setting for 0 < t < 1, (x, Y) E D Pi(x, Y, t, h) _ (x, Y+tH1, z(x,Y)+t[z(x,Y+H1)- z(x, Y)], zy(x, Y) +
+ t[zy(x, Y+Hi) - y(x, H)]), we have obviously
(4. 5) kim Pi(x,Y(x), t, h) _ (x,y(x), z(x,Y(x)), zY(x,Y(x))) = (x,Y(x), z{x), Q(x)) -38-
uniformly with respect to x and t. Applying Hadamard's mean value theorem to the extreme right-hand member of (3. 5) we can write dgi(x,h)
dx
(5.5)
1
1
J fy (Pi (x,Y(x), t, h))dt + J fz(Pi(x,Y(x),t,h)) dt qi(x, h) 0
0
1 1
zyj(x,Y(x))]h-1
fq(x, Y (x), z (x),
Since zy(x, Y) satisfy a Lipschitz condition with respect to Y the expression i
IzyJx, Y(x) +Hi)- zyJ(x,Y(x))] h 1 J
is bounded and hence by (2. 5) and (4. 5) the right hand member of (5. 5) tends
uniformly with respect to x to the limit fy(x,Y(x), z(x),Q(x)) + gi(x)fz(x,Y(x), z(x),Q(x)) i
when h -> 0. It follows that the left hand member of (5. 5) is uniformly convergent also and therefore by (2. 5) we have
um -dgi(x, x h)
h --,. 0
_dgi(x)
Thus we get finally from (5. 5) dg
- = fyi(x,Y(x), z(x), Q(x)) +gi(x)fz(x,Y(x), z(x), Q(x))
The last formula means that functions (1. 5) satisfy the last n equations (5. 4). The first n equations (5. 4) are satisfied because of (8. 4) and (1. 5). Finally we verify the n+l equation (5. 4) by simply differentiating the identity z(x,Y(x)), which completes the proof.
In theorem 1. 5 the assumption that
-39-
(x, Y)
z
yi
satisfy
a Lipschitz
z(x) _
(34
)].
condition with respect to Y is essential. The following example due to A. Plis [ 6' ] shows that a solution of class Cl may fail to have property P. 4
Example 1. 5. f(x, y, z, q)
Put f (x,
x
z,
3
and
zx
=x
+
3
and z(x, y) are of class Cl everywhere and it is easy to check
that z(x, y) satisfies everywhere the equation zx = f(x, y, z, z
y
)
Since zy = x + y1/3 and fq = 3(q - x) 2, the equation (8. 4) takes the form
V_
-3y2/3
and is satisfied by y(x) - 0. But, since fy = fz = 0, the characteristic equations (5. 4) have the form
dy = -3(q-x)2,
_ (q-x)3- 3q(q-x)2,
=0
and are obviously not saitisfied by y(x) = 0, z(x) = z(x, y(x)) ^ 0, q(x) =zy(x, y(x)) aF X.
Example 1. 5 does not exclude the possibility that every solution of
(1. 4) which is of class Cl is generated by characteristics. The first proof of this theorem given in 1928 by A. Haar [1] was not correct. It was only in
1954 that A. Plie succeeded in obtaining a correct proof of the theorem in case of two independent variables. The case of an arbitrary number of in-
dependent variables requires some additional assumptions besides the hypothesis that z(x, Y) be of class C1.
Example 1. 5 shows that in order
to prove the property of a solution of class Cl to be generated by characteristics, one has to use a different method from that consisting in establishing
property P. In connection with the problem stated above we shall prove first the -40-
Theorem 2. 5. [ 3 ]. Let f(x, Y, z, Q) be of class C1 in some convex do-
main I and let z(x, Y) be a solution of (1. 4) which is of class Cl in a Let (x*, Y*) be an arbitrary point in D. Then, for
domain D.
each fixed i there exists a solution Yi(x)
(ylx),..., yn ())
of (8. 4) satisfying the initial condition (6. 5)
Yi(x*) = Y*
and such that the functions zi(x) = z(x,Yi(x)),
Yi(x),
(7. 5)
ql(x) = zy(x,Yi(W))
i
satisfy the system of equations dye (x)
dd 1
(8.5)
(j = 1, 2...., n)
fq (x ,Yi(x), zi(x), zy(x,Yi(x)))
(rx
(x) = f(x,Y'(x), zi(x),zY(x,Yi(x))) (x,Yi(x)) f (x,Y1(x),zl(x), zY(x,Yi(x)))
y2
d
(x)
x
Proof.
qp
= fyl(x,Yi(x). zi(x), z1,(x,Yi(x)))
Pi(x, Y, t, h),
where i is fixed and h + 0, having the same
meaning as in the proof of theorem 1. 5, denote by y1(x, h)
(j = 1, 2...., n)
the solution of the system of ordinary differential equations dyj (9. 5)
J fq (P. (x, Y, t, h)) dt 0
satisfying the initial condition -41-
(j = 1, 2,..., n)
Yi(x*, h) = Y*
(10.5)
Now, setting 4i(x, h) = [z(x,Yi(x, h) +Hi)
zi(x, h) = z(x,Yi(x, h)),
(11.5)
- z(x,Yi(x, h))]h 1
we have dy3(x h) dx
(1Z.5)
1
_ - S fq (Pi(x,Yi(x, h), t, h)) dt
(j = 1, 2,..., n)
0
i dz(xxh)
(13.5)
= f(x,Yi(x, h), zi(x, h), zY(x,Yi(x, h))) 1
-
z
(x,Yi(x, h)) Yl
0
1
= jzx(x, Y'(x,
d i;Ix
f q (Pi(x,Yi(x, h), t, h)) dt 1
h) +Hi) - zx(x,Yi(x, h))
r
+ L l zy (X,Yi(x, h) +Hi) - zY1(x,Yi(x, h))]
dY1
' h) j h 1
LLL
= [f (x, Y1(x, h) +Hi, z (x, Yi(x, h) + Hi), z Y(x, Yi(x, h) +Hi) )
- f(x,Yi(x, h), z(x,Yi(x, h)), zY(x,Yi(x, h))) +
i
i
[zy1(x, Y (x, h) +Hi) - zy1(x,Y (x, h))]
dy1(x, h)
- - 9,x- r h
1
J
Applying Hadamard's mean value formula we obtain i 1 (`1 dX' h) = J fyi(Pi(x,Y1(x,h),t,h))dt + c fz(P (x,Yi(x,h),t,h))dt 9i(x,h) dyi(x h)
+
[zY1(x,Yl(x,h) +H1) -- zY1(x,Y(x,h))
)
and finally by (1Z. 5) -42-
c1
5 f g1(Pi(x,Yi(x,h),t,h))dI
1
i
(14.5)
dqaX,) =
1
J fy (P1(x,Yi(x,h), t,h)) + g1(x,h)$fz(Pi(x,Y1(x.h),t,h)) dt 0
i
0
Since the right-hand member of (9. 5) tends uniformly to -fq (x, Y. z(x, Y), zY(x. Y))
when h -a- 0, there exists a sequence
by - - 0 such that
lim Yi(x, hv) = Y1(x)
(15.5)
V -- oo
uniformly with respect to x, where Yi(x) is a solution of (8. 4) satisfying by (10. 5) the initial condition (6. 5). By (11.5) and (15.5) we have lim zi(x, hv) = z(x, Yi(x)) = zi(x)
(16.5)
V __?.co
lim qi(x, h) = z (x. YI(x)) = qi(x) V
(17.5)
yi
On the other hand, because of (15.5), (16.5) and (17.5) the right-hand members
of (13.5) and (14.5) with h = hV tend uniformly respectively to the right-hand members of the last two equations (8. 5). Hence
- ddzi(x,hy)
dgi(x,hv)
-fix
x
are uniformly convergent too, and by (16.5) and (17.5) we have lim V -0- 00
dzi(x, dx
hv)
= dzi x)
'
lim v --f OD
dgi(x, dx
hY)
=
x
Thus we see that functions (7. 5) satisfy the system (8. 5), which completes the proof.
The following two theorems are immediate consequences of theorem 2.5. -43-
Theorem 3. 5. [3]. Suppose f(x, y, z, q) to be of class Cl in a convex domain SE and let z(x, y) be a solution of equation
zx = f(x, y, z, z
)
Y
which is of class Cl in a domain D. Then z(x,y) is generated by characteristics.
Proof. Theorem 3. 5 follows from theorem Z. 5 applied to the particu-
lar case n =1. Theorem 4. 5. [3]. Let all the assumptions of theorem 2. 5 hold true and
suppose in addition that for each point
(x*, Y*)e D the system of equations
(8. 4) admits a unique solution passing through that point. Then the solution z(x, Y) has property P and consequently (see re-
mark 1. 5) is generated by characteristics.
Proof. Let (x*, Y*) be an arbitrary point in D and denote by Y(x) the unique solution of (8. 4) passing through (x*, Y*). Then obviously for
every i we have Yi(x) = Y(x)
(18.5)
(i = 1, 2,..., n)
where Yi(x) denotes the solution of (8. 4) whose existence is guaranteed by theorem 2. 5. By (7. 5) and (18. 5) we get zi(x) = z(x, Yi(x)) = Z(x, Y(x,) (if
z(x)
(i = 1. 2,..., n)
(19.5)
qi(x) = z yi
(x, Yi(x)) = z (x, Y(x)) yi
(i = 1. 2,..., n)
From (8. 5), (18.5) and (19.5) it follows that the functions Y(x), z(x) = z(x,Y(x)), Q(x) = zY(x, Y(x)) satisfy the characteristic equations (5. 4), which completes
the proof.
-44-
According to theorem 4. 5 the uniqueness of solutions of the system
(8. 4) is a sufficient condition for the solution z(x, Y) of class Cl to have
property P; but this condition is not a necessary one. We will give now
a necessary and sufficient condition for property P to hold true. To this effect let us introduce first the following
A solution z(x, Y) of (1. 4) which is of class C1 in a domain
Definition 3, 5.
D is said to have property Pl if for every point (x*, Y*) E D and for any two solutions Y1(x), Y2(x) of (8, 4) passing through (x*, Y*) we have
xlix * I zyi(x, Y1(x)) - z(x, y Y2(x)) I
(20.5)
I x-x* I-i = 0
(J
n)
Theorem 5. 5. [6]. Suppose f(x, Y, z, Q) to be of class C1 in some convex
domain a and let z(x, Y) be a solution of (1. 4) which is of class Cl in a domain D. Then property P for z(x, Y) is equivalent to the property Pl. Proof.
