On some class oî integral-functional equations in locally convex spaces

ROOZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PEACE MATEMATYCZNE X X (1978)

M a r i a n K w a p i s z a n d J a n T u r o ( G d a n s k )

O n some class oî integral-functional equations in locally convex spaces

At the first part of the present paper we describe a general setting of the equation solving problem and we show a general pattern of proofs of the existence, uniqueness and convergence of successive approxima­

tions based on the idea of comparative method.

Further we analize, following the above mentioned pattern, a class of a non-linear comparative operators and we give for such operators an effective conditions being sufficient to ensure the existence, uniqueness and convergence of successive approximations for some class of integral- functional equation with an unknown function of n real variables defined on an unbounded domain with a range in an arbitrarily fixed topological locally convex space (see equation (11)). We also give an effective theorem on continuous dependence on the right-hand side of a solution of this equation.

In the last part of the paper we consider in a topological locally convex space with a family of generalized seminorms an integral-functional equation with upper limits of integrals appearing in considered equation depending also on an unknown function.

Our result are generalizations of some results contained in [1]-[11].

Because the method used in the present paper is very close to th a t used in [2]-[8] we shall not give here the detailed proofs of theorems stated.

1. General assumptions and theorems. Let R m denote some m-dimen- sional-real linear space. If x e R m and x = (aq, ..., xm), then x > 0 means th a t Xi > 0, i = 1, .. . , m, however, x < y when у — x > 0, \x\ = ( |®х| , ...

..., \xm\), R™ - [x: x e R m, x > 0].

Suppose th a t a set X is given. Let in X a family of semimetrics Qt>i\ t e T , l e A (T, Л — arbitrary sets), be defined. We assume th a t this family has the following properties:

1° Qui00* У) > ° j t e T, 2 6 Л,

2° 6tA æ > У) = °» t e T > 1 e A iff x = У,

378 M. K w apisz and J. Turo

3° Qi,dx i y) = QtAVi x, y e X , < e Tf l e A 1

4° е и (х ’ У) < QtJVy z ) + QtAVy z)i a>,yt z e X , t e T , l e A.

For a fixed l e A, x, у e X we shall denote the function defined on T with the values of>l(x, у ) by the symbol д{.л){х, у ).

We mean for xn, x e X , xn->x iff gt>i(xn, x)->0, n-+oo for any t e T, l e A.

Suppose th a t X is a sequentially complete, i.e., if {a?n}, xn e X , the condition Qt,i(œ’m xk)-> n ,T c-+ o o , is satisfied for any f e T, l e A, then there exists x e X such th a t xn->x, n->oo.

We consider in X an equation

{1) x = f(ae),

where / : X-> X.

Let F ( T , R™) be a set of mappings from T into R™ and let G (T, R™) a F (T , R™). We mean for u, v e F (T , R™), u ^ v iff u(t) < v(t), t e T , and un, и e F ( T , R™), un-+u iff un(t)-+u(t) for any t e T . Suppose th a t G (T , R™) has the properties: a) if u, n eG (T , R™), then u + v eG (T , R™), b) if un e G(T, R™), un+l < un and un->u, then и e G (T , R™).

A s s u m p t io n A . Suppose that

1° there exists a fam ily of operators Qt : G{T, R™)-+G(T, R™), l e A, which has the following properties:

(a) i f u ,v e G (T ,R ™ ) and u ^ v , then й г{и) < &i(v), I e A,

(b) i f un, и e G (T , R™), un+1 < un, n = 0,1, and un->u, then O ^ u J-^Q ^ u ), l e A,

2° the inequality

Q tA fW ’f W < Qi( 6{;Ax ’ У))(*), t e T , l e A , holds for any x, y e X .

A s s u m p t io n В (ht). Suppose that

1° for a fixed function ht e G (T , R™) for any l e A there exists a solu­

tion дг e G( T , R™) of the inequality

&i(9i) + ^ дг}

2° for any l e A, in the class of functions belonging to G(T, R™) and satisfying the condition 0 < gt < gt , l e A, the function gt = 0, l e A is the only solution of the equation

9i = ®i(9i), l e A .

Let us define the sequence {xn} by the relations

{ 2 ) ® n + i ~ f ( x n ) i n = 0 , 1 ,

where x 0 is an arbitrarily fixed element of X .

