..., J ; ), j ; ...) (m = 1, 2, ...). On integro-differential equations in para-normed space

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXIV (1984)

J. P o p io l e k (Bialystok)

On integro-differential equations in para-normed space

Ugowski [ 6 ] proved the existence theorem of solutions of the first initial-boundary value problem for the system of parabolic integro- differential equations. The proof of this theorem is based on Schauder’s fixed point theorem and Friedman’s a priori estimates for a single parabolic equations ([3], p. 65 and p. 191).

Author [4] considered the first initial-boundary value problem for the following infinite system of parabolic integro-differential equations:

(1) £ а% (x, t)DÎ и "+ £ ЬТ (х, î)DXjum + cm (x, t)u " -D ,u "

i,j — 1 J i = l

= f m (x , t, u \ u2, ..., DXl u \ DXl u2, ..., DXnu \ DXf u2, ...

..., J u1 (y, t)nl (x, t ; ^dy ), j ^u 2 (y, t)\i2 (x, t ; ^dy), ...) (m = 1, 2, ...).

g , g ,

Method of proof of the existence of solutions is similar to the method applied by Ugowski in [ 6 ].

In this paper we are dealing with the existence of solutions of the first initial-boundary value problem for the system of type (Î). We take advantage of the Schauder-Tichonov fixed point theorem for a locally convex space [5].

Let G be a bounded open domain of Euclidean space En+1 of the variables (x, t) = (xl 9 x2, ..., x n, t), where n ^ 2, enclosed by two domains R 0 and R T lying on the planes t = 0 and t — T = const > 0 respectively, and by a surface S located in the strip 0 < t ^ T. By I we denote the parabolic boundary of domain G, i.e. E = R 0 u S .

Let us introduce the following norms ([3], p. 61)

. iG I , 0 4 1 I ,G î |G î \ U ( P ) - U ( Q )I

Mo = sup |u(P)|, « = Mo+ sup r -- ,

P e G

P,QeG

Mf+a = Ыа + J \DxM\S> M2 + « = MÎ + « + Z l ^ x M S + \Dtu\S, i,j= 1

i= 1

where d(P, Q) — the parabolic distance of points P, Qe G; 0 < a < l

By Ck+a{G) (к = 0, 1, 2) we denote the set of all functions u(x, t) for which \u\f+a < oo.

Set и (x, t) = (u1 (x, t), u2 (x, t), ...), û(x, t) = (n 1 (x, t), Ü2 (x, t), ...), where um(x, t), й™(х, t)eCk+a{G) (m = 1 ,2 ,...; к = 0, 1, 2; 0 < a < 1). We define the following operations:

w + u = (w 1 +Ы1, m 2 + m 2, rju — ( щ 1, щ 2, ...)

(rj is a real number).

Then the set Q°+a(G) is a linear space of all vector-functions u(x, t). In the C£°+a(G) we define the function

( 2 ) NlF+« = I ^ 2m Mm + \u ^{l m[G} k + a where constants Mm > 0 will be specified later on.

The function || ||fc+a is a para-norm, because it satisfies conditions (F l)~

(F4) ([1], p. 103). The space C^°+a(G) with the para-norm (2) is a locally convex space ([1], p. 313).

For every 0 < t ^ Tlet Bm (m = 1, 2, ...) be an operator defined on the set of all vector-functions u(x, t) regular in GT with values belonging to the set of all functions defined in GT\ I \ where Gr = G n {(x, t): 0 < t < t } and Г = I n |(x, t): 0 < t ^ t }.

We shall consider the problem

(3) L" и" s £ aTj (x, t) D1 u " + £ b? (x,t) Dx. u” +

i j = l i = l

+ cm(x, t)um- D t um = Bmu, (x, t)E~GT\ r , (4) um(x, t) = cpm(x, t), ( x , t ) e r (m = 1 ,2 ,...) .

By a solution of equation (3) we shall always understand a regular solution, i.e. continuous in the domain GT and possessing in Gr \ I z continuous derivatives appearing in Lmum.

The following assumptions are introduced (/, j = 1, 2, ..., n; m

= 1, 2, ...; 0 < т ^ T).

I. For any (x ,t ) e G and ÇeEn we have a” (x, t) — а™(х, t), П

а%(х, t)£i£j ^ K 0\Ç\2, where K 0 is a positive constant.

i,j= 1

II. The coefficients of Lm satisfy the uniform Holder condition with exponent a in G, and, moreover, a ^ e C 1^ 0(S) ([ 6 ], p. 256).

III. The sufrace S belongs to C 2 +an C 2- 0 ([ 6 ], P- 256).

IV. The functions (pm(x, t) are of class C1 +P(ZT) n С 2 +(Х(ГГ), a < fi < 1.

V. If <PeC?+f({Gx) n C ? +a{Gx) and Ф = (p on Г , then ВтФ = Ьт(рт on dR0.

VI. Operators Bm are continuous in the space C f +a(Gx), i.e. if u, use C f> +a(Gx) and lim || ms —и||?+в = 0, then lim \\Bmus —Bmu\\oT = 0.

s - * o o s-> oo

VII. Operators Bm map the space C*+a(Gr) into the space C 0 (GT) and furthermore the following inequality holds

(5) |B"u|g

where Mm > 0 is such that

( 6 ) К (a) Tr \Lm Фт|<>* + l^ m|i+ a « Mm( l - £ ( a ) r ')

for a constant K(a) depending only on a, K 0, K 2, К ъ and domain G ([ 6 ], p.