1)
Assume property P to hold true and let Y1(x), Y2(x) be
two solutions of (8. 4) such that Yl(x*) = Y2(x*) = Y*
(21.5)
Then, in virtue of property P the functions zyj(x, Y1(x)) and zyj(x, Y2(x)) satisfy respectively
dzy(x,Y1(x)) J
fyi(x,YY(x), z(x,YJ(x)), zY(x,Y1(x))) + dx +
yj(x,Y1(x))fz(x,Yl(x), z(x,Y1(x)), zy(x,Y1(x)))
dzyJ (x,Y2(x)) dx
=
yj
z(x,Y2(x)), zY(x,Y2(x))) + + zyi(x,Y2(x))fz(x,Y2(x), z(x,Y2(x)), zY(x,Y2(x)))
Hence, by (21.5) we have -45-
dzy (x, Yl(x))
dx
dzy (x, Y2(x))
Ix=x*
lx=x*
dx
z(x*, Yl(x*)) = z(x*, Y2(x*)).
The last two formulas imply (20.5) and thus we have proved that property Pl
follows from property P. 2) Assume now that property PI holds true and let Y(x) be an arbitrary solution of (8. 4). In order to prove that z(x, Y) has the property P,
it is sufficient to show that for an arbitrary x* the functions
satisfy characteristic equations (5. 4).
Q(x) = zY(x, Y(x))
z(x, Y(x)),
Y(x), z(x) = To
this effect denote Y(x*) = Y* and observe that by theorem 2. 5 for every fixed i
(1 < i < n) there exists a solution Y'(x) of (8. 4) satisfying (6. 5)
and (8. 5), where zt(x) and ql(x) satisfy equations (7.5) and (8.5). But, Y(x) and Yi(x) being two solutions of (8. 4) passing through the point (x*, Y*) we have by the property PI lim
x}x*
z
(x. Y(x)) - zy (x, Y'(x)) I I x- x* I = 0
Since zy (x*, Y(x*)) = zy (x*, Yi(x*)),
(i, j = 1, 2, ..., n) .
it follows from the last formula applied
j
for j = i that for every i the derivative dzyi (x, Y(x))
x=x*
dx
exists and that dzyi (x, YW())
(22.5)
dzyi (X, YiW()) dx
dx
Furthermore it is easy to check that -46-
x=x*
x=x
dyi(N) (23.5)
dx
x=x*
Since Yi(x*) = Y(x*) , zy (x*, Y(x*))
dzi x)
dz (x, YI(x) )
dz(x,Y(x)) (x)) (24.5)
(i,,)=lr2r...,n)
dx
x=x*
(i=1,2,...,n)
,
(x*,Yi(x*)) z (,,x*,
_
x=x*
x=x*
zl(x*) = z(x''`,Yi(x*)) = z(x*, Y(x*)),
r
and, in particular, q(x*) i = zy (x*, Yl(x*)) = zy
Y(x*))
i
J
it follows from (8, 5), (22. 5), (23. 5) and (24. 5) that the functions Y(x), z(x,Y(x)), zy(x, Y(x)) satisfy characteristic equations (5. 4).
This completes the proof
that property P1 implies property P. Remark.[6].
One can easily verify that property P1 holds true if f(x, Y, z, Q)
and z(x, Y) are of class Cl and satisfy the Hdlder conditions
Izy(x,Y)-zy(x,Y)I < K(LIyk - ykI)a k
j
r
ifq(x,Y,z,Q) - fq(x,Y,z,5)I < K[( Iyk - yk1)R+Iz-z1R +( Iqk-gkl)J1 i
J
where 0 < a < 1,
k
k
0 < y < 1, min[a(1+9),a(l+ay)] > I.
0 < (3 < 1,
We will give now, in the case of two independent variables, a sufficient
condition for the property P, involving only the right-hand member of equation (1. 4).
First, we will prove the following
Lemma 1. 5. [8]. If q(x) is continuous in a < x < b and (b
a
>m
then there exist numbers c, d, 0 < c < d < b -47-
such that
q(c) + m (x-c) < q(x) < q(d) + m(x - d) for c < x < d .
Similarly, if q(b) - q(a)
< m
b-a
then there exist numbers c*, d*, a < c* < d* <_ b q(c*) + m(x - c*) >_ q(x) ? q(d*) + m(x - d*)
Proof.
such that
for
c* < x < d*
For c, d it is sufficient to put, for instance, c = max {x
a < x < b, q(x) = q(a) + m(x - a)
d = min {x ; c < x < b, q(x) = q(b) + m(x - b) } Theorem 6. 5. [8]. Suppose f(x, y, z, q) to be of class C 1 in a domain IZ
and to satisfy the following conditions (25.5) (26.5)
Ifq (-, y, z, q) - fq(x, Y*, z*, q) I
fq(x, y. z, q)
fq(x, y, z, q*)
I Y - y* I + I z - z* I
for q > q*
Then, property P holds true for every solution z(x, y) of zx = f(x, Y, z, zy)
(27.5)
which is of class C 1 in a domain D.
Proof. Let z(x, y) be a solution of (27.5), which is of class C1 in some domain D and let y(x) be an arbitrary solution of (28.5)
dy
3x
fq (x , Y, z ( x, Y), zy (x , Y))
It is evident that the functions y(x), z(x) = z(x, y(x)), q(x) = zy(x, y(x))
satisfy the first two characteristic equations -48-
)
d
(29.5)
= - fq(x, y, z, q)
,
= f (x, y, z, q) - gfq(x, Y, z, q)
dq = fy(x, Y, z, q) + gfz(x) Y, z, q)
We have to show that the last equation (Z9.5) is also satisfied.
Suppose
the contrary. Then, there exists a point x0, (x0,y(x0))e D, for which this equation is not satisfied. Hence, if we put 1 = fy(x0,Y(x0), z(x0), q(x0)) + q(x0)fz(x0,Y(x0), z(x0), q(x0))
,
there exists a number R > 0 and a sequence xn --1 x0 siich that q(xn) - q(x0) (30.5)
-1 >R>0
xn _ x0
(n= 1, 2,...
Because of the continuity there is a neighborhood of he point (x0, y(x0)), say
V C D,
(31.5)
such that
Ify(x,Y)z(x,Y), zy(x,Y)) + zy(x,Y)fz(x,Y,z(x,Y), zy(x,Y)) - I I < R
for (x,y) e V
Now, there exists an interval containing x0 in its interior a < x < 9
(0 )
such that every solution y(x, t) of (28.5) passing through a point (t, y(t)),
where a < t < p, is defined in 0 and is contained in V when x e 0. For a fixed xn e 0 we have by (30.5) either q(xn) - q(x0)
xn-x0
>
1 +R
q(xn) - q(x0) or
xn -x 0
< 1-R.
We will consider the first case only. Now, by lemma 1. 5 there exists an -49-
interval [c, d] C [xn, x0] C e
such that
q(x) > q(c) + (1 + R) (x - c) _ q(x) < q(d) + (1 + R) (x - d)
(32.5)
for c < x < d. On the other hand, by theorem 3. 5 there are two basic solutions of (28.5)
(see definition 1. 5)
u(x), v(x)
satisfying the initial
conditions u(c) = y(c),
(33.5)
v(d) = y(d)
By the choice of the interval e, u(x) and v(x) are defined for x E [c, d]
and are contained in V for x E [c, d].
We will prove now that (34.5)
v(x) > y(x)
u(x) > y(x) ,
for
c<x
We will restrict ourselves to the proof of the first inequality. u(x) being a basic solution of (28.5) the functions u(x), z(x, u(x)), zy(x,u(x))
Hence, by the third equation (29.5)
satisfy the system (29.5) in [c, d]. and by (31.5) we get zy(x, u(x))
But, by (33.5) we have
<
zy(c, u(c)) + (I + R) (x - c)
.
zy(c, u(c)) = zy(c, y(c)) = q(c) { therefore, by
the last inequality and by (32.5) we ob.ain (35.5) Now,
(36.5)
c<x
zy(x. u(x)) < q(x) ,
.
z(x, y) being of class Cl there exists a constant K such that
z(x,y(x)) - z(x,u(x))I
KIy(x)-u(x)I,
The function r(x) = y(x) - u(x) fulfills the identity -50-
c<x
r1(x) = fq(x,n(x), Z(x, ulxl), 7, y(x,u(x)) - fq(x,y(x), z(x,y(x)), q(x))
and hence, by (26.5) and (35.5) we have r'(x) < fq(x,u(x), z(x,u(x)), q(x)) - fq(x,y(x), z(x,y(x)), q(x))
From the last inequality if follows by (25.5) and (36.5) that r'(x) < L(1+K) r(x)
for every x e [c, d] such that r(x) > 0. But, since by (33.5) r(c) = 0, it follows from the theorem on differential inequalities [17] that r(x) < 0 for x E [c, d], which completes the proof of the first inequality (34.5).
From (33.5) and (34.5) we conclude that there is an e, c < e = d
such that u(e) = v(e). X(x) = u(x),
If we define
c < x < e,
X(x) = v(x),
e < x < d
,
then X(x) is a basic solution of (8. 4) contained in V and (31.5) being
satisfied for y = X(x) we get from the third equation (29.5) (fulfilled by X(x), z(x, X(x)), Zy(x, X(x)))
q(d) - q(c) = zy(d,y(d)) - zy(c,y(c)) = zy(d,v(d)) - zy(c,u(c)) _ zy(d,)L(d)) - zy(c,X(c)) < (1 +R)(d - c)
.
But, from (32.5) it follows q(d) - q(c) > (1 + R) (d - d)
and thus we obtain a contradiction to the last inequality, which completes the proof.
Remark. From example 1. 5 it follows that assumption (26.5) cannot be omitted in theorem 6.5 even if
f
is a polynomial. However, it may be
replaced by the assumption that f be non-increasing with respect to q. q
Assumptions of theorem 6.5 are satisfied, in particular, for
f = a(x, y, z)q + b(x, y, z),
where a, b are of class
Cl.
Contact points of two solutions.
Let us introduce the following Suppose f(x, y, z, q) to be of class Cl in a domain St
Assumptions C.
and zl(x, y), z2(x, y) to be two solutions of (27.5) which are of class Cl
in a neighborhood of (x0, y0).