To prove the convergence of the sequence {xn} to the solution x of equation (1) we define the sequence {gi>n}, l e A, by the relations (3) 9i.о = 9n 9i,n+1 = Qi(9i,n), l e A i n = 0,1, ...

Similarly as it was done in [3], [7], by induction, we can easily prove the following two lemmas.

L emma 1. I f Assumption B(7q) and condition 1° of Assumption A are satisfied, then

0 ^< 9itn+i < 9 i , n < 9 u 0 , 1 , . . . , ^{1 e Л ,} 9i,n- >0 for oo, l e A.

L emma 2. I f Assumptions A and B(7iz) are satisfied for hft)

= Qtjfi® o), a?0), 7 e T, l e Л, ht e G{T, Й” ), l e A, and i f x is a solution of equation (1) sueh that Qt>i{oc, x Q) < 9i{t), t e T , l e Л, then the estima­

tions

Qi,i(xn+p 1 Xn) ^ 9i,n(t) i Vi ^ 1? • • • i I ^ T, I g A, Qt,i(xn i x ) < 9i,n(t) > n = 0, 1, ..., t e T , l e Л, hold true.

By Lemmas 1 and 2 we obtain

T h e o r e m 1. I f Assumptions A and В(& г) are satisfied with ht(t)

= Qt,i{f(x o)i x o)i l g T, \ e G( T, R™), l e A, then there exists a solution x of equation (1) being the limit of the sequence {xn} defined by (2) and the following estimations

(4) Qt)l(x, x 0) < gj(t), t e T , l e Л,

Qt,i(v > xn)< 9г,п(1) i n = 0 , 1 , . . . , t e T , l e A,

hold true, where the sequence {дг>п} is defined by (3). A solution x of (1) is unique in the class of functions satisfying (4).

2. The main lemma and some remarks. If M is an m x m matrix, then M > 0 means th a t all elements of the matrix M are non-negative.

We denote by q (M) the spectral radius of the matrix M.

How we give conditions under which the family of non-linear opera­

tors Qi defined by the formula

n ns ô S,Vs(i) p

№iU)(t) = JTw ?’rs ( 7 , r - e&?*rs(i) f u(T)drj + £ À u (t)u(Pi(t)),

s = 1 rs=l 0 i=l

t e R +y e A, where cof^, Ц,Гз, Ku, are non-negative, continuous vector-functions and matrix-functions respectively, and <5s,rs,

& are continuous functions on R +, satisfies Assumption B(7q). By the

10 — Roczniki PTM Prace Mat. XX 2

380 M. K w ap isz and J. Turo

general considerations it follows th a t these conditions will be sufficient for the existence, uniqueness and the convergence of successive approxi­

mations for equation (1) or for its special cases considered in adequately- chosen spaces.

Let

5S,rS(t)

8 = 1 r- = l

2o> î-r’ ( t , t n- 3kpr4t) / «(T)dr), I e R , 0

t

W ’ n m = j u ( г » * ) , 0

(X,«)(«)= «(&(*)), г е Л .

».

г - 1

P u t Xf = ХгХ*-1, к = 0 , 1 , . . . , X° = I, l e Л, where 1 denotes the identity operator in F ( R +, R+).

From the definition of the operators L u l e Л, it follows th a t

P P

№*«)(<) = У ... .... •*№),

i i = l i fc= l

where

A (<) = A(«), /& -* * + • «) = A 1 '* (Ai+1 <*)) ’

&(*) = A„(t), *;•;<*+■№ = k ik+1 W A r - ik( K +i W>

Ь ~ 1» • • • » V > & = 0 ,1 , ...

P ut

00

j ¥ , « =

k — 0

with the pointwise convergence of the series in R +.

By the same way as in [2], [6], [8] and [10] (being some generaliza­

tion of the method use in [3]) we get

L e m m a 3 . I f

1° the vector-functions o)i,rs: R + xR™->R™ are continuous, non-de­

creasing, cof’r« (t, 0) == 0, and there exist constants df'**, g 8^ 8 e R +, v\,r* 6 [0 ,1 ] such that

«rv.,rf’r*

(5) yv) < df’^ fi ||y|| coïr»{t, v) for t e R + , /ле (0 ,1 ]

and for any constant matrix y > 0, s = 1, ... , n, r8 = 1, ..., n9, l e A,

2° the functions \ e C{R+, R™), Xu e C ( R +, R f ) , ôs’r*, ft e C { R +, R +), are non-decreasing, ô8,fa{t), ft(t) e [0, t], t e R +, s = 1, n, rs = 1, ...

