On the existence of solutions for some integral-functionalequation

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXIII (19831

Krysîyna Nowicka (Gdansk)

On the existence of solutions for some integral-functional equation

Introduction. Let C{X, Y) denote the set of all continuous functions defined in X taking values in Y, where X, Y are arbitrary metric spaces.

For ^{jc = (x j,} xn), у = (уь . . . , yn) e R n, Rn^— real space of dimension n, we define x ^ y as x{ ^ y( for г = 1, n. We denote by | *| the norm in Rn and for the set A c R" by Ln(A) the n-dimensional Lebesgue-measure of A. Let

E(x) = {Ç eR n : g ^ x} for x e R n.

We assume that G <= Rn+ (R+ = [0, +oo)) is connected, bounded and closed set. Let S = {xeG : L„(£(x)nG ) = 0}.

Suppose that the set G0 is given and it fulfils the following conditions:

(a) G0 c: U E(x) and G0 is connected, bounded and closed set,

xeS

(b) G0 n G = S.

Let В be an arbitrary Banach space with the norm || |i.

Suppose that the function (peC(G 0, В) is given. We denote by CV(G, B) the set of continuous functions defined in G u G 0 taking values in В and identical with (p on G0.

We consider the integral-functional equation (1) u(x) = F (x, f f 1( x ,s ,u ( a 1(s)))(ds)Pl9...

H t (x,u(-))

J f m(x,s, u(ocm(s)))(ds)Pm, u (^ (x , u( ))),..., и (ft (x ,u ( •)))), x e G , and u{x) = ф(х) for x eG0, where

F eC (G x Bm x Bk, B),

a = ( « ! , . . ., О e C (G u G 0, (G u G 0)m), P = ( A , • • -, Pk)sC(G x Cy(G, B), (G и G 0)‘ ),

Hj. G x C,(G , B) - 2°uG», j = 1, . . m,

280 K. No w i c k a

are given functions and ocj(x) ^ x for j = 1, m, x eG , /?,•(x , w(-)) ^ x for i = 1, k, x e G , weC^iG, B). We shall assume that equation (1) is of Volterra type, i.e., Hj(x, w( • )) c: Ê (л:) for every x e G , weCv(G, В). Further­

more we assume that the consistency condition is satisfied (p(x) = F (x, j f t (x, s, <jo(ai(s)))(ds)Pl, ...

H^x,</>())

J fm (x,s, (p(ctm(s))){ds)Pm, (p(p!(x, <p( ))),..., <p(fik(x, q>(•)))) for x e S . We omit this assumption when L„(G0) = 0.

We assume that the set H j(x ,w ()) for every x e G and w eC v(G, B) is contained in a p,-dimensional hyperplane parallel to the coordinate axes and it is Lebesgue-measurable considered as a p^-dimensional set. Let x, w(* )) denote the Pj-dimensional Lebesgue measure of Hj(x, w( )).

We assume that pj does not depend on (x, w(*))- If p;-dimensional hy­

perplane containing the set Hj(x, w(-)) and parallel to the coordinate axes is defined by the equations

*r2 = xf2, ..., xtr = x£, r = n -p j, then

J g{x,s)(ds)p , s = (sj, ..., s„)

H.(x,w(-))

the space x°

л ,г.

0xmixm2 denotes the py-dimensional Lebesgue integral in

e {{1, n } - { t lt ..., tr}} and sh = x °, ...

Let A' = {/e{l, ..., n}: p,- = «}, B' = {/e{l, ..., n}: 1 < pt < n}. By changing the notation if necessary we may assume that A' = {1, k0}, B'

- {k0 + l, m}. We define the sets Cj c= {1, n} in the following way:

ie v j if and only if the axis Ox, is parallel to the p,-dimensional hyperplane in which the set Hj(x, w(-)) is contained. Put <ту- = {1, ..., n} — Oj.

We adopt the following notations:

J f ( x , s, и (a (s))) ds = ( J fi( x , s, u(ail (s)))(ds)Pl, ...