262); y = î ( l —a).

T heorem 1. I f assumptions I-VII are satisfied, then there exists a solution u(x, t) = (u 1 (x, t), u2(x, t), ...) of problem (3), (4); moreover, ueCf+p{Gx) n

п С 2 *+£(С'), 0 < £ < 1.

P ro o f. Denote by A the set of all vector-functions u e C f +a(Gx), such that |мт|?+а ^ M m and um(x, t) = q>m(x, t) on I х for every positive integer m.

Now for u e A consider the following problem

(7) L mvm = Bmu (x , t ) e G x\ r ,

( 8 ) vm{x,t) = (pm{x,t) (x , t ) e Z x {m = 1 , 2 ,...).

By virtue of assumptions I-VII and Lemma 1 ([ 6 ], p. 258) problem (7), ( 8 ) possesses a unique solution v(x, t) belonging to C 2 +e(Gx); Furthermore by (2.5) ([ 6 ], p. 262) u eC ? +ls(Gx).

Now, we define on A a transformation Z setting v — Zu. Applying a topological method based on the theorem of Schauder-Tichonov ([5]) we shall prove that Z has a fixed point.

At present we show that Z(A) cz A. In view of inequality (2.5) ([ 6 ], p.

262) we have

(9) И ? '+, « K(«r»(|B"a|g' + |L" 0 4 f ) + |<2>"|f+(1.

Assumption VII applied to (9) yields the relation

(Ю) \ v l ?+, « м т.

It also follows from (10) and assumption IV that veA. Note further that Z is continuous, i.e. lim ||ue — к||?+в = 0 implies lim ||Zus — Zu\ff+a = 0.

s oo .s oo

It is easy to see that the set A is a convex and closed set. To finish the

Proof of theorem it remains to show that A is a compact set.

Let N be a fixed natural number and let A N be a set of all u e A for which um — 0 for m > N.

Since the functions uN — (u1, u2, . . uN, 0, 0, ...) are uniformly equicontinuous the by theorem of Arzèla ([1], p. 371) it follows that A N is relatively compact in Cff+p(Gx), whence by theorem of Fréchet ([2]) A is a compact set.

Thus, as all assumptions of Schauder-Tichonov theorem ([5]) are satis­

fied we are confirming that the transformation Z has a fixed point, i.e. there exists u 'e A such that Z i i —u'. Observe that u' satisfies (3), (4) and u'EC f+ji(Gx) n C£+S(GX), for some 0 < e < l . Q.E.D.

Now we derive a corollary from Theorem 1 concerning a special case of operators Bm. We introduce some more assumptions:

VIII. Let the functions f m(x, t, p, q, r) (m = 1, 2, ...) be defined on G x £ x+nx + x satisfy a uniform Holder condition in every bounded set G x H (H c : £ x+nx + x) and inequality (2.13) from [ 6 ], p. 264; furthermore

| / m(x, t, p, q, r)i ^ Mm (m = 1 ,2 ,...),

where Mm > 0 are positive constants and satisfy an inequality similar to ( 6 );

functions f m are defined by relation ( 1 ), i.e.

p ~ u = (u \ u2, ...),

q ~ Dxи = (DXi u1, ..., DXnu \ DXx u2, ..., Dx u2, ...),

r S j u{y, t)fi{x, t; dy) = (J u'iy, r)/d(x, t; dy), j u2{y, t)p2{x, t;dy), ...).

Gt Gt Gt

IX. The measure pm(x, t : D) (m = 1 ,2 ,...) defined on . # (•// the <5- field of all Lebesgue-measurable subsets of the domain D0 = (J Gt) satisfies conditions (1H3) from [ 6 ], p. 259-260.

C orollary . Let assumptions I-IV and VIII-IX befulljied; then Theorem 1 is true in the case

Bmu = f m(x, t, p, q, r).

Proof of this corollary follows immediately from Theorem 1 and Lemma 4 ([ 6 ], p. 260).

References

[1] A. A le x ie w ic z , Analiza funkcjonalna, PWN, Warszawa 1969.

[2 ] M. F r é c h e t, Quelques propriétés des ensembles abstraits, Fund. Math. 12 (1928).

[3] A. F r ie d m a n , Partial differential equations o f parabolic type, Prentice-Hall, Englewood

Cliffs (1964).

[4] J. P o p io le k , O pewnym zagadnieniu granicznym dla ukladu rôwnan rôzniczkowo-calko- wych, Zeszyty Naukowo-Dydaktyczne Filii UW w Bialymstoku.

[5] A. N. T ic h o n o v , Ein Fixpunktsatz, Math. Ann. I l l (1935).

[6] H. U g o w s k i, On integro-differential equations o f parabolic and elliptic type, Ann. Polon.

Math. 22 (1970).

INSTITUTE O F MATHEMATICS, UNIVERSITY O F WARSAW

BIALYSTOK BRANCH