. . . , n s, i = 1 , ..., p, l e / 1 (the case p — +oo is also possible), and (6) $i(t) — (AIihx)(t) < c+ °°> l e Л, t e R +,

(7) 8 ‘/ - ( t ) ü ( M l a‘l^ ) ( t ) < + o c , l e A , t e R + ,

where fff’sff) = [tn- ’ l}tf’r>(l)\\Ss’r° (t)y and sup№'»(«)«

< +oo,

3° for any l e A, there exists a non-decreasing function gt e C{R+, R™) being a solution of the integral inequality

n n s

£ d y * W r‘m + s„

8 = 1 Ts= 1 where ё8,Га = df,r*sup [Sf’rst 1 ];

4° for any l e Л the function дг — 0 is the unique solution of the equation

' П n8

ft = Z ] ? d r ‘Vl-r*9„

s = l rs = 1

in the class C (R +> , R™), then

(a) for any l e A there exists a non-decreasing solution щ e G( R+, R™) of the equation

(8) щ = MjQfUi + Mihi,

(b) for any l e A the function ux is a solution of the equation

(9) щ — QiUi~\-LiUi-j-hi,

(c) any solution щ e MX( R+, R™, щ), l e A, of equation (9) is a solu­

tion of equation (8), where

M X( R+, Jt“ , щ) = {и: и ^e M ( R +, R™), fw ]< + oo}, M = inf {1^1 < сщ, c e R +}

C

and A I(R +, R™) is the class of all non-negative and non-decreasing vector- functions defined on R +,

(d) for any l e A the function щ — 0 is in the class M f R +, R™, щ) the unique solution of the inequality

щ < йУщ+ЪгЩ.

R e m a rk 1. Condition (6) assumed for any у > 0 strongly restricts

the class of fun étions cof,rs satisfying the assumptions of Lemma 3. This

382 M. K w a p isz and J. Turo

condition can be weakened if we assume th a t (5) is satisfied only for any constant у 6 (0 ,1 ].

Now the functions oj,rs in conditions (7) are of the form af,r 3(t) =

s , r 4 ® Qj » „ s,rR

= [(5s,rs(t)] -t 1 and therefore instead of the operators Qf,rs we must t

take (Di,rsu)(t) = а)£*г«(<, tn~skf,rs(t) f u(r)dr).

0

E e m a r k 2. The examples of vector-functions cof,rs satisfying the assumptions of Lemma 3 are:

(a) the functions wi’rs(t,v ) — A%,rs(t)v, where A \,Vs e G(R+, JR+2) are non-decreasing and JL£,r*(Z) < Cf,rst 1 , Gi,rs ^ 0, qfrs > 0, t e R + (now con­

dition (5) is fulfilled with the constants d f rs ^ 1, vf,rs = 1 ) , (b) the functions defined by the formula

0 for v = 0, (Osy rs(t, V) = «|ln®| for 0 < v < e_1,

V for v > e~l. s = I, . . , n, T 8 = 1, .. . , n s, l e A (condition (5) is now fulfilled with the constants Щ’г> > 2/Ve, = 0, vf’rs — J and y e (0 ,1 ] — see Eemark 1).

E e m a r k 3. Now we give some effective conditions under which conditions (6) and (7) are fulfilled.