..., f f m(x, s, u(am(s)))(ds)pJ ,

H J x M))

u(p(x, и (‘))) = (и (^ [х , и(■ ))), ..., u(pk(x, u(-))))

. and define the operator $

(1) (ftu)(x) = F (x , J f ( x , s, u(<x(s)))ds, u(fi(x, u( •)))),

H ( x M ) )

x e G .

Using these notations and the definition we can write equation (1) in the form

(2) u = %u

and u e C v(G, В).

The initial problems for ordinary differential equation z'(t) = f ( t , z(t)), t e [ t 0, t 0 + d]

and the classical boundary problems for almost linear hyperbolic equations (3) uxy(x, y) = f ( x , y, u{x, y), ux(x, y), uy(x, y)),

(x, y) e [0, d] x [0, b] c= R2 can be reformulated in terms of Volterra integral equations. These differential equations and integral equations corresponding to them are generalized in various directions. Different kinds of equations are taken into consideration for example: in particular the differentiàl-functional equations of the neutral type and equations of type (3), where a function of n variables is introduced.

Moreover, the equations may be investigated with respect to various con­

ditions for example: Carathéodory type, and in some abstract spaces.

We do not present any complete review of these problems. A very rich bibliography can be found in references given in [2], [5], [6], [15], [22], [23] and monographs [19], [25]. The results of this paper extend those of previous works [4], [20], [21], where the integral-functional equation in which the functions /?f and the sets Hj in (1) do not depend on the unknown function was investigated. Moreover, some boundary problems for almost linear hyperbolic equations with deviated argument where this argument depends on the unknown function and these equations are of neutral type, can be reduced to an equation of the form (1). In view of these problems the main results are more general than those in [2]. Furthermore in the present paper we consider integral-functional equation which covers as a particular case some boundary problems for a partial differential equations of hyper­

bolic type with time lag, where there is an n-dimensional initial set of non-zero n-dimensional Lebesgue measure (see example in I). From this viewpoint the main results of the present paper are generalization of the results in [11]- [13], [23]. The equation of type (1) with n — 1 has been investigated in [3], [9].

The particular case of equation (1) is also the functional equation which is discussed by many authors for example [1], [17]. In the present paper we consider the existence of solutions of equation (1) in the class of functions fulfilling a Lipschitz condition. The questions of existence of solutions should be always considered with respect to some class of regularity. For example in

282 K. No wi c k a

[10], [15], [18] there are considered integrable solutions of some integral functional equations.

We prove the existence of solution of equation (1) in a finite dimensional Banach space. We shall use the Schauder fixed-point theorem and some results of the comparison method (see [8]).

I. An example of a boundary problem which leads to equation of type (1).

We adopt some notations and assumptions of [2].

Let G be a subset of R". Suppose that the set G fulfils the following conditions :

(i) G c [0, b], where [0,b] is an n-dimensional interval, b = (bl , . . . ..., />„), 0 — (0, ..., 0), G is connected and closed,

(ii) for every /, 1 ^ i ^ n, we have

[0, b j x ... x [0, bf_ t] x {b,} x [0, b( J x ... x [0, b j c= G,

(iii) if xeG and х Ф Ь , then there exists tGRn such that [x, x-H] c= G.

Moreover, we characterize the set G introducing the following notations and assumptions. Let

5 = G n [0, b]\(int G)\{x gR" : exists г, 1 ^ i ^ n such that x, = />,], Zi = S n { x e R n: xf = 0}, / — l,v..., n.