Assume that zl and z2 are tangent at
the point (x0, y0), I. e. 21(x0, YO)
= 22(x0, YO) = z0 ,
z1 (xO, Y0)
= z2(x0, Y0) = q0 ,
and let y(x), z(x), q(x) be a solution of the characteristic equations (29.5) satisfying the initial conditions
Y(x0) = y0,
(37.5)
z(x0) = z0,
q(x0) = qO
.
Now, the question arises whether the identities zl(x, y(x)) = z2 (x, y(x)),
(38.5)
z1 (x, y(x)) = z2 (x, y(x))
are satisfied in a neighborhood of x0, in other words, whether the two solutions are tangent along the characteristic y = y(x) in some neighborhood of x0.
Under the more restrictive hypothesis that f, zl and z2 are of class CZ the answer to the above question is positive and the proof is immediatel
but, the following example [9] shows that under assumptions C this is not
necessarily true, even for f being a rational function and zl, z2 being polynomials. Example.
Put
f(x, y, z, q) = 4x3y {Z - 2[1+g2(x4+ 3y2)-2]
f(0, 0, z, q) = 0. -52-
for (x, y) ++ (0, 0)
It is easy to verify that
f
is a rational function of class C1 everywhere
and that zl = 0, z2 = x4y + y3 are two polynomial solutions of (27.5)
tangent at the point (0, 0), which are not tangent at any other point. However, we can prove the following [9] Theorem 7.5.
Under the assumptions C and under the hypothesis that
the solution y(x), z(x), q(x) of (29.5) satisfying (37.5) is unique the iden-
tities (38.5) hold true in some neighborhood of x0.
Proof. By theorem 3.5 there exists a solution t7(x), (x),,y (x) of (29.5) satisfying the initial condition ri(x0) = y0 and such that (39.5)
5(x) = z1(x, rl(x)),
in a neighborhood of x0.
4(x) = zy(x, rl(M))
In particular, for x= x0 we have
5(x0) = zl(x0, y0) = z0, and hence by our hypothesis r1(x0) = y0,
(40.5)
Y1(x) = y(x),
C,(x) = z(x),
y(x0)
zy(x0y0) = q0
y (x) = q(x) .
From (39.5) and (40.5) it follows that (41.5)
zl(xi y(x)) = z(x),
zy(x, y(x)) = q(x)
In a similar way we show that (41.5) holds true for z2(x, y) and thus (38.5) is proved.
In the case of an arbitrary number of independent variables we can prove by a similar argument and by using theorem 1.5 the following [8] Theorem 8.5. Let f(x, Y, z, Q) be of class C2 in a domain Cl and let z1 (x, Y), z2 (x, Y) be two solutions of equation (1. 4) which are of class Cl -53-
in a neighborhood of (x0, YO) and whose derivatives zY, zy satisfy
a Lipschitz condition with respect to Y. Denote by Y(x), z(x), Q(x) the unique solution of the characteristic equations (5. 4) satisfying the initial conditions Y (xp) = YO, z(xO) = zl(x0, YO),
Q(x0) = 4Y (x0, YO)
Under these assumptions, if the relations z1(x, Y(x)) = z2(x, Y(x)),
z1 (x, Y(x)) = z2 (x, Y(x))
are satisfied for x = x0, then they are satisfied in a neighborhood of x0.
CHAPTER III.
J6. Characteristic strips of second order. In D4 we have given certain necessary and sufficient conditions for a solution of the equation (1. 4) to exist and to be of class C2 "along a given
characteristic". We will now treat the same problem by means of characteristic strips
of second order introduced by A. Plie [7]. In order to introduce the notion of characteristic strip of second or-
der we first prove the following Theorem 1.6. [7]. Let W*(Y) be of class C3 in a neighborhood of the point Y°`, and let f(x, Y, z, Q) be of class C2 in a neighborhood of the point (xx', YW*(YW* (Y*)).
Denote by Y*(x), z*(x), Q*(x) the solu-
tion of characteristic equations (5. 4) satisfying initial conditions
-54-
Y*(x*) = Y*,
(1.6)
Q*(x*) = W* (Y*)
z#(x ) = W (Y*),
Let z*(x, Y) be a solution of the equation,(1.4) which is of class C3 in a neighborhood of the point (x*, Y*) and satisfies the initial condition z*(x*, Y) = oil,(Y)
(z. 6)
.
Under these assumptions the functions tij(x)
z* Yj(x, Y*(x)) i
satisfy the following system of equations n
dtij
n
=
k,m=1 gkgm
fgkyi(P*) + fgZ(P*) qi`] tkj
tkitm
+
(3.6)
k=
fgkyj(P) + fgkz(P )gj ] tki+ fZ(P*) tij
+
+ fyiYj(P*) + fyiz(P*) q3 + fyjz(P*) qi + f5 Z(1'*) qi qj where P* = (x, Y* M, z*(x), Q* (X))
Proof.
.
We have by (5.4) dy*
dtij Zy
1i
(x, Y*(x)) -y V z* k=1 yiyjyk n
(4.6)
Z* Yiy
(x, Y*(x)) -
fq (P*) k_1 k
Z*
i jk(x, Y*)
On the other hand, substituting z"(x, Y) into the equation (1.4) and
differentiating twice with respect to yl -55-
and yj we obtain
,
zyiyjx
fylyj(R``)'+ fyiz(R*) z* + fy,z(R*)z* + fzZ(R*)zyiz*
=
+
n +
k =1
(5.6)
[fq y (R*) + fq (R*)zy ] zy + fz(R*) z* Yiyj ki kz i kyj
+
n + L [f q k=1
(W`) + fq (R*) z* ] zy YJ kz kyi
kyj
+
rn f
(R*)z*
q
z*
ykyi ymyj
km=1
+ L f (R*) z* yiyjyk k=1 qk
where R*(x, Y) _ (x, Y, z*(x, Y), z* (x, Y)). Now z*(x, Y) being a solution
of class C3 it follows by theorem 1.5 that the functions Y*(x), z*(x,Y*(x)), 4(x, Y*(x)) satisfy characteristic equations (5.4) and by (1.6), (2.6) they
satisfy initial conditions
Y*(x*) = Y*,
z*(x*, Y*(x*)) = W (Y*),
zY(x*, Y*(x*)) = wY(Y*)
.
But, the solutions of (5.4) being unique we get by (1.6) that z*(x, Y*(x)) = z*(x),
(6.6)
4(x, Y*(x)) = Q*(x)
Hence we have (7.6)
R*(x, Y*(x)) = (x,Y*(x), z*(x,Y*(x)), z* (x,Y*(x))) = (x,Y*(x),z*(x),Q*(x))
= p* , Substituting Y = Y*(x) in (5.6) we get by (4.6), (5.6), (6.6) and (7.6) the equation (3.6).
Before proceeding further we will prove Lemma 1. 6. Let w*(Y) be of class C2 in a neighborhood D* of Y* and ).)v(Y)
(v = 1, 2, ...) be of class C3 in D*. Assume -56-
(8. 6)
IDkw*I, IDkw1I < M
lim DkwV = Dw''
(9. 6)
(k = 0,1,2; v=1,2,...) in D* (k = 0,1,2)
uniformly in
D*
V-p.co
where Dk denotes any derivative of order k. Suppose f(x, Y, z, Q) to be of class C2 in a neighborhood I and
(x#. Y*, w*(Y "), (* (Y*))
C3 in n*.
fV(x, Y, z, Q) (v =1,2, ... )
of
to be of class
Assume
(10.6)
IDkfI' IDkfVI < M
(11.6)
lim D k fV = Dkf
(k= 0,1,2; v=1,2,...) in 12*, (k = 0,1, 2)
uniformly in SZ'
V --- 11-0c)
Consider the equation (1.4) with the initial condition (2.6) and the equation zV = fv(x, Y, zV, zV)
(1.4v)
with the initial condition zv(x*, Y) = WV(Y)
(2.6v)
Under these assumptions there exist unique solutions z(x, Y) and zv(x, Y) of (1.4), (2.6) and (1.4v), (2.6V) respectively, which are of class
C2 and C3 in a common neighborhood G* of (x*, Y*). Furthermore, we have
lim Dkzv = Dkz
(12.6)
(k = 0,1,2) uniformly in G':
V --> oo
Proof. By (8.6) and (10.6) it follows from theorem 4.4 that there
exists a common neighborhood G* in which z(x, Y) and zv(x, Y) are defined.
These solutions are constructed by means of characteristics
in the following way (see theorem 2.4). Denote by -57-
Y*(x, H), zl`(x, H), Q*(x, H) ,
(13.6)
where H = (rill..., nZ), the solution of characteristic equations (5.4) satisfying initial conditions Y*(x*, H) _
z"(x*, H) =
Q"(x", H) = w* (H)
(H),
Functions (13.6) are of class Cl in a neighborhood V*of the point x = x H = Y". Denote similarly by Y*"(x,
(13.6")
Q*'(x, H)
z*v(x, H),
H),
the solution of equations (5.4) with f
replaced by f', satisfying the
initial conditions Y*"(x*,
H) = H,
z" (x", H) = WV (H),
Q*"(x*, H) = tvY(1i)
Functions (13.6') are of class CZ in V*. By (9.6) and (11.6) we have (14.6)
lim D.z*" = D z*, lim D Q*" = DQ*
lim D.Y*V = D
V - co
V -s- m
7
j
"+co j
J
(j = 0, 1)
uniformly in V*. Since 8 y"`
Det (
J
88 rJ i
)
x = x*
we can assume, by (14.6), V" to be so small that *
(15.6)
Det (anj ),
Denote now by (16.6)
respectively by (16.6")
k"
Deta (te) > i
(" = 1, 2.... ) in V*.
the inverse transformations of Y = Y*(x, H) and Y = Y*v(x, H) respectively. H*(x, Y) and H*v(x, Y) are of class Cl and C2 respec-
tively in G* and we may assume G* to be so small that the image of G* by (16.6) and by (16.6v) be contained in V*. By (14.6) and (15.6) we have litn DJH*v. = DJH*
(17.6)
(j = 0,1) uniformly in G*.
V->oo Moreover, the relations
(18.6)
z(x, Y) = z*(x, H*(x, Y)),
zv(x, Y) = z"V(x, H*v(x, H))
(19.6)
zy(x, Y) = Q*(x,H*(x, Y)),
zY(x, Y) = Q*V(x,H*V(x, H))
hold true in G*. From (19.6) it follows that z(x, Y) and zv(x, Y) are of class C2 and C3 respectively in G*. By (14.6), (17.6), (18.6) and (19.6) we get (12.6).