(a) If we assume th a t

(io) *„(*)< i„ , № ’rsm < M > r*t, ôs’rs ( t ) ^ ô s’r*t, д о o < A * , t e n +>

ôs,rs, Â g [0 ,1 ], g l l +, Xl{ are some constant matrices s = 1, n, rs = 1, . . . , n s, i — 1 , l ^g A, then conditions (6) and (7)-are satis­

fied if

oo p - p к к

1 2 - Ш 1 К Н * П ^ ) < + ° ° ’ ( e K + >

and

k=

P

0 i1=l ik= 1 m==x m — 1

É>(2 ^ l « i—1 ) < i , i G Л, X ~ min[(?i

s,rs ~ S + 2) vs,rs + qs,rs^ •

i? = 1, .. • 1 ^ t ~ • * • ? >

where we mean

oo p P oo p p 0

2 2 - ■■ 2 j Wk >4 = fo + V V _j L j j L j ■ 2 & ■•Л / K - L -

k —0 ii=l {k=l k= 1 *x=l tk==1 m=l

(b) If лн (t) < tin, (Oil < îcï’r*t, 0s’rs( t ) <à в,Гв<,

l*>rs 6 [0,1], A e [o ,i)i and if hi(t) < tHi, н г> ъ, Z ^g yl, then conditions

(6) and (7) are both fulfilled.

(c) Finally, if we suppose (10) and h^t) < A ffz, l e A , t e R +, for some

p M вг e R + and vector I I г > 0, then (6), (7) are satisfied if о (]? i= 1 < 1 with i = min{0z, min[(w —s + 2)rz’rs + </z’,'s]}, l e A, s = 1, 2, ..., n, rs

= 1, . . . , n s.

Particular cases of this conditions are the conditions appearing in [5], [7], [8].

3. Integral-functional equation with an unknown function of n variables. Let Y denote some real linear topological Hausdorff space, locally convex and sequentially complete with the topology given by the family seminorms {|-|z}Ze/1.

Let {II • ||}Zs/1 be a family of generalized seminorms in Y, the values of which lie in R ™, such th a t there exists numbers bx > 0, b2 > 0 sat­

isfying the condition

v l M i i |U » < for any v e Y ’ l e A -

We suppose th a t X from Section 1 is now a topological space С (I , Y) of all continuous functions defined on 1M = [ж: 0 < æ < -f oo], со, 0 e R n, with range in Y, with the topology given by the family seminorms (||* ||Zjfc}, where

INI/,* = max IMaOII*, l e A , z e C ( I eoiY ) f I k = [oo: 0 < æ < Щ,

x e I k

0 e R n, Tc = {( a , ...,/i) e R n, p = 1 ,2 ,...

If Y is sequentially complete, then 0 (1 ^ , Y) is also of such type.

In this case family of semimetrics (see Section 1) can be introduce by a different manner. We take for z ,z e X = 0 (1 ^, Y)

QU (z,z) = \\z-z\\ht, where ||z -z ||M = max ||«(®)-g(®)||„

l e A , t e R + , and

IN = l*xl+--. + l*»l for any * = ( * ! , . . . , sn) e R n.

For the class 0 ( T , R™) now we shall consider the class of all non-negative and non-decreasing vector functions defined in й + .

Now we consider the integral-functional equation in the unbounded domain I TO of the following form

v\’n4x) v]ln4x)

(11) z(co) = f [ x , f ... J f ’ni(x, rlf ..., Tn , z[%x, ..., rn))drx ... drn,

о 0

У 2,1(х) -4’V)

/ ... J f 2,1 (oc, T2, ..., rn, z(y\>l {x), r 2, ..., rn))dt2... drn, ...

о 0

384 M. K w ap isz and J. Turo

vl'n4*) v2 n-i№

J ... J / 2’П2(®,г1,...,т п_1,«(т1,...,г п_1, й п*(®),)Дт1...Лгп_1,...

о 0

Vnn ’ \ x )

..., J f n,1(co, Tn, z(y^l (SG), . . . , y £ i ( ® h rn))drn, .

•.., / / n,”n (ж, T1? 2 r(r1? y l’n4x), ... (Ж))) dTj,

о

*(a}(®), •••, ai(®)), . ..,« K ( a > ) , <*£(<»))) =

where тг3 = an^ ^ e known functions f s,r«: I ^ x [0, + oo)M 8+1 x xY ->Y , loo X Y V+P->Y, v = 2n - l , A » 1: г. = 1 , . . . , л „ s = 1, . . . , n, are continuous.

The integrals appearing in equation (11) are generalized Riemann integral being defined by applying the ordinary defining procedure.

R e m a rk 4. If instead of !«, we take I a — [0, a] and Y = E being a Banach space, then we have the equation of the form (11) which was

considered in [5], [8].