Let us consider the following projection mappings

projfc: Rnэ (x j, ... x „ )^ (x 1, ..., хк_ ь xk+1,..., х „ ) е Г -1 . In particular we have

projid-,)

= [(Xj, ..., X,-_ !, xi + 1, ..., x„)eRn~1 : (xj, ..., x,-^, 0, xi + 1, ..., x je l ',} . Let -S, = proj,(S) — projfd,-), i = 1, n. Suppose there are given n functions (pt: Si -* [0, b j, i e { l , ..., n} with the following properties:

(i) for every i e [ l , ..., n}, </>,• is continuous in and <p,-(x) = 0 for x e S .-n p ro j.d ),

(ii) for every i e { 1, ..., n], q>i is stricly decreasing with respect to each variables,

(iii) for every j e { l , . .., n] the following condition is fulfilled : xf

= <MX?’ •••’

- (рк{х°и ...,

o+oiX? x

1 t X * X?)

. , х й

if and only if for every k e { 1, n], x°k Let Г 0 = {(xt , ..., ^{X „ )}i eS : there is an j e { l , ..., n) such thatП XJ

= (pj(Xj, ..., Xj_1? xj+1, ..., x„)}. We assume that S = Z°vj( 1J Г,).

!= 1

Let /„ = (/* = (/*!, ц„): н е {0, 1}J and I'n = /Д[(1, ..., 1)}. For це1„

and for sufficiently regular и we define differential operator Dд by the formula

+ ... + nn

DMu = ‘ (/>(0," " 0)м- = и).

СХД ... cx„n

Let G0 be a connected ana closed set and such that ^{G 0} с (E (b)\G) u S, G0 n G = S, where E{b) = {x e Pn: x ^ b}. We suppose that the functions G - ^ G u G 0, уд: G x R " - > G u G 0 for jUg/„ and /: G x Rq

R, q = 2n, (p: G0 -+ R are given and there exist all partial derivatives D^cp for ц е 1 п and they are continuous. We consider the following Cauchy- Darboux problem

' D(1” ”1)m(x) = /(x, Du(y(x, w(a(x))))), xeG,

(4) u(x) = <p(x) for x gG0,

D^u(x) = D^(p{x) for x gS, /i g/Д where

Du (у (x, и (a (x>))) - {D»u (y„ (x, и (a„ (х))))}д6/(1

and Ядм(у„(х, z)) denotes the partial derivative Яд of the function и at the point y„(x, z).

We shall denote by the unique solution of the problem, D(l....uu(x) = 0, x eG ,

(5) u(x) = (p(x) for x gG0,

О^и(х) = D^(p(x) for x e S , ц еГ п.

For p e l n we denote by proj(0....0) the composition functions proj, for / such that ^ = 1 (proj^ denotes the identity mapping, i.e., proj(0... 0)x = x).

Let G (x) = [0, x] n G and GM (x) = projM G (x) for ц e Гп, Я д (x) = [s g Rn : projMsGGM(x), s^. = x^. for i such that /г, = 1}. The set Я^(х) is contained in a hyperplane sц = xц for i such that /г,- = 1 and it is n — |/i|-dimensional set, where |/r| = /q+... + /V Then j z(s)(ds)fl denotes the n — ^-dimensional

Hn{x)

integral for such s for which /r, = 0 and s^. = хц. for ^ = 1.

We shall accept the substitution Я(1," д>м(х) = z(x). Then we have u(x)= J z{s)ds + Xtp(x),

H(0...0)<*>

D^u(x) = J z(s)(ds)fl + Dfl ^ (x )

H (x)P

for ц е Г п,

284 K. No w i c k a

and

IT и (y„ (x, и (a„ (x)))) = J ^Z (s)(<fo)„ + D* (y„ (x, и (aM (x)))), ц еГ п.

H Ц(У ц(х,и(а ^(x))))

Problem (4) is equivalent to the equation

z (x )= / (x , { { z(s)(ds)„}^, z(y(lf...fl)(x, z(-)))), x eG ,

Нц(уц(хМ ) ) )

•z(x) = <p(x) for x e G 0, where

Ун(Xj z( ■ )) = Уц(x, J z(s)rfs + ^ (a |i(x))) for ц е1 п.

"(0...