Definition 1. 6. A sequence of functions Y(x), z(x), Q(x), T(x) _ (til(x),..., tnn(x))
satisfying equations (5.4) and (compare (3.6)) n
n
dtij
:' m=1fgkgmtki tmj + k=1 (fgkyi + fgk. qi) tkj
+
n
(20.6)
+
(fgkyj + fgkz gj) tki + fz tij + fyiyj k=1
+ fyiz qj + fyjzgi + fzZ gigj
+
(i, j = 1, 2,,.., n)
is called characteristic strip of second order of equation (1.4). Remark 1. 6.
Equations (5.4) are obviously independent of equations (20.6)
-59-
and can be integrated separately. If we have a solution Y(x), z(x), Q(x)
of (5.4), then, substituting in (20.6) Y = Y(x), z = z(x), Q = Q(x) we get for the unknown functions tij(x) a system of equations whose right
hand members are polynomials of second degree in tij with continuous coefficients depending only on x. Therefore, the solutions of the system
thus obtained are uniquely determined by the initial data.
Theorem 2.6. [7]. Suppose the assumptions B (§4) to be satisfied. Let z(x, Y) be a solution of (1.4) and (2.4), which is of class C2 in a do-
main D containing the characteristic (29.4). Under these assumptions functions (21.6)
Y(x),
ii
tij(x) = zy y (x, Y(x))
z(x), Q(x),
satisfy equations (5.4) and (20.6) in the interval (27.4) and the initial conditions (28.4) and (22.6)
ti,)(x)
= wYiy j(Y) .
Proof. It is sufficient to show that for any x*e (ai, a2) functions (21.6) satisfy system (20.6). To this effect, set w*(Y) = z(x*, Y) .
Y* = Y(x*),
The function w*(Y) is then of class C2 in a neighborhood D* of Y*. f(x,Y,z,Q) is of class C2 in a neighborhood a* of (x*,Y*,w*(Y*),wY(Y*))
and we can assume that for some constant M > 0 we have (23.6)
Dkw* I
<M
(k = 0, 1, 2)
in
D*
(24.6)
IDkfi < M
(k = 0, 1, 2)
in
Sl*
On the other hand, there exist sequences wv(Y) and fv(x, Y, z, Q) of class
-60-
C3 in D* and 11 * respectively such that (9.6) and (11.6) hold true. By (9.6), (11.6), (23.6), and (24.6) we can assume that (8.6) and (10.6)
are satisfied. Thus, all assumptions of lemma 1. 6 are fulfilled and hence, there exists a neighborhood G* in which z(x, Y) and the solutions
zv(x, Y) of (1.4v) and (2.6v) are defined.
zv(x, Y) are of class
C3 in G* and we have (12.6). Now,
z(x, Y) being of class C2 we have by theorem 1. 5 and by
the uniqueness of solutions of (5.4) z(x) = z(x, Y(x)),
(f being of class C2)
Q(x) = zy(x, Y(x))
,
and hence, the functions Y(x), z(x), Q(x) satisfy the conditions (25.6)
Y(x')
Ay(x*,Y*) =w* (Y*)
*, ?(x*) = z(x*,Y*) = w*(Y*), Q(x*)
Denote by Yv(x), zv(x), QV(x) the solution of system (5.4) with f re-
placed by fv, satisfying the conditions Yv(x*) =
(26.6)
Y#0
zv(x*) = wv(y*),
Qv(x*) = wYV (Y*)
These solutions exist in some common neighborhood of x* (independent
on y). Now, zv(x, Y) being of class C3 we have by theorem 1. 6 that the functions (27.6)
YV(xl, zv(x), Qv(x), til(x) = 7V (x, Y''(x)) yiyj
satisfy (20.6v) where (20.6v) is obtained from (20.6) by replacing f by fv. In virtue of (9.6), (11.6), (25.6) and (26.6) we have (28.6)
lim Yv(x) = Y(x),
lim zv(x) = z(x), v -- co
-61-
lim Qv(x) = Q(x)
v-
Co
(29.6)
lim tij(x) = lim zy v
ijy (x, Yv(x)) = zyi jy(x,Y(x)) = tij()
Now, by (11.6), (28.6), and (29.6) the right-hand members of (20.6v) with
the substitution Y = Y'(x), z = zv(x), Q = QV(x), tij = tij(x) tend uniformly to the right-hand members of (20.6) with the substitution Y = Y(x), z = z(x), Q = Q(x), tij = tij(x). Hence it follows that functions (21.6) satisfy (20.6) in a neighborhood of x*. Theorem 3. 6. [7]. Let the assumptions B (94) be satisfied and suppose
that the solution tij(x) of the system (20.6) with the substitution Y = Y(x), z = z(x), Q = Q(x), satisfying initial conditions (22.6) exists in the inter-
val (27.4). Under these assumptions there exists a solution z(x, Y) of
(1. 4) and (2. 4), which is of class C2 in a domain D containing the charac-
teristic (29.4). Proof. (30.6)
In view of theorem 3. 4 it is sufficient to prove that Dot (uij(x) + 0
in the interval al < x < a2 (see the notations introduced in theorem 3.4). Now, since Det (uij(x)) = 1,
inequality (30.6) holds true in some neighborhood of z. Let cl < x < c2
be the largest subinterval of (a,, a2)
n which (30.6) is satisfied. To com-
plete the proof, we have to show that cI = a1, c2 = a2. Suppose for instance that (31.6)
c2 < a2
.
Since all the assumptions of theorem 3.4. are satisfied in the interval
cI < x < c2, there exists a solution z(x, Y) of (1.4) and (2.4) which is of -6z-
class C2 in a domain containing the characteristic
cl <x
Y=Y(x),
.
Now, by theorem 2.6 the functions Y(x), z(x), Q(x), zyivj (x, Y(x)) satis-
fy the system (5.4) and (20.6) in the interval (cl, c2). (Hence, since the
solutions of the above system are unique (cf. remark 1.6), we have zyj (x, Y(x)) = qj(x),
zvi yj(x,Y(x)) = tij(x),
Therefore, by (45.4) for each fixed
j
cl < x < c2
the functions uij(x) (in)
satisfy in (cl, c2) the linear system n
dui (32.6)
ax =
n
r
f
k=1 L giyk
(Z) - f z(Z) qk(x) - f q q (Zl tik(x)] uk qi m=l i m
But, the coefficients of (32.6) are continuous by hypothesis in the interval ( al, a2).
Hence, there exist n solutions of (32.6) (v1j(x), ..., vnj(x))
(j = 1, 2,..., n) in the interval (a1, a2) satisfying initial conditions (33.6)
vij(z) = 6ij
and the inequality (34.6)
Det (vjj(x)) + 0,
al < x < a2
vij(x) = uij(x),
c1 < x < c2
Due to (33.6) we have
and hence, because of continuity in (a1, a2) and by (31.6) we get (35.6)
vij(c2) = ttij(c2)
From (34.6) and (35.6) it follows that
Det (uij(c2)) + 0 , which contradicts the definition of the interval (c1, c2) -63-
Remark 2. 6. It follows from the proof that under assumptions of theorem
3.6 functions ui](x)
(i = 1,2,..., n) satisfy system (32.6).
Theorem 4.6. [7]. Under the assumptions B (§4) the necessary and sufficient condition for a solution of (1.4) and (2.4) to exist and to be of class
C2 in a domain D containing the characteristic (29.4) is that the solution tij(x) of the system (20.6) with the substitution Y = Y(x), z = z(x), Q = Q(x),
satisfying initial conditions (22.6), be defined in the interval (al, a2).
Existence theorem and estimation of the existence domain.
7.
We consider a simplified equation (the right-hand member does not depend on z) (1.7)
zx = f(x, Y, zY)
with the initial condition
(2. 7)
z(ic, Y) = W (Y)
.
Theorem 1. 7. [4]. Suppose that f(x, Y, Q) and its derivatives fl. fQ
are continuous and bounded for (S)
yl < X < y2,
Y,Q
arbitrary
,
where yl < z < y2. Assume that fY, fQ satisfy a Lipschitzz condition with a constant K with respect to Y, Q and (3.7)
IfYI < A,
IfQI < C.
Let w(Y) and WY(Y) be continuous and bounded for arbitrary Y. As-
sume that WY satisfies a Lipschitz condition with a constant J and (4.7)
IWYI < D -64-
.
Denote
dl = max( Y1,x
1
n
d2 = m '-('Y2,,1"
+-n
1
n
+nJ
)
J+K(1+nJ)Ix-xI 1- nK(1 +nJ) I x-z I
Under these assumptions there exists a unique solution z(x, Y) of (1.7)
and (2.7) which is of class Cl in 61 < x < 62, Y arbitrary.
(5.7)
This solution is bounded in (5.7) and satisfies the conditions n (6.7)
I zh(x, Y) - zY(x, Y*) I < T(x)
j=1
Yj - Yj
(7. 7)
Izy(x,Y) - zy(x*,Y)l < ('A+,nCT(x))Ix-x*I
(8.7)
Izy(x,Y)lD+AIx-zd
.
Proof. Let us adopt the additional assumption that f and a are of class C2. Then, the assumptions of our theorem imply that (9.7)
IfYYI' IfYQI' IfQQI < K.
IWYYI < J.