(b) The Darboux problem for the differential-functional equation ux1 (v2,1(^)), •••» % ^п ( у 2,П2И Ь •••

• • • > (yn>1 И » • • • » ^ 2...*п(уп'п” И , %...■* (а1И » —

can be reduced to a particular case (f8’r3{ot>,...,z ) = z) of equation (11).’

If the function F depends only on the first n - f l variables, then we have the differential-functional equation which has been considered in [9].

(c) The particular cases of equation (11) are also the integral-func­

tional equations which were considered in [4], [7] if n — 2 and in [3]

if n — 1.

A ssu m p t io n 0. Suppose that

1° there exist continuous and non-deereasing rector-functions <о8,Гз:

R+ x R™ ->R™, and matrics-functions l u : R +->JR™2, such that

|| F ( x , 21,ni, ..., zn,nn, v 1, ..., v p ) —F( %, z1,ni, ..., z“,n», v 1, ..., Vp)|||

n n e P

< Z Z m‘’r‘ <il*">ii2‘,r' - 2‘,r'ii<>+ Z д« 1 (i w i ) - 1 s'il> l e A >

s = l r a= 1 i = l

f o r e Y, s = 1 , r, = l , . . . f n 8, i = 1 ,

2° the functions f 8,rs satisfy Lipschitz condition with respect to the last variable, i.e., there exist continuous and non-decreasing matrix-functions 1c8,rs: R +-+ R f such that

\\f8'r*{x, . . . , z ) - f s’r*(x, z)||*< fc?,r*(INI)ll*-«ll/» l e A , s = 1, ..., n, rs = 1, ..., na, x e l ro, z , z e Y ,

3° there exist non-decreasing functions è8'*8, f}. e G(R+, R +), d 8,r9(t), Pi(t) e [0, t], t e R +, such that

ll/'r*(®)ll<^r*(INI), * = 1, ...,n , re = l, ...,we, w, = ( g"j), lla 'H IK РЛ\М), i = 1 , х е 1 ж.

P u t

(12) ht(t) = \\SFzQ- z 0\\ht, l e A, t e R +,

where 3F is defined by (11) and z0 is an arbitrarily fixed element of 0,(1^, Y).

Let ns define the sequence {zk} by the relations (13) zk+l = &як, f c = 0 , 1, ...

By induction and Lemma 3 we obtain

L emma 4. I f assumptions of Lemma 3 are satisfied, then 0 <Щк+1 < ипс<Щ, Те = 0 , 1 , . . . , l e A,

ulkz£ 0 for Tc-+oo, l e A ,

where the sign i t denotes the uniform convergence in any compact subset of R +, and

UlQ -— 'll] •

(14) Щк +1 = щЩк + Ц и 1к, Тс = 0,1, ..., l e Л,

and щ is the solution of equation (9) and operators Q\, L t are defined in Section 2.

L e m m a 5. I f Assumption О and assumption of Lemma 3 are satisfied, then

II**-*olli,i <«*(*)> Tc = 0 , 1 , . . . , l e A, t e R +, and

I t e k + j - z A t ^ u M , h ,j = 0 , 1 , . . . , l e A, t e R +.

From Lemmas 3, 4 and 6 it follows

T heorem 2. I f Assumption C and assumptions of Lemma 3 are satisfied for ht defined by (12), then there exists a solution z e 0 (1 ^ , Y) of equation (11)

and the estimation

^ uik(t)i f c = 0 , 1 , . . . , t e R +f l e A ,

386 M. K w ap isz and J. Turo

holds, where zk and ulk are defined by (13) and (14). The solution z of equa­

tion ( 1 1 ) is unique in the class of functions Z ( I Т, щ) , Z ( I œ, У, щ)

= l>: z ^g СЦы, Y), df z , z0) < + oo], where dt(z, z0) = inf [ \\z -z 0\\lt <

C

< cut{t), c e R +, l e Л].

4. Continuous dependence of solutions on the right-hand side. Let ns now consider another equation

? { > « !( * ) y ] l n 4 x )

(16) v{x) = F (a?, J ... f f hni(æ, т1} ..., rn, v(rlt ..., rn))dr, ... drn,

о 0

p 2' \ x ) у % \ х )

f ... / f 2,1(®, r 2} .. . , rn, v(y\>x{x), r2, ..., Tn))dr2 ... drn, ...