The equation is of type (2). A similar problem of Z. Szmydt for the partial differential-functional equation of hyperbolic type R 2 can be re­

formulated in terms of the system Volterra integral-functional equations. The problem of Z. Szmydt contains the classical problems of Darboux, Cauchy, Picard and Goursat (see [7]).

II. Assumptions and lemmas. In analogy with paper [20] we use the following notations and definitions. For arbitrary elements w, DeH', w = (wj, ..., wf), v = (vi , ..., Vi) we define

i

< W , V } t = £ W j V j .

v i

Furthemore we define the linear operator 2 and a sequence of its iterations {£ r} in the following way:

{2°h)(x) = h(x), (2 ih)(x) = (2h)(x) = <L(x), Л(Д (*))>*, (2r+1h)(x) = (2 (2 rh))(x), r = 0 , l ... xeG, heC (G , R + ),

where L = ( L 1, . P., Lk)eC (G , R\), Д = (Д1? ..., fik)eC (G , Gk) are given func­

tions and h(fi(x)) = (h(jMx)), ..., h(Pk{x))).

Below we quote Lemma 1 (see [20]) which characterizes a comparison equation for (1).

We assume that H f : G -» 2G, j = 1, ..., m and the set H f(x) is Lebesgue-measurable considered as a pj-dimensional set (1 n) for any x e G and j = 1, ..., m.

Let xeG, JieR n, x + heG, ieB'. Suppose that the set H*(x) is contained in a p,-dimensional hyperplane (1 < pt < n) which we denote by S, (x) and the set Щ (х + И) is contained in a pr dimensional hyperplane 5,(х + й) parallel to the hyperplane Sf(x). There exists a vector t^(x, h )eR n such that the set

— ïi(x, h) + H?(x + h) is contained in St (x).

We introduce

A^ssumption Hx (see [24], p. 970; [25], p. 134). Suppose that:

Г for ie A' we have lim L„ [Hf (x) - (x + îï)] = 0 uniformly with respect to xeG (the sign — denotes the symmetric difference of two sets),

2° for ieB ' we have (a) lim f (x, E) = 0,

о

(b) lim [H f (x) — ( — Т{(х, Е) + Щ (х + Я))] = 0 uniformly with respect to h-*0

xeG.

Assumption H2 ([20]). Suppose that Г the functions heC (G , R + ),

К = ( K lt ..., K m)eC (G , R ï), â = (â1? ..., âM)eC(G, Gm), L = { L l , Lt) e C(G, R%), P = (Д„ R')eC(G, Gk) are given and аДх) ^ x, Д,(х) ^ x for xeG, j = 1, ..., m, / = 1, ..., /с,

2° for xe G we have

m(x) = Y (£lE){x) < + o o , i — 0

3° for xeG we have

M(x) = Y (£‘к)(х) < ^{+ o o ,}

i = 0

where k(x) = Y K j(x)L Pj{Gj(x)) and j= i

Gj (x) = { s e R n: st. — x° for tt e Gj, 0 < st. < <pj^(x) for г,- еегД, (pyeC iG , R + ), tiEOj,'j = 1, ..., m,

4° Л/, m eC (G , Л + ) and the function

M ( * ) = L { I * , ) - }

i = 0 j = l se<r

is bounded for xeG.

Lemma 1 ([20]). I f Assumptions H1? H2 are satisfied, then : (a) there exists a solution zeC (G , JR + ) o f the equation

(6) z(x) = Y (&h)(x)+ Y £ ‘(<Ж(*), I z(â(s))ds}m), x e G ,

i=⁰ i — 0 H*( x)

286 K. No wi c k a

and the solution z o f (6) is unique in the set M(G, R +) o f the non-negative and upper semi continuous functions in G,

(b) the function z is the unique solution o f the equation

(7) z(x) = <K(x), J z(â(s))ds}m+ (L (x ), z(p{x)))k + h(x), x eG ,

m x )

in the set M(G, R + ) = {z e M (G , R + ): \\z\\0 < + oo}, where ||z||0 = inf {c e R + : 0 ^ z ^ cz} and

(8) lim (£rz)(x) = 0 uniformly with respect to x e G ,

r - * oo

(c) the function z (x) = 0 for x e G is the unique solution o f the inequality

(9) z ( x ) ^ ( K ( x ) , f z(x{s))ds)m + (L {x , z(p(x))\, x eG ,

H ( x

in the set M(G, ).