Under these assumptions we will prove that there exists a solution z(x, Y) of (1.7) and (2.7) which is of class C2 and bounded in (5.7) and satisfies conditions equivalent to (6.7), (7.7) (10.7) (11.7)
I ZYY I
I zxy I
S T(x) ,
< A + nCT(x)
and inequality (8.7) . Since f does not depend on z, the characteristic equations
-65-
(5.4) and (20.6) take the simpler form
ax --fqi'
(12.7)
k,m
Tx- =
f qkgm tki
tmj +
ki
k (f q y tkj +
fq y tki) + fY
kj
iY j
Denote by Y(x, H),
z(x, H), Q(x, H),
t1y(x, H) .
the solution of (12.7) and (13.7) satisfying the initial conditions Y(x, H)
(14.7)
Q(x, H) = wY(H).
z(x, H) = w(H),
H,
tij(x, H) = wy
It is obvious that the functions Y, z, Q are defined and of class for
and arbitrary H. Setting
yl < x < -y2
P = (x, Y(x, H), Q(x, H)) we have dyi
.-
_ -fg(P)I
dgi dx
= fy(P) i
Differentiating the last identities with respet:t to nj we get d
- 8yk - 8gk + fq q (P) ail. J (P) an. [f4 Y k ik ik
ay
(8rij
ay
3x 8 nj) _
_
[fyiykP) 8n3 t fYigk P)
aq
k
Hence, it follows by,(9.7) d Tx-
(a - l
<
KL(ln_ k -66-
I
+
Ia
I)
ij
Y (H)
C1
d 8qi
(8n )I
8qk
By'
<_
+ Ior)
I
k
)
Since
8yi 6-,1j
X =x
8qi
= 161iI = 1,
Ollj
X=x
= Ikyy(H)I <
ij
we obtain by the theorem on differential inequalities [16] that (15.7)
8i 8
I < I (x)
'l < x < ryl , H arbitrary
for
where I'(x) is the solution of the equation
V=
ZnKy
satisfying the initial condition r (z) = max (1, J). In view of (9.7), (13.7), (14.7), we have for M(x, H) = max itij(x, H) I
i,j
ID+MI < K(n2M2+ ZnM+1), I M(z, H) I
J,
<
where D+ denotes the right derivative. Since for
< x < x + nK +n
T(x) satisfies the equation
E = K(n2T2+ 2nT+1) and the initial condition T(x) = J, it follows by the theorem on differential inequalities [16] that (16.7)
I t ij(x, H) I <
M(x,H) < T(x)
for arbitrary H and all x in the interval 61 < x < 62 for which
-67-
tij(x, H) are defined. Hence, by the theorem on the extension of the solution of a system of ordinary differential equations to the boundary of the domain ([2], p. 135, Satz 2.) we conclude that tij(x, H) are defined and satisfy (16.7) for 6l < x < 62 and arbitrary H. Now, in virtue of
remark 2.6 the functions syi(x, H)
(i = 1, Z,..., n)
uij (x, H) =
satisfy in 61 < x < 62 for every fixed du (17.7)
(Z) -
_x _ k LL
giyk
j
the system
f
m gi g m
(Z) tk(x, m H)] Uk
where Z = (x, Y(x, H), Q(x, H)) . Since
uij (x, H) = 6ij and the coefficients of (17.7) are bounded (see (9.7) and (16.7) ) by
K[1+nT(x)], it follows by Liouville's theorem that there exists a posi-
tive function P(x) such that 8yi(x, H)
(18.7)
Det (-
)
> ON >
0
for 61 < x < 62 and arbitrary H. From (15.7) and (18.7) we conclude
by Waziewski's theorem on implicit functions [15] that the transformation
Y = Y(x, H) admits a unique inverse transformation H = H(x, Y) which
is of class Cl in (5.7). Hence, puting z(x, Y) = (x, H(x, Y))
we get by remark 1.4 a solution of (1.7) and (2.7) which is of class C2 in (5.7). Boundedness of z(x, Y) follows immediately from (1.7), (2.7), and
-68-
from the boundedness of f and w. By theorem 2.6 and by the unique-
ness of solutions of the system (13.7) we get zy
y
(x, Y(x, H)) = tij(x, H)
and hence (10,7) follows from (16.7).
Differentiating equation (1.7) with respect to Y we get (11.7) by (3.7) and (10.7). z(x,Y) being of class C2 property P holds true by theorem 1.5 and hence
zy(x,Y(x,H)) = Q(x,H)
From the last identity we get by (3.7) Tx- zy(x, Y(x, H)) I =
I
TX_ Q(x, H) I
= I fy(P) I
and therefore, since
Izy(x,Y(x.H))I = IQ(x,H)I = I)y(H)I S D, we finally obtain
Izy(x,Y(x,H)I < D+AIx-xI
.
From the last inequality follows ($.7) because for every point (x*, Y'`)
in (5.7) there is an H* such that
Y* = Y(x*, H*). The uniqueness of
the solution z(x, Y) is an immediate consequence of corollary 3.2.
Now, let us turn back to the original assumptions of our theorem.
Under these assumptions there exist sequences fv(x, Y, Q) and
wv(Y)
(v =1,2,... ) such that fV is of class C2 in (S), Wv is of class C2 in the whole space, and (19.7)
lim fv = f, V --+ao
lim wv = w V -s 00
uniformly. Moreover, these sequences can be chosen so as to satisfy (3.7), (4.7)
amd (9.7).
Then, by the first part of the proof we conclude that there -69-
exist solutions
zv(x, Y) (v = 1,2,...) of (1.7), (2.7) with f,w replaced
by fv, wv, which are uniformly bounded and of class C2 in 61 < x < 62
(20.7)
Y arbitrary .
,
Moreover, these solutions satisfy (6.7), (7.7), (8.7). Hence it follows that the sequences
and
zv(x, Y)
zvl,(x, Y)
are uniformly bounded
and equicontinuous in (20.7). Therefore, by Arzela's theorem there
exists a subsequence
v
z
µ(x, Y)
(µ =1,2,...) such that
lim z u(x, Y) = z(x, Y),
(21.7)
4-1-- m
lim Y (x, Y) = zY(x, Y) µ---P. oo
uniformly in (20.7). By (19.7), (21.7) and from the equation satisfied by v
z µ(x, Y) we get (22.7)
v
lim zX (x, Y) = zx(x, Y) µ-* 00
It follows from (19.7), (21.7) and (22.7) that the limit function z(x, Y) is a solution of (1.7), (2, 7), which is bounded and of class CI in (20.7), and
satisfies (6.7), (7.7), (8.7). Remark. The strip (5.7) cannot be replaced by a larger one. Indeed,
one can construct f and w satisfying the assumptions of theorem 1.7 such that (5.7) is the exact domain of existence of the corresponding solution. F38.
Generalized Cauchy problem for a system of partial differential
equations of first order. In this paragraph we will deal with the system (1.8)
zi = f1(x, Y, Z, zY) -70-
m)
where the i-th equation involves derivatives only of the i-th unknown function .
For system (1.8) we will consider the following generalized
Cauchy problem (i= 1,2....m) ,
zi(ai, Y) = Wi(Y)
(2.8)
where ai are given constants. In particular, for aI = a2 = ... = am=;, we have the classical Cauchy problem.
Before treating the questions of uniqueness and of existence of the solution of the generalized Cauchy problem for the system (1.8), let us introduce the following Assumptions D.
fi(x, Y, Z, Q) are defined for IxI <0, z, Q arbitrary
and satisfy the Lipschitz condition m (3.8)
if i(x, Y, Z, Q) - fl(x, Y, Z, Q) I
B
j=1 where the constants B and
13
n
l .j - ;j j
+
C lgk - gk l k=1
satisfy the inequality
ZmB(3 < 1.
(4.8)
The constants
ai (im) fulfill the inequalities
(5.8)
Jail
<
13
.
Preliminary considerations. Denote by I' the Banach space of sequences of functions U(x, Y) (u1(x,Y),
... , um(x,Y))
(6.8)
continuous and bounded for
IxI < R,
Y arbitrary
with the norm -71-
IIUll
= max lsl sup lul(x.Y)l i
x
P
Assuming that assumptions D hold true consider for a given U(x,Y)4E and for a fixed i the equation with one unknown function zi.
1 = fi(x. Y, U, zi )
(7.8)
and the initial condition zi(ai, Y) = wi(Y)
(8.8)
.
Denote by zi(x, Y; U) the bounded solution of (7.8) and (8.8), which is defined and has Stolz's differential in (6.8), provided that such a solution
exists. By corollary 3.2 this solution is uniquely determined.
Now, let U and U be two elements of r for which the corresponding solutions zi(x, Y; U) and zi(x, Y; U) do exist. Then, putting
v(x, Y) = zi(x, Y; U) - zi(x, Y; U)
we easily verify by (3.8) that the following differential inequality n
lvxl
< mBllU-UII + C ) Iv k=l
holds true in (6.8) and that v(ai, Y) - 0.
yk
I
Hence, by corollary 2.1
(making L in formula (4.2) tend to zero) we get
Jv(x,Y)l < mBIIU`UII Ix-ail whence, since lxl, Jail < Q, it follows that (9.8)
I zi(x, Y; U) - zi(x, Y; u) I <
2mB/3ll'U -UII
If for a given U E F there exist for each i corresponding solutions zi(x, Y; U), then, we denote -72-
Z(U) _ (zI(x, Y{ U), ... , zm(x, Y; U)) .
We have obviously Z(U) E I, for each u c r
for which the transforma-
tion Z(U) is defined. From inequality (9.8) follows Proposition A. Transformation Z(U) satisfies the inequality IIZ(U)-z(U)II < 2mBplIU-U11
(10.8)
for any couple of elements U, U from r' for which it is defined. Obviously, we have moreover
Proposition B. Any solution U(x, Y) of (1.8) and (2.8) bounded in (6.8)
is a fixed point of Z(U), i. e.
Z(U) = U . Suppose that assumptions D hold true. Then, there
Theorem 1. 8. [5].
exists at most one solution of (1.8) and (2.8) which is bounded and has
Stolz's differential in (6.8). Proof. In virtue of proposition A and B for any two such solutions U. U
we have
IIU-511 < 2mB)I1U-UII and hence by (4.8) we get 11U - UII = 0 and consequently U = U. Theorem 2. 8. [5]. Suppose that the functions fi(x, Y, Z. Q) and their
derivatives fY, fZi
,
fQ are continuous and bounded in the strip IxI
1.
< a; Y, Z, Q arbitrary,
Under assumptions D there may exist different unbounded solutions
of (1.8) and (2.8). -73-
and satisfy the inequalities (11.8)
IfYI < A,
If'
< B,
I
IfQI
C
fy(x,Y, Z,Q) (12.8)
Ifi (x.Y,Z,Q)-fi (x,Y,Z,Q)I Ifi (x,Y,Z,Q) -fi (x,Y,Z,Q) I
t,(Y) be continuous and bounded for arbitrary Y
Let (ft(Y) and
and satisfy the inequalities (13.8)
IWYI < D
(14.8)
L)Y(Y) -4(Y)I
IYJ-YJ1
Put M = 21) + A min(4a,
= min(a,
y;-j)
,
1
4n(1+nJ)[(1+mM) IL+mB(1+2J)]
)
,
and assume that the constants ai satisfy (5.8). Under these assumptions there exists a unique solution ui(x, Y) (i = 1,2,..., m) of (1.8) and (2,8)
which is bounded and of class Cl in the strip (15.8)
IxI < R,
Y arbitrary
.