о 0

у \ п Ц х ) У п И \( х )

..., J ... J ^{/ 2' П 2( л ? , т} 1 , . . . , T n _ 1 , ® ( т „ . . . , т га_ 1 , Й ’ и2 ( а ?) ) ] й г 1 . . . й г п _ 1 , . . .

о 0

. .., J Z”’1 (®, Tn , « (y ”'1 (®), ..., y l L1! (a>), т J ) drn, ..., 0

У^'ПЩх)

j p,nn (a, , Tl, v (Tl, У»•»» (®), . . . , y (®))) dr,, 0

®(5}(0), ...,ô i(® )), ...,«(5f(a?), ...,o*(®))) = (£v){< 0),

where / s,r«, ys’r», аг, are the same kind as F, f s>rs} ys,rs, s = 1, ..., n, r8 = 1 , i = 1 , . . . , ^ .

Let г; be a solution of equation (15). P u t

<Pi{t) = < e R + , 1 е Л,

where 3F is defined by (11), and let wl e G{R+, JR+), I g A be such th at P — v||w < t e R +, l e A .

P ut

ht(t) - max{to,(«), 9 >i(«),M 0 b ÎG /1 -

T h e o r e m 3. I / Assumption C, relations (12) assumptions of Lemma 3 are satisfied with ht replaced by then there exists a continuous, non-negative solution щ of the equation

Щ = Qi щ -\-LiUi + 9 Oj, I g A , such that

\\z-v\\ i.* < « i(0 , t e R + t l e A .

5. Another integral-functional equation. Kow we consider the case cof (t, ®) = Ц (t)v, I e A, where Ц are non-decreasing matrix-functions (now the double index s ,r s, s = 1 , ..., n, rs = 1, ns, ns = is replaced by single index s, s — 1, ..., S, S = 2n— 1).

Since the vector-functions cof are linear (with respect the second variable) therefore assumptions 1°, 3°, 4° of Lemma 3 are obviously ful­

filled.

We have

L em m a 3 . I f h l E(J(R+,R%), Jcf, Xu e 0 (R +, R f ) , Ô3, ft e 0 (R +, R +) are non-decreasing, 6s(t), p{(t) e [0, <], t e R +, s = 1, S, i = 1, ..., p, l ^e A (the ease p = + oo is possible), and

$i(t) — (ilfj/ij)(l) 4“ °°? t

R +, l ^e A ,

$i(t) — < + 00> i £ R+ > ï e /1, where hpt) = (^)<5s(i) <md i f su p --- < +oo, then

s^l t

(a) for any l ^e A there exists a solution щ ^e C( R+, R™) of the equation

8 às(t)

щ — where (Кгщ)Ц) = ^ Jcf(t) J ut(x)dx, l ^e A,

S = 1 0

(b) for any l ^e A the function щ is a solution of the equation иг = К г щ + L tUi A'hi, l e A ,

(c) assertion (c) of Lemma 3 is satisfied,

(d) for any l ^e A the function щ = 0 is in the class M t(R +, R™, Щ) the unique solution of the inequality

Щ < K ^ + L ^

(the operators М г, L l and the class M t(R +, К™, щ) are defined in Lemma 3, where Щ appearing in definition of this class is defined in (a) of Lemma 6).

К олу we consider the integral-functional equation of the form ÿî(x,s()) Vln { x , z {))

(16) z ( x ) = f ( x , f ... f f 1 (aff r1 J . . . , T„f e(rl t . . . f Tn))dv1 ...

о ⁰

' Vfx.zV)) Ÿrn(x’eA)

... d tn ... ; J* ... J f r ^ x , Ti , . . . , Tn, z ( x i , . . . , xn)^ dx\ ... dxn,

о о

г (a1 (a?)), . . . , z ( ap(x))} = (^z)(x),

388 M. K w ap isz and J. Turo where

v \(x ) q3n (x)

*(•)) = У(®, J • •• / Ф и •••> *n)d*i ••• drn, *К(®)))»

О О

and the functions F : I œ x Y r+p->Y, fc: I ^ x I ^ x Y - ^ Y , y3: I ^ x Y x x Y-> Iœ, a \ rf, aj : I ^ I ^ , j = 1, ..., r, i = 1, . . . , p , are continuous.