Remark 1. Some effective conditions under which assumptions of Lemma 1 are fulfilled can be found in [20].

A^ssumption H3. Suppose that there exist functions kj, 'lj, /,-, Pi e C (G, R + ), j = 1, . . m, i — 1, . . к such that

(a) ||F{x, uu ..., um, vu ..., vk) - F { x , щ, ..., Um, vu ..., nk||

m к <

^ X kj(x)\\uj - üi\\+ X //(x)||uI-r,||,

j = i ; = i

(b) • \\fj{x, s, v).-fj(x, s, v)\\ ^Tj{x)\\v-v\\, 7 = 1,..., m,

(c) Pi (x, w( • )) ^ Pi(x), / = 1, ..., k, for any x eG , s e G и G0, s < x, Uj, ûj, Vi, G, v, v eB , weCy(G, B).

Assumption H4. Suppose 'that there exist functions Hy G 2G, j = 1, . . m such that Hj(x, w ( • )) n G c Hj(x) for (x, w( • ))eG x C(fl(G, B).

Lemma 2. Suppose that

1° Assumptions H3 and H4 are fulfilled,

2° there exists a junction u0e C (p(G, B) such that assumptions o f Lemma 1 are satisfied with Kj(x) = kj(x) Tj(x), ôcj(x) = ocj(x), j — 1, ..., m,

Li(x) = /,• (x), p ;(x) = Pi(x), i = \ , ..., k,

m к

h (x )= X k j(x )y j(x )r j+ Y h (x) шо (2 |Д,- (x)|) + y (x),

j= i ,= i .

where

y^{x^{) =} sup f(i), y(x) ^ ||(îÇmo)(jc) -m0(x)||, y}(x)= sup fj(x),

’yj{x) ^ \\fj(x > s’ wo(aj(s)))l|’ 7 = 1 ,..., w, xeG , s gGuG0, Гj = sup Гj (x), Г,- (x) ^ L (Hj (x, w( • ))), j = 1, . . m,

x e G *

co0 is file modulus o f continuity for the function u0 and H f(x) = H j⁽x).

Then the operator ft defined by (Г) maps

D{G, B ,z) = {ueC^iG, В): ||m(x) -m0(x)|| ^ z(x)}, where z is defined by Lemma 1, into itself

Proof. Let ueD {G , B, z) and w(x) =(^u)(x). Then wgC</)(G, B) (see [24]). We prove that ||w(x) — «0(x)|| ^ z(x). We have

IIvv(x) Uq(x)|| ^ ||( ftu ){x)-{ftu 0){x)W + \\{ftu0){x )-u 0(x)\\

m

^

Z

j = l н . ( х , и ( ) )

-

j

I

H .{ x ,u 0( ) )

+ Z

и0(‘)))|| + У(*) i= 1

< Z

1 Ilw(a7(s))~w0(«J(s))||-h

7 = 1 H . ( x , u ( ) ) n G ' ;

m к

+

Z 7=1

(*)

(*)

+ Z i=l

(*) I

( ■ ))) ■ -

-U 0(Pi{x, M(*)))||-:

+ Z №)«о(|(М*>

M 0(-))|) + y(x)

i=

1

m к

^ J Kj{x) J z(<Xj{s))(ds)p.+ Z M *) * (ft (•*)) + M*) <z(x).

7 =1 ^ й.(дс) l = l

Hence it follows that w eD (G, B, z). Thus the Lemma is proved.

Suppose that the function ip is given and satisfies the Lipschitz condition with a constant X ^ 0 (we shall assume certain additional conditions on A).