Moreover this solution satisfies the inequalities
Iuyl < M
(16.8) (17.8)
(18.8)
4Y(x,Y) -uY(x,Y)I IuY(x,Y)
-
uY(x,Y) I
(1+2J)
YJ - YjI
< [A+mBM+nC(1+2J)] Ix-XI -74-
Proof, Since obviously
4mBp <
(19.8)
1,
the uniqueness of the solution follows from theorem 1. B. To prove the
existence of the solution, filet us denote by S the set of elements
U = (ul, ... , um) a I' (see preliminary considerations) for which u i are continuous in (15.8) and satisfy (16.8), (17.8), (18.8). Since for the functions uY are uniformly bounded and equicontinuous, it follows
that the set S is closed in r. Let U(x, Y) be an aribtrary element of S and consider for a fixed i
the equation
zix = gi(x, Y, zY) ,
(20.8) where
gi(x, Y, Q) = fi(x, Y, U(x, Y), Q)
Put
K = (1+mM)2L+mB(1+2J)
(21.8)
Functions gi(x, Y, Q), gY, gQ are continuous and bounded in the strip Ixl
< P; Y, Q arbitrary.
Now, since
9Q (X, Y, Q) = fQ(x, Y, U(x, Y), Q) r
we have by (12.8) and (16.8) that gQ satisfy a Lipschitz condition with
respect to Y, Q with the constant K and
l gQ I
< C. Furthermore
we have k(x, Y,U(x,Y), Q) uY
gY(x'Y,Q) = fY(x,Y,U(x,Y), Q) + k
whence by (11.8), (12.8), (16.8), and (17.8) we get
-75-
A+ mBM
I gY I
yj-yjl +mMjy-Y31)+Iq,-qj 1)
IgY(x.Y.Q) - gY(x.Y,Q)I
y,I +mML( XIYj-YjI+mM Ivy-yjl+Iq,-q,l) i
J
]
J
Iyj -yjI
[L(1+mM) +mB(l+2J)+mML(l+mM)]
+
+(1+mM)LIq,-q,I < K(IYj-YjI +Iq,-q,l ,)
)
J
,)
Thus we see that for the equation (20.8) with the initial condition z1(ail Y) = wl(Y)
(22.8)
all the assumptions of theorem 1.7 are satisfied with x = ail
y2 = p, A = A + mBM and K given by the formula (21.8). Hence, in virtue of theorem 1.7 there exists a solution zi(x, Y; U)
of
(20.8) and
(22.8) which is of class Cl and bounded for
max(- Q, ai
nK
1+nJ ) < x < min(Q, a1+ n 1+nJ )' Y arbitrary.
But since ai
R
1
1
ai + nK +n
nK +n
zl(x, Y; U) is defined in (15.8). On the other hand, for
have by (5.8)
Ix-ail < 20
<
-76-
1
2nK -+-n J
>
IxI < 0
we
whence, by (6.7)
J+K(1+nJ)Ix-ail
I zy(x,Y; U) - 4(x,Y; U) I <
_YjI
1- nKnJ x
-aiYJ-
J
(1+2J)
I,, - YJI
Therefore, for any U E S the transformation Z(U) (see preliminary considerations) is defined and Z(U) _ (zI(x,Y; U),..., zm(x,Y; U)) satisfies (17.8). Since in our case A = A+mBM, T(x) < (1+2J), it follows
from (7.7) that (18.8) is satisfied too and from (8.7) we get 1z' (x,Y;U)I < D + (A+mBM) Ix - aiI
(23.8)
For
IxI
< 9 we have obviously Ix - ail <_ 2/3 < min(Za,
1
79 +j
)
and hence, by the definition of M and by (19.8) we get (24.8)
D+(A+mBM)Ix-ail < D + Amin(2a, + ZmBf3Amin(4a, BI
)
B+
34) + 4mB(3D +
< 2D+Amin(2a,
2
B1TLT )
It follows from (23.8) and (24.8) that Z(U) satisfies (16.8).
=M.
Thus, we
have proved that for any U E S we have Z(U) E S and since, according to proposition A, Z(U) satisfies (10.8) with 2mB3 < 1, it follows by
Banach's fixed point theorem that there is a U E S such that Z(U) = U
This U is a solution of (1.8) and (2.8) satisfying all the conditions of our theorem. -77-
CHAPTER IV Systems of equations with total differentials.
9.
Let us consider the following system of equations zi _ Piv(X, zi,..., zm) x v
(1.9)
(1 = 1,2,..., m; v = 1,..., p)
where X = (xl,..., xp), which can be written in the differential form dzi = L Plv(X, Z) dxv v=1
(1.9)
(i = 1, 2,..., m)
r
and initial conditions (i = 1, 2,..., m)
z1(X) = zl
(2.9)
System (1.9) is overdetermined (contains more equations than unknown
functions) and is called system of equations with total differentials.
Geometrical interpretation.
Let (X*, Z*) be a point of the domain A
in which Piv(X,Z) are defined and consider in the space (X, Z) the ndimensional plane zi _ zi*
(3.9)
Piv(X°'`.
Z*)(xv - x*)
(i
m)
v=1
Suppose zi(X) (i =
m) is a solution of (1.9), which is of class Ci
B
in some domain D. The n-dimensional surface in the space (X, Z) (4.9)
zi = zi(X),
m),
(i
XED
is called integral surface of (1.9). If X*E D and Z* = Z(X*), then the -78-
surface (4.9) is tangent to the plane (3.9) at the point (X*, Z*). Now, it is obvious that the necessary and sufficient condition for a surface (4.9) of class C1 to be an integral surface of (1.9) is that it be tangent
at every point to the corresponding plane (3.9). The geometrical meaning of the initial conditions (2.9) is that the
corresponding integral surface (4.9) passes through the point (X, Z).
In the particular case p =1, system (1.9) reduces to a system of ordinary differential equations. System of ordinary differential equations associated with (1.9) with a
e
respect to a given point (X, Z) and a given vector A =
x ). P
Suppose zii(X) (i= 1, 2,..., m) to be a solution of (1.9) and (2.9)
which is defined in a starlike domain with respect to )t, and introduce the following Mayer's transformation xv = xv + avx
(v = 1, 2,..., p)
,
where A = (X1,..., Xp) is a given vector. Put zi(x) = zi(X + A x)
(i = 1, 2,..., m)
The functions zi(x) are then defined in some interval
0Sx< aA and satisfy the following system of ordinary differential equations i (5.9)
= L X Pµl(l + Ax, Z)
(i = 1,2,..., m)
µ=1 and the initial conditions (6.9)
0
z (0) = zi
(i =I, -79-
M)
,
System (5.9) with the initial conditions (6.9) will be called associated with the equation (1.9) with respect to the point (X. Z) and the
vector A. We can prove now the following uniqueness theorem. Theorem 1.9. Suppose Piv(X, Z) satisfy a Lipschitz condition with re-
spect to Z in a domain containing the point (X, Z). Then there exists at most one solution of (1.9) and (Z.9) which is defined and of class Cl
in a starlike domain (with respect to X) D. Proof. zi(X), (7.9)
z1(X),
We have to prove that for two solutions of (1.9) and (2.9)
which are of class Cl in D, and for any X'ke D we have zi(X*) = 21 (X*)
To this effect put A = X* - )
(i = 1,2,..., m) .
and
zi(x) = zi(X + Ax),
z1(x) =
zl(X+Ax)
zi(x) and zi(x) satisfy system (5.9) in the interval 0 < x < 1 and the initial conditions (6.9). But, in view of our assumptions, system Then,
(5.9) admits unique solutions whence
zl(1) = zl(1)
.
which is equivalent to (7.9). Compatibility conditions. Suppose Plv(X,Z) are of class Cl in a domain SZ ; and let zl(X) be a solution of (1.9) which is of class Cl in a domain D.
Then, it follows from (1.9) that zl(X) is necessarily of class C2 in D whence -80-
(8.9)
in D.
zxVxµ = zxµxv
Substituting zl(X) in (1.9) and differentiating the v-th equation with re-
spect to xµ and the ii-th equation with respect to xv we get by (8.9) Pxv(X,Z(X)) - P1µ(X,Z(X)) + µ V
m
+ L [Pi°(x,Z(x)) P)µ(x,Z(x)) - P1µ (X,Z(X)) PJv(X,Z(X))] =0, zJ zJ i=1 If
Piv
(9.9)
-
µ
XE D .
m
p1µ + v
Pif Piv) = 0 (i = 1,2,,.., m; µ,v =
(PiV j=1
zJ
z
= 1,z..... P)
holds true for every (X, Z) E Sl , then system (1) is said to be completely
integrable in it . Relations (9.9) are called compatibility conditions. Remark.
We have actually proved that (9.9) is a necessary condition
for the existence of a solution of (1.9) passing through any given point of
it. We will prove further that it is a sufficient condition also. Geometric interpretation of the compatibility conditions.
We will give a geometric interpretation of the left-hand member (9.9) at a given point in the case p = 2,
[12].
It is obviously sufficient to restrict ourselves to this particular case. Consider, therefore, the equations (10.9)
dzi = Pi(x,y,Z) dx + Qi(x,y,Z) dy
and a given point (x,
Z).
(i= 1,2,..., m)
Denoting by Pi = Pi(x,y,Z) and similarly -81-
Ql the left-hand member of (9.9) at the point (c,y,Z) can be written in the form m
j =1
Consider now in the space (x,y,Z) the cylindrical surface x = x + r coo cp ,
(11.9)
where cp ,
y = y+ r sin T,
zl = zl + his
r > 0 fixed,
H = (hl,..., h"') are parameters, and let (q,, hl,..., hm) be a
point on (11.9).
The 2-dimensional plane determined by (10.9) at this
point has the equations zl
- (zi+hi) = Pi(x+rcosq Qi(z+rcoscp,y+rsi'tsp,Z+H)(y-y-rain;)
This plane intersects with the surface (11.9) along the curve whose equations on the surface (11.9) are hi
- hi = Pi(x+ r coo , y+ r sin;, Z +H) r (cos
cp - cos
+ Qi(x+rcoscp,y+rsincp,Z+H)r(sin cp - sin;). The tangent vector to this curve at the point cp = 4
has the components
r[Q1(x+rcoo q,,y+rsinT,Z+H)coo cp - Pi(x+rcoo 9,y+rsin;,Z+H)sincp]
Thus, the system (10.9) defines on the surface (11.9) the following system
of ordinary differential equations i (12.9)
d
= r[Q1(x+r coo 9, y+rsinT,Z+H)COST
-
- Pi(x+r coo T,y+rsinp,Z+H)sincp]. -82-
Denote by h1((p, r) the solution of (12.9) satisfying the initial condition
h1(0, r) = 0
(13.9)
.