R e m a rk 5. The differential-functional equation

(17) иХх_ Хп{х) u[yx{x7 u ^ x ) ) , ...

w (/(® , « ф г(®)Ь \ . ÆJ f f r (a;)))),

^ X j . . . a ; n ( a * ( ® ) ) ) • • * ) '^'xx...x n ( a ^ ^ l o o 1

with conditions of Darboux type can be reduced to a particular case (/i(®, t j , rn, *) = e, j = 1, ..., r) of equation (16).

If n = 1, then equation (17) has the form

•

(18) u'(x) = f ( x , u[yx(x7 и{г)х{х))7и'{вх{хЩ, . . . , « ( / ( x, u(r)r{x)), ц' (ar (ж)))), и' (аЧ®)), ..., и' ( а * » ) ) , of which a particular case: r = 1 and F does not depend on the last p variables was considered in [11].

Theorem on the existence of solutions of equation (18), in the case where r = p = 1 and y does not depend on the last variable, and with assumption th a t the function F satisfies Lipschitz condition with respect th e last variable with constant less than 1 can be found in [1].

A s s u m p t io n D. Suppose that

1° there exist non-decreasing matrix-functions klj7 klj7 mlj7 e C (R +, B ? ) , such that

IIF ( x 7zX7 . . . 7zr7v17 . . . 7vp) - F ( x 7zX7 . . . , z r , t>1 ...,t>1,)||1

Г p

« 2 % ' (M)ll%- З Д + ÿ% <ll*ll)ll«i-0A ,

i = 1 i = 1

Щ х 7 rlf ..., xn7z ) - f i {x7 xx, ..., тп, з)||г< ^ ( И ) Н 2!-2|1г, j = 1, . . . , r ,

ll/jf(®, *11 •••, *n,z)\\i< Шу{\\х\\)-е, j = 1 , . . . , r ,

for x, x e /«j, ^ е Л , zj7 zj7 v{7v{ e Y, j = 1, . . . , r , i = 1, . . . , p ,

2° there exist non-decreasing functions /л^, vy g G{R+ , II™ ), yt , щ , 5if a{ 6 C( R+, R +), such that

ly»(®> z)|e < fitj (||a?||)||w — Щ\г -f Vy(||o ?||)\\z — z|||, l e Л,

||у*(я,и,г)||< у,||(® )||, У ( х ) \ \ < ъ ( М ) ,

H^(®)ll<ôry(|HI), 3 = l,...,r,

||а>)11 < « < ( » , * == 1,...,JP,

the norm ||*|| is defined in Section 3, <wd e e R m is the unit vector, i.o.f Put

M f) = + j = 1 , r, ï e A ,

(19) AK(t) = tn_1 (t)vH{t)mu (t) -Ь Ди(t), i — 1 , ma x( r , p) = g, PAG = max(5<(«), 5,(t)), i = 1 , . . . , m a x (r,p ),

< W = m a x(£;(<)> fy(*)b 3 = 1, ••*,

мЛеге AK(i) = 0 , аг(<) = 0 /or i = p - f l , . . . , r i f r > p and <n_1fcw(#)X X ^ (t)% (« ) = o, aw(t) = 0 /or i = r-f-1, . . . , p i / r < p .

Let us construct the sequence {zfc} by the relations

(20) 0fc+1 = «^3*, = 0,1,

where the operator is defined by (16) and z0 is an arbitrarily fixed element of 0 (1 то, Y).

From Lemma 6 it follows

T h e o r e m 4. I f Assumption D and assumptions of Lemma 6 are satis­

fied for ky, pif ôj, j = 1 , r, i = 1 , ma x (r,p ) = g, de/med % (19), then there exists a solution z g G(Iж, Y) of equation (16) and the esti­

mations

Ш - ZkWi.t < ЩкУ) » fe = o, 1, ..., Ï G A, t e R + , hold true, w/iere zk are defined by (20),

%> = %fc+i ^ к = 0,1, ..., I g A,

and u{, K lf L t are defined in Lemma 6. The solution z of equation (16) is unique in the class Z (ITO, Y , щ) (see Theorem 2).