We denote by Lipv(A, G, B) the set of all functions defined in G u G 0 identical with the function tp on G0 and satisfying the Lipschitz condition in G u G 0 with the constant X.

288 K. N o w i c k a

Assumption H5. Suppose that assumptions of Lemma 2 are fulfilled and 1° the function u0eLip4,(A, G, B),

2° there exist non-negative constants k0, fhj, r j,j = 1, m, such that HF(x, uu ..., um, vu ..., vk) - F ( x , щ, ..., um, vu ..., yk)|j ^ £0l * - * b

\\fj{x, s, v ) - fj(x , s, r)|| ^ m jlx -tf + r jls -s ], j = 1, m, for x, xeG , ^s, s eGuGq, II^H ^ qp ||у.-|| < gi9 ||p|| ^ q0, where

Qj = sup T j(x) sup z(x) sup Lp.(Hj(x))+ Г j sup fj(x),

x e G x e G x e G x e G

Qi = sup z(x) + sup<w0(2|^(x)|) + sup u0(x),

xeG xeG xeG

Q0 = SUp z(x) ⁺ SUp U0(X),

xeG xeG

where z is defined by Lemma 1,

3° there exist non-negative constants Cj,j = 1, m such that

\ctj(x)-0Lj(x)\ ^Cj\x-x\, 7 = 1 ,..., m, x, x e G u G 0,

4° there exist non-decreasing functions ф{: R + ->R+ and non-decreasr ing functions rjeC(G , R +), i — such that if w, iv e L ip ^ , G, B), x, x e G , then

(a) \Pi(x, w(-))-pi(x, w(-))| < ф М \ х-х\ ,

(b) \Pi(x, w (-))-ft(x, w(- )| ^ г({х) sup ||w(s)-vv(s)|| for f = l , . . . , k.

seG n E (x )

Rem ark 2. If /?,(x, w(-)) = Д (х, w (<!>,• (x))), weLip^A, G, B) and we sup­

pose that

Г there exist non-negative constants tt such that

|<5,(х) — <5,(x)| < ?(|x — x|, i = l , . . . , k, x, xeG ,

2° there exist non-negative constants b{ and non-decreasing functions rjeC(G , R +) such that

(a) 1/5,-(x, v)-Pi(x, v)\ < h, |x-x|,

(b) \Pi{x, v )-p i(x , ^й)| ^ ri(x)|i; —û] for i = 1, ..., k, x, xeG , |H, \\v\\

< g0, then ipi(À) — bi + rftiA, where rf — sup f|(x).

xeG

We adopt the following assumptions on the sets Hj(x, w( )), 7 = 1 ,..., m. Let xeG , x + h eG , w, w + de Lipv(/l, G, B), ieB'. Suppose that the set Я, (х, w( )) is contained in a p,-dimensional hyperplane (1 ^ p,

< n) which we denote by Sf(x, w( )). Moreover, the set Я,(х + Я, (vv+ */)(■)) is contained in a p.-dimensional hyperplane S, (x + /i, (vv+ */)(•)) parallel to the hyperplane S,(x, vv(*)). There exists a vector f,(x, vv(-), E, d ())e R n such that the set — r,(x, vv(-), E, d(-)) + H{(x + E, (vv+ */)(•)) is contained in S,(x, w()).

Assumption H6. Suppose that Г for ieA '

lim Ln[Hi(x, w {-))-H i(x + h, (w+ </)(•))] = 0

h~*0

*(•>- 0

uniformly with respect to (x, w (-))eG xüpv(l, G, B), 2° for ieB '

(a) lim t,(x, w(-), E, d{-)) = 0, d()-0Й-0

(b) lim Lp. [Я, (x, w(•))- ( - tf (x, w (•), E, d (•)) + Я, (x + E, (w + d) (•)))] = 0

uniformly with respect to (x, w( ))gG x Lipv(2, G, B),

3° there exist non-decreasing functions s}: R + -+ R + , k j,b j,C je eC (G , R+), j = 1, m such that if x, xeG , w, vveLip^A, G, В), then