Theorem 2.9. Pi(x,y,Z) and Qi(x,y,Z) being of class Cl in a neighbor-
hood of (1,y,Z) we have m (14.9)
Qi PJ
Qx - Py
Proof.
zi
-
Pi QJ) = lim hl(Zr, r)
r+0 r x
z.)
The right-hand members of (12.9) are, for small r > 0,
of class C1 for arbitrary p and H from some neighborhood of 0. Since for r = 0 we have by (13.9) (15.9)
hi((p, 0) ° 0
,
and the solution h1((p, 0) . is defined for arbitrary
cp,
it follows that for
sufficiently small r > 0 the functions h1(cp, r) are defined and of class C1 in the interval 0
cp
!
Zr.
Let us denote by 3Ci(cp,H,r) the expression in brackets in the righthand side of (12.9).
Then by (13.9) we have
i
h (Z7r, r)
r 2 ?r
Zr
= 1
r7r
S 3C1(cp,H(r,cp), r) dy 0
Since by (15.9) 3C1(cp,H(p,0), 0) = Q1 cos cp - P1 sin cp and consequently 27r
3C1((p'H(9,0), 0) dy 0
we can write -83-
=0
h(2r,r) r
la =
1
_ [C1(cp,H(p,r), r)
- XCi(W,H(p,0). o)] dcp
0
Now, JCi(cp,H(p,r), r) being of class Cl we have
rlim0
(16.9)
r
[3i(p,H(wr),r)
lim hl( r --r 0 r
- 3i((p,H(p,o),o)] = d 3C i(p,H((,,r),r)
r= 0
rZrr dcp
= a J0
r
.
r= 0
But, by the definition of Xi we get m (17.9)
3C1(9,H(p,r),r)
= cos p[di coe x (P
r=0
y
sin
J=1
Oi h)r (cp,0)]
z
m^^
- sing[ PXCoecp+Pyeincp+ L
hr (q),0)]
J-1
On the other hand, by (12.9) dhr(cp,0) =
'ti Cos cp - °Pi sin cp ,
dcp
and hence, since by (13.9), (18.9)
hr(0, 0) - 0,
h=(rp,0) = Qi sin cp + Pi(cos cp - 1)
From (16.9), (17.9) and (18.9) follows (14.9).
We will prove now the following existence theorem. Theorem 3.9. Suppose that Piv(X,Z) are of class Cl and that the compatibility conditions (9.9) hold true in the domain -84-
X(xv
- xv)2 < r
,
vx(-i
- zi)2 < R.
i
V
Assume, furthermore, that (Piv)2 < M
(19.9) i, v
Under these assumptions there exists a unique solution zi(X) (i=1,2,...,m) of (1.9) and (2.9), which is of class C2 in the domain (xv - xv)2 < 6 =min (r,
(20.9)
)
V
Proof. Uniqueness follows from theorem 1.9. To prove the existence, denote by zl = zl(x, A)
the solution of system (5.9) satisfying the initial conditions (21.9)
zi(O, A) = zi
We have obviously (22.9)
zi(x, O) - zi
The right-hand members of (5.9) are of class C1 for (23.9)
0 < x < aA ,
yiw-
i
and (24.9)
L( v)2
<6
V
where
-85-
zi)2
aA =
r
>
r >1.
()Iv)2
V
We will prove that for A = ()L1...., Ap) satisfying inequality (24.9) the functions 21(x, A) are defined in the interval
0<x<1
(25.9)
.
Indeed, let 0 < x < PA be the largest interval in which zi(x, A) are defined.
It is sufficient to show that PA > 1, Suppose the contrary, then
PA < 1
(26.9)
Then by (5.9), (19.9), (21.9), (24.9), (26.9) and by the definition of 6 we
get successively for 0 < x < PA
x (' 0
x)
2
[z'(x,A)
<
2 [
I avPiv(X+Ax,j Z(x, A))I? i
dx
x (x(x,A)]
dx
V
PAI
PIv)2dx < PAM6 < R
Setting PA
R'=
=
(X)2 S 0
T(Piv)2 dx
i,v
we have obtained the conclusion that for
0<x
A)
-
.i]2
< R' < R .
i
This contradicts the fact that zi(x, A) can be extended to the boundary of the domain (23.9) (see [2], p. 135, Satz 1). Therefore, the functions
zi(x, A) are defined for A satisfying (24.9) and for x in the interval (25.9).
They are, furthermore, of class Cl (see [2], p. 161, Satz 2).
Now, set zi(X) = zi(1, X - X)
(i = 1, 2,..., m)
.
Functions zi(X) are of class C1 in the domain (20.9) and satisfy (2.9) by (22.9). In order to prove that they satisfy the system of equations
(1.9), put for a fixed v and A cpi(x) = z (x,A)
-
xP"'(X + Ax, Z(x,A))
V
Since
zx(X) =z'X (I,X-X) v
V
it is sufficient to prove that (27.9)
cpi(1)
=0
p1(0)
=0
Now, by (21.9) we have (28.9)
and by (5.9), (9.9), putting
Q = (X+Ax,Z(x,A)) we get succesively
i
= ;i x- Piv((I) - x
Piv(Q)
= piv(Q) + x xxµ
Xi(i
(Q)
+
µ
L.PJ (Q)zjt - Piv(Q) - xL Z
+
µ
Pxv(Q) µ
v
J
- XR1Q) DL Pµ(Q) = xY XµPiµ (Q) "v
J
x P)z (Q)
+
µ
Vxv
)
PiJ (Q) PJV(Q) _ z
xv
_
xµ µ
P41(Q) [zX - x P° (Q) ]
i
Thus we see that the functions
Lpi(x)
satisfy the syntem of linear ordinary
differential equations with continuous coefficients
Hence from (28.9) it follows that pi(x) vanish identically and consequently we have (27.9).
As an immediate consequence of theorem 3.9 we get the following Corollary 1.9. If Piv(X,Z) are of class Cl and the compatibility con-
ditions (9.9) hold true in a domain Q, then for every point (A,Z)e SI there exists a unique solution (1.9) passing through that point. N10. Involutory systems of equations.
Let us consider the following system of equations _88_
zX = fiv(X,Y,Z,4Y)
(1.10)
m; v = 1,..., P)
V
i
where X 4 (xl,..., xp), Y = (yl,,,. Yn), Z = (zl,..., zm), i
(zyl,.
zY =
with the initial conditions
i
z Yn ),
zl(X,Y) = Wi(Y) ,
(2.10)
Suppose zl(X,Y) to be a solution of (1.10) and (2.10) which is of class Cl
in a domain D whose projection on the space X is starlike with respect to X. Put z l(x,Y,A)
= zl (X+Ax,Y)
where A = (Xi,...I Xp). It may be easily verified that the functions
zl(x,Y,A) satisfy the following system of partial differential equations (3.10)
zX =
cP
L xµf14(X+ Ax,Y,Z, 2Y)
(i = 1,2,..., m)
µ=1
and the initial conditions zl(0,Y,A) = wl(Y)
(4.10)
System (3.10) contains as many equations as unknown functions and is of
the form discussed in paragraph B. We will prove now the following uniqueness theorem. Theorem 1.10. Suppose flv(X,Y,Z,Ql), where Ql = (g,...,gn)
,
are
defined for arbitrary Z. Ql and for p (5.10)
X Ixµ-
µ1 < a,
IYj - YjJ
µ=1
bj- M
Ixµµ
-89-
1
(j
n),
bj
where a < min (),
and satisfy a Lipschitz condition with respect
to Z, Qi with the constant M. Under these assumptions there exists at most one solution of (1.10) Iyj - yj I < bj), which has first derivatives in the interior of (5.10) and possesses Stolz's differentials on and (2.10) (wi(Y) being defined for
the side surface of (510). 1
Proof. We have to show that for two such solutions zi(X,Y) and
zi (X,Y) and for any point (X*,Y*) from (5.10) we have
liz
(6.10)
= z' (X*,Y*)
To this effect, put A = X*- X and 2.
2 (x,Y,A) = zi()ZtAx,Y)
i
1 i;
z (X }Ax,Y)
2 (x,Y,A)
The functions zi and zi have first derivatives in the interior of the domain (7.10)
0 <x
jx (jn)
IYj - YjI
where
a aA
> 1
and possess Stolz's differentials on the side surface of (7.10). They satisfy, furthermore, the system (3.10) and the initial conditions (4.10). Now, by our assumptions the right-hand members of (3.10) satisfy Lip-
ICI for
schitz condition with respect to Z, Qi with the constant M µ -90-
(x,Y) belonging to (7.10) and for Z, {jl arbitrary. Hence, by theorem 3.2 we get
zi(x,Y,A) = zl(x,Y,A) in (7.10), and in particular
z l(l,Y*,A)
= zi (1,Y*,A)
since the point (1,Y*) belongs to (7.10). The last equality is equivalent to (6.10).
An immediate consequence of theorem 1.10 is the following Corollary 1.10. Suppose fiv to satisfy hypotheses of theorem 1.10 for
arbitrary Z, Ql and for
) Ixµ- 'c
(8.10)
(1=1
< a,
Y arbitrary
Under these assumptions there exists at most one solution of (1.10) and ( 2.10 )
i (Y) being defined for arbitrary Y), which possesses StolzIe (IV
differentials in (8.10) Compatibility conditions.
Suppose fiv(X,Y,Z,Q1) to be of class C1 for (X,Y.Z)E 1 and
Q'E G' and let zi(X,Y) be a solution of (1.10) which is of class C2 in some domain D.