P ro o f. At first we prove the following estimations (21) ll**-*olli.i< W ) , & = 0 ,1 , , t e R +, l g A , (22) II %k+m zk%,t ^ uikW)i ifc, w = 0 ,1 , .. . , t G R + , I g A.

2

390 M* K w a p isz and J. Turo

I t is obvious th a t (21) holds for к — 0. If we suppose th a t (21) holds for some к > 0, then we have for x e 7^, ||a?|| < t

r v \ \ x , z k {-)) Pn ( x ,s k (-))

ll**+l(®)-g!o(®)lll<^, ^(ll®ll)^(ll®ll) J ••• / IM* 1 , T„)-

J = 1 о 0

? • • • i rn)^i^ri • • • dTn -j- Г

+ tn~x ^ к^(\\х\\)ти(\\х\\)е\у3п(х, zk( ‘)j - f n(x, zQ(-))\ + j=i

p

4- ^ J u (\\x\\)\\zk (ai (x))--z0(ai (xj)\\l ^ h l(t) i=l

y { ( x , z k ( . ) ) y

3

X k j W h j i t ) / ••• / ûl{r1+ ... + Tn)dT1 ... drn +

3=1 о 0

n-j ( v\ d I

+ tn- 1^ k lj(t)/iij(t)mij ( t ) j ... J \lzk (T1, . . . , Tn) - z 0(T1,...,T n)\lidT1...dTn+

3=1 о 0

r p

+ tn~x £ кцМщЮ' ГцУ) |K (M ^)) - *о(о* (®))\\i + Уз *МИа<И11) + MO

j = 1 г = 1

r _ llv-'(a:»«*(*))ll * »• 11^(011

< Г"1 £ кц{г)ТсцЦ) j üz(T)dT + (tn- 1)2^ f c ?i(t)Wy(f)/iy(t) J %(r)dT +

+ *'

r JJ

' '1 £ h (0 Щ (0 (0 K (0) + 4 (0 щ (â (0) + (t)

j = i i=i

r

< tn~l { M(0[M(0 + <n_1^ ( 0 ^ j( 0 ] J 5г(г)йг} +

i«a о

+ ^ P” = ( Kj i i i ) ( t ) ( L^i ) ( t ) — иг{ 1).

г=1

Now (21) follows by induction.

Similarly we prove (22).

From estimation (22) and from ulk->0 for k-+oo, l e Л follows the

existence of a solution of equation (16). The uniqueness p art of the theorem

follows from (d) of Lemma 6. Thus the proof of the theorem is completed.

R e f e r e n c e s

[1] L. J. G-rimm, Existence and uniqueness for nonlinear neutral-differential equa­

tions, Bull. Amer. Math. Soc. 77 (1971), p. 374-376.

[2] Z. K a m o n t and M. K w a p isz , On the Cauchy problem for differential-delay equations in a Banach space, Math. Nachr. 74 (1976), p. 173-190.

[3] M. K w a p is z and J. T uro, On the existence and convergence of successive approxi­

mations for some functional equations in a Banach space, J. Diff. Eq. 16 (1974), p. 289-318.

[4] — — On the existence and uniqueness of solutions of the Darboux problem for partial differential-functional equations in a Banach space, Ann. Polon. Math.

29 (1974), p. 89-118.

[5] — — On some integral-functional equations with unknown function of n variables, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astr. Phys. 22 (1974), p. 1001-1009.

[6] — — Existence, uniqueness and successive approximations for a class of in ­ tegral-functional equations, Aeq. Math. 14 (1976), p. 303-323.

[7] — — Some integral-functional equations, Funkcialaj Ekvacioj 18 (1975), p. 107-162.

[8] — — On a class of functional-integral equations treated by the comparative method, unpublished.

[9] A. P e lc z a r , Some functional differential equations, Diss. Math. 100 (1973), p. 3-110.

[10] J. T uro, On the existence, uniqueness of solutions and convergence of simple and approximate iterations of the integral-functional equations with an unlcnown function of several variables (Polish), doctors thesis, Faculty of Mathématics

Physics and Chemistry, University of Gdansk, 1974.

[11] L. A. Z y v o t o v s k iï, On the existence and uniqueness of solutions of differential equations with delay dependent on the solution and its derivative (Russian), Diff.

Urav. 5 (1969), p. 880-889.