(al) b j LHJ (*’ w (')) - Hi (*> w (•))] ^ Sj (A) |x - x|,

(a2) Lp [Hj(x, w ( ) ) - H j ( x , vv())] ^ kj(x) sup ||w(s)-vv(s)||,

seG n E (x)

j = 1, k0, ||w||, ||w|K Q0,

(bi) Lp.[Hj(x, w (- )) - (- tj ( x , w(-), x - x , 0) + Hj(x, w(•)))] ^ s}(A)|x-xj, (b2) LPj[Hj(x, w(*))•— ( tj(x, w(-), 0, (w-\v)(-)) + Hj(x, vv (•)))]

^ kj{x ) sup ||w(s)-vv(s)||, j = k0+ l , ..., m, ||w||, ||vv|K Qo,

seG n E (x )

(c) |r,-(x, w(■), x - x , (w — vv)('))| ^ ^(х)|х-х|+с;-(х) sup ||vv(s) — vv(s)||.

seG n E (x)

Rem ark 3. If Я ,(х, vv(-)) = ЯДх, w(yj(x))) and Г there exist non-negative constants pj such that

I \yj{x)-yj{x)\ ^ Pj\x-x\, j = 1, ..., k0, x, xeG ,

2° there exist non-negative constants s}, j = l , . . . , k 0 and non-de­

creasing functions kjeC (G , R + ) such that Lp [ H j ( x , v ) — È j ( x - f E , v + d)\

< s}|ft| + k/(x)||<i|| for j = 1, ...,k 0, x, x+ EeG, v, v + d e B, then condition 3°(a) is satisfied for s}(A) = Sj + kfpjX, where k f = sup kj(x).

xeG

7 — Prace Matematyczne 23.2

290 K. No wi c k a

In a similar way we can give some sufficient conditions for 3°(b), where Hj(x, w(■)) = Hj(x, w(y;(x))), j = k0+ 1, m.

Let

*0 m _

Г ( Я = k 0 + Z k j L „ ( G u G 0) r h j + £ k f [ m j + ( r J + Acj t f ) b f ] r j +

j- 1 j=kQ+l

m к

+ Z kJ(FfQo + m ¥ * ) + Z W . W ,

j= 1 i=l

where

k f = sup /Cj(x), Ç = sup Tj(x), b f = sup bj(x),

xeG xeG xeG

yf = sup y7(x), /f = sup li(x) for j = 1, . . m, i = 1, . . k.

xeG xeG

We suppose that there exists a non-negative constant Я such that Г(Я) ^ Я. We introduce the following class of functions

D{G, В, z, Я) = {weD(G, B, z): ||w(x) — w(x)|| ^ Я|х-х|}, /

where the constant Я satisfies the condition Г(Я) ^ Я.

Lemma 3. // Assumptions H5, H6 are satisfied and Г(Я) < Я, r/icn the operator ft defined by the right-hand side o f (1) maps D(G, B, z, Я) into itself.

Proof. Let ueD (G , B, z, Я) and w(x) = (ftu)(x), x e G . From Lemma 2 it follows that w eD (G , B , z).

We prove that ||w(x) —w(x)|| ^Я|х —x|. We have

||w(x)-w(x)|| = ||(ftu)(x)-(ftu)(x)||

m

^ £0|x-x| + Z M*)|| J fj ( x , s, u(xj(s)))(ds)Pj~

j= 1 ^{H .(x ,u (}))

f

H j( x M ))

+ Z li W I\U № (* ’ U < '))) " U (A (*, M ( ■)))[I • i = 1

Let for /еЛ', x, x + h eG , w, w + d e Lip^,(Л, G, B)

Я?(х, vv (■), Я, d(-)) = Я у(х, vv()) —Я7(х + Я, (w + t/)( )), Я] (x, w(*), Я, </(•)) = Я^(х, vv(;)) n Hj(x + h, (w + d)(-)),