Substituting zl(X,Y) into (1.10) and differentiating
the v-th equation with respect to xµ and the)i-th equation with respect to xv we get, by the relation zx x = zx x v µ
µ v
m µ
n
c-
--=I `
m
r
j =I qi
k +
L Yj
'i
k= I z
n
hp) zy5Y
+
s=1
j .J
s
n
m
m
f"4(p) fkv(P) + fiµ(P) rfiv(P) + Z fik(P) zk f''(P) + LYj j=1 qj v k=1 z k=1 z yj
+
n
zi I +f(P) qs s=1 j where P = (X,Y,Z(X,Y),z4(X,Y)), and hence
m
fiV(P)
k
m
n
gi(P)Iyµ(1') +I fZk(P)zk(X,Y)]
l fzk(P)f '(P) +
j=1 j
m
fkv(P) +
f41(P) +'
n
k=1 m
fiiP) rfyv(P) +I flV(P) Zk (X, Y)]
x
j
j =1 qj
for (X,Y)E D.
k= 1
j
If
m fiv x (9.10)
j
n
µ k=1 m
k=1
m
n
fi4 + c fik fkv +
xv
m
fii (yµ+ fZk q ) j=1 qj j k=1
fZk fkµ +
+
z
fiµ(fiv + j=1q,
Yj
iv qk)
k_1
zk
hold true, for all indices i,µ,v andbr all (X,Y,Z)E
Qk=
(ql,.
then system (1.10) is said to be involutory. for (X,Y,Z)E0. QkEGk. -92-
gn)EGk,
Conditions (9.10) are called compatibility conditions.
Now, we will prove an existence theorem for an involutory system (1.10).
Theorem 2.10. Assume fiv(X,Y,Z,Qi) are of class Cl and bounded
together with their first derivatives for p
- zµ I < a
(10.10)
Y, Z, Qi arbitrary ,
,
µ=1
and satisfy the inequalities (11.10)
'
If
I
< A.
B,
I fi I
Suppose, further, that fY , fZ , fii satisfy, with respect to X, Y, Z, Qi Q
a Lipschitz condition with the constant L and that compatibility conditions (9.10) hold true in (10.10).
Let Wi(Y) be of class Cl in En and bounded together with their first derivatives and satisfy the inequalities
(12:10)
Assume finally that WY satisfy a Lipschitz condition with a constant J. Put
M = 2D + A min (4a, -.) min (a,
1
4n(1+nJ)[(1+mM)2L + mB(1+2J)]
)
Under these assumptions there exists a unique solution of (1.10) and
(2.10), which is of class C1 in the strip rP
(13.10)
Ixµ- zµI
< 9,
µ=1 -93-
Y arbitrary.
This solution satisfies inequalities n (14.10)
4 < M,
I4Y(X,Y) - 4(X,`) I < (1+2J) I IYJ -YJ I J=1
Proof. Uniqueness follows from corollary 1.10. Now, to prove the
existence, observe that there exist sequences of functions fiq(X,Y,Z,Qi) (.1iq(Y),
(q= 1,2,...) which are of class C2 in (10.10) and En respec-
tively such that (15.10)
lim fivq =fiv q-aco
lim fiv = fiv X X
,
lim fivq = fiv Y Y
q-).co
lim fiq = fivi
q
Q'
Q
uniformly in (10.10) and lim WY = WY
lim Wiq = Wi ,
(16.10)
q-aoo
uniformly in En.
q-*- co
Moreover the functions fivq,
Wiq
can be chosen so as
to satisfy all the hypotheses of our theorem concerning fiv and Wi, and to be bounded uniformly with respect to q. Consider now the system of equations zix =
p`
xfiµq(X+ Ax, Y,
,
Y)
df = g iq(x, Y,
-i zY, A)
t=1
with the initial conditions
(4.10q)
2i (0, Y, A) =
iq(Y)
By the properties of fivq and Wiq functions giq(x, Y, Z, Qi, A) are of -94-
class C2 with respect to all variables) for p
ICI
Ixl
(17.10)
a
<
Y, Z, Qi arbitrary
,
ICI µ
and L) iq are of class C2 in En. giq and (jiq satisfy in (17.10) all the assumptions of theorem 2.8 with constants A, B, C, L, a replaced by
AIAI, µ
c2II. µ
µ
and are bounded uniformly with respect to q. a µ
a IXµi
µ
Now, since
) = ZD+Amin(4a, B+') = M,
1
iX1I
we conclude by theorem 2.8 that for every vector A = (X1,..,, X ) 11
P
satisfying the inequality (18.10)
a
there exists a unique solution
ziq(x,Y,A) of (3.10q):and (4.10q), which
is of class C1 in the strip (19.10)
Ixl < II
,
Y arbitrary
By theorem 2.8 we have, furthermore, that ziq are bounded uniformly 95-
with respect to q and satisfy (14.10). It can be proved that 2iigY ,
zA
are continuous and bounded uniformly with respect to q.
zYA
Let us emphasize here the following property of functions 2iq(x,Y,A). Since for A satisfying (18.10) we have p
>
the functions 2iq(1, Y,A) are defined and of class Cl for (20.10)
L
By the properties of
I
I
Y arbitrary
<
iq, stated above, the functions z1q(1, Y,A) and
2}q(1, Y, A) are uniformly bounded and equicontinuous in (20.10). _iqs
by Arzela's theorem there exists a subsequence z (21.10)
lira
z
qs (
(Y, A),
1, Y,, A)) =
8 _-4- GO
uniformly in (20.10).
Hence
such that
q
lim z a(1,Y,A) = zY(Y.A)
Moreover, the functions
(Y, A) satisfy inequali-
ties (14.10).
Now, for fixed q, v, and A, and A satisfying (18.10), put for (x,Y) in (19.10) p1(x,Y) = zi (x,Y,A) - xfivq(R) V
where
R = (X+ Ax, Y. 2q(x,Y,A), ziiq(x,Y,A))
The functions zA , zYY, and zYA being continuous 1. we get from (3.10q)
The approximate functions flvq Wiq were introduced just in order to guarantee the continuity of these derivatives. 1.
-96-
fivq(x) + x
(22.10)
X f'xq(R) +
fz4 q(R) zj
v
X fq q(R) ziigk,v -
fivµ(R)
)LµfiXq(R)
-x
k
fiµq(R) +
k q.
k
µ
µ Yk
µ
- )Ll ,ivq(R) µ
µ
-xLfivq(R)
fiµq(R) zjq Yk
zJ
µ
4
+
v
J
L
r
f,1µq(R)
J
fiµq(R) zlq q1 ykyr r
.
By (9.10), (15.10) and (14.10) (satisfied by ziq) we can write
jq]
(23.10)
f1
(R) +
f(R) f,lµq(R) +fivq(R) Ifi kq(R) +fzµq(R) zY k k
f1
(R) +
fl iq(R) I ykq(R) k q
fi q(R) f'(vq(R) +
LL
,ivq(R) zjqk
+
where
+nµv(x,Y)
,1
(24.10)
v
,Y)I < Eq
urn Eq= 0
,
q--I, ao
for (x, Y) in (19.10). By (22.10) and (23.10) we get
fi q(R)z
xfivq(Ri1 v
µ
+
Ifigk(R) Yk
JJJ
[v -97-
x
-x
Ak
µ
17kv
Y)
in (19.10) and finally by the inequalities (11.10) satisfied by the functions fiµq and by the inequality (18.10) and (24.10) we obtain Icy
PC
Jp B
'Px
k
i
+
q
k
in (19.10). On the other hand, we have by (4.10q) (P
i (0, Y) = 0,
From the last two relations it follows by corollary 1.2 that
iI
<<
(P
emPBIxl
Eq
mB
C
-1
in (19.10) and in particular for x = 1 (25.10)
<j[emPB-1].
Iza (1,Y,A) V
By (15.10), (16.10), (21.10), (24.10), (25.10) and (4.10q) we get in the limit
zx (Y,A) = fiv(X+A,Y,Z(Y,A),zY(Y,A)), ;1(Y,0) = w1(Y)
(26.10)
V
for
I
I
<9
and arbitrary Y. Now, putting z l(X,Y)
= zl (Y,X -
we see that functions zi(X,Y) are of class C1 in (13.10), satisfy (1.10) and (2.10) by (2610) and that inequalities (14.10) hold true since the functions
zi satisfied (14.10).
BIBLIOGRAPHY [1]
Haar, A. , "Zur Charakteristikentheorie", Acta Litterarum ac
Sci. Univ. Szeged, 4 (1928), pp.103-114. [2]
[3]
Kamke,E. , Differentialgleichungen reeller Funktionen, Chelsea Publishing Go., New York, 1947.
Plis, A. , "Characteristics of non-linear partial differential equations", Bull. Acad. Polonaise des Sciences, Cl.III, 2, No. 9, (1954), pp. 419-422.
[4]
------
"On the estimation of the existence domain for solutions of a non-linear partial differential equation of the first order", ibid. 4, No. 3, (1956), pp, 125-129.
[5]
------
"Generalization of the Cauchy problem for a system of partial differential equations", Ibid. 4, No. 11, (1956) pp. 741-744.
[6]
------
"On characteristics of partial differential equations", ibid. 5, No. 10, (1957), pp. 951-958,
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------
"A method for determining the existence domain for solutions of partial differential equations of the first order", Annales Polonici Math. 3, 2(1957), pp. 183-188.
[8]
------
"The characteristic equation for partial differential equations of the first order", Colloquium Math. 6 (1958), pp. 223-226.
[9]
------
"On contact points of two integrals of a partial differential equation of the first order", Bull. Acad. Polonaise des Sciences, Cl. III, 6, No. 1 (1958), pp. 11-14.
[10]
Szarski, J., "Sur certains systemes d'inegalites differentielles aux derivees partielles du premier ordre" Annales de Is. Soc. Polonaise de Math. 21, 1 (1948), pp. 7-25.
[11]
------
"Systemes d'inegalites diffe'rentielles aux derives partielles du premier ordre, et leurs applications", Ann. Pol. Math. 1 (1954), pp. 149-165. -99-
[12]
Szarski, J. and Wazewski, T., "Interpretation geomdtrique des conditions d'intdgrabilite complete pour un systeme d'6quations aux differentielles totales" to appear in Ann. Pol. Math.
[13]
Wazewski, T., "Sur 1'unicite et Is. limitation des intdgrales des equations aux derives partielles du premier ordre", Rendiconti d. Accad. Lincei, 18 (1933), pp. 373-376.
[14]
------
"Ueber die Bedingungen der Existent der Integrale partiel]er Differentialgleichungen erster Ordnung",
Math. Zeit, 43 (1938), pp. 522-532. [15]
------
"Sur 1'evaluation du domaine d'existence des fonctions
implicites rdelles ou complexes", Ann. $oc. Pol. Math. 20 (1947), pp. 81-120. [16]
------
[17]
------
"Sur les systemes des equations et des inegalites differentielles aux seconds me mbres monotones et leurs applications", ibid. 23 (1950), pp. 112-166. "Certaines propositions de caractere 'epidermique'
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