(10)

_ /

and for jeB ', x, x + h eG , vv, w + d e Lipv(A, G, В)

H j( x, vv(•), Л, d(-)) = Hj(x, w ( - ) ) - ( - t j ( x , w(-), h, <:/(*)) + + Hj(x + h, (w-f-<■/)(•))),

H) (.y, w('), h, d(-)) = Hj(x, w(-))n(~tj(x, w(■), h , ( ! ( • ) ) + ,

(11) +H j(x + K,(w + d) (•))),

Я?(х, vv (■), h, d()) = ( - t j( x , vv (■), h, </(•)) +

+ Hj(x + h, (w + d)(-)j)--Hj(x, w(-)), H j(x, vv(■), Я, d{-)) = H j(x, w( • ) ) - ( w(-), h, d{')) +

+ Hj(x + Jt, (vv + d)(•))).

For j e A' we get substituting ds in the place of (ds)n

(12) || J f j ( x , s, u ( o ij(s)))d s- J f j ( x , s, *<(ay(s)))ds||

H .(x ,u ()) H .( x M ))

^ I ||/}(л:, s, M(aj(s))) —/}(x, s, м (a^ (s)))|| y/s+ Hj(x,u(-),x— x,0)

+ n ^! OfQo + fj)ds Hj(x,u(),x-x,0)

[L„ (G и G0) ^{r f i j} + ^{( I f q}0 + y f ) S j (Я)] |x - x|.

We estimate now for je B '

|| J f j ( x , s , u(ctj(s))){ds)Pj- j j ) ( x , s, ы(а7(5)))(^ )р.||.

H .(x M )) H .(x,u ())

Because

{ g(x, s)(ds)Pj = j g (x,s + tj(x ,u (),h ,0 ))(d s)pr

H .(x + h M )) - r ( x , u < ) , / i , 0 ) + W.(x + fc,u())

then

(13) || J fj(x , s, u(ccj{s))){ds)Pj- { fj(x , s, u(a,(s)))(ds)pJ|

H f x M ) ) H j(x,u())

^ j ||/}(лг, s, u ((X j{s)))-fj(x, s + tj(x, u(-), x - x , 0),

н 9 ( х , и ( ) , х - х , 0 )

u(xj(s + tj(x, u(-), x - x , 0))))|J(ds)p.+

292 K. No wi c k a

+ з H j(x ,u ( -),x — x, 0)

f

H j ( x ,u ( ) ,x - x ,0 )

^ Lp.(Hj(x, w(-), x —x, 0))[m;-|x — xj +rj\tj(x, u{), x — x, 0)| +

H j(x ,u (-),x — x ,0 )

< { Гj [nij + (r'j + Xcjlj (x)) bj(x)] + (IfQo + yf)Sj{X)}\x-x\.

Because Г(Х) ^ X, then by (12) and (13) it follows that

m

+ X k j F ) № j + {rj + t f j l f ) b f ] +

m k

+ 1 k f ( T fe o + m W ) + I W i W } |x-x|

i = 1

= Г(Я)|х — Зс] < Я|х — x|.

Hence it follows that w eD (G, B, z, X). Thus Lemma 3 is proved.

Now we can formulate

Theorem 1. I f В is finite dimensional Banach space and the assumptions o f Lemma 3 are satisfied, then equation (1) has at least one solution ueD (G , B, z, X).

P roof. In view of Lemma 3 it follows that the continuous operator defined by (L) maps the bounded, closed and convex set D(G, B, z, X) into itself. Because D(G, B, z, X) is a compact set, then from the Schauder fixed- point theorem follows the assertion of the theorem.

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[5 ] M. K isie le w ic z ,' Som e properties o f partial differential functional equations o f hyperbolic type (Polish), Lubuskie Towarzystwo Naukowe, Poznan-Zielona Gôra (1971).

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