• Nie Znaleziono Wyników

# On the existence of continuous solutions of nonlinear integral equations in Banach spaces

N/A
N/A
Protected

Share "On the existence of continuous solutions of nonlinear integral equations in Banach spaces"

Copied!
8
0
0

Pełen tekst

(1)

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series 1: COMMENTATIONES MATHEMATICAE XXX (1990) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXX (1990)

Ja n u s z Ja n u s z e w s k i (Bydgoszcz)

## On the existence of continuous solutions of nonlinear integral equations in Banach spaces

1. Introduction. Let £ be a Banach space and let D = [0, d] be a compact interval in R. In this paper we shall prove some existence theorems for the Urysohn integral equation

(1) x(t) = p(t) + À f f ( t , s, x(s))ds,

D

and for the Volterra integral equation

(2) x(t) = p(t) + J /( t, s, x(s))ds, о

where f , p and x are functions with values in E.

Our fundamental tool are measures of noncompactness. There have appeared a lot of papers using the Kuratowski measure of noncompactness in proving existence theorems for ordinary differential equations [1, 5, 6, 8, 10]

and integral equations in Banach spaces [9, 11]. In [3] Banas and Goebel have introduced an axiomatic approach to the notion of a measure of noncompact­

ness. The usefulness of their axiomatics was illustrated by an existence theorem for the Cauchy problem [3, Th. 13.3.1]. In our paper we shall show that axiomatic measures of noncompactness are useful for integral equations too.

Our notation is the same as that in [3] or [2]. Let ME denote the family of all nonempty and bounded subsets of E and NE the family of all nonempty and relatively compact sets in E.

De f in it io n 1. A function p: M£ -+[0, + oo) is said to be a measure of noncompactness if it satisfies the following conditions:

1° the family ker/x — { X e M E\ p{X) = 0} is nonempty and ker/x c NE, 2° if X c Y, then p(X) ^ p(Y),

3° p(X) = p(X), 4° p(convX) = p{X),

5° /х(лЛг+ (1 -л )Г )^ л /х (Х ) + (1-/)/х(У) for Ae[0, 1].

For a given measure p in the space E set Ец = {xeE: {xjeker^}.

(2)

86 J. Ja n u s z e ws k i

In what follows we shall need the following lemmas:

Lemma 1 [2]. I f ||X|| = sup{||x||: x e l } < 1, then li{X+Y)<n(Y)+\\X\\n(K(Y, 1)), where K(Y, 1) = {xeE: dist(x, 7) ^ 1}.

Lemma 2 [2]. Let tx, t 2, . . . , t n be nonnegative real numbers such that

£ " =1£г^ 1 and let р е Е ц. Then I

Kp+ t iix è ^ i t M p + x i)-

i = 1 i = 1

Lemma 3. I f X is a bounded equicontinuous subset of C(D, E), peE^ and 0 < Я mes D ^ 1, then

p(pA Я j X(s)ds) ^ Я j p(p + X(s))ds,

D D

where X(t) = {x(t): x e l } , jDX(s)ds = {Ji>x(s)ds: x e X ).

Lemma 3 is a modification of Lemma 7.2.4 from [2].

Moreover, in this paper we shall apply some modification of Sadovskii’s fixed point theorem [1 1].

Theorem 1. Let В be a bounded, closed and convex subset of a Banach space, ре В and let G be a continuous mapping of В into itself I f the implication (3) F c conv(G(K) и {p}) => V is relatively compact

holds for every subset V of B, then G has a fixed point.

2. The Urysohn integral equation. We consider the integral equation (1).

Assume that

I. p: D-*E is a continuous function;

II. (t, s, x) —>/(t, s, x) is a function from D2 x E into E which satisfies the following conditions:

(i) / is continuous in x and strongly measurable in (t, s);

(ii) for any h > 0 there exists a measurable function mh: D2^ R + such that ||f ( t , s, x)|| ^ mh(t, s,)(t, seD, ||x|| ^ h) and §Dmh(t, s)ds ^ a(h) < oo;

(iii) for any h > 0 there is a function dh: D3->R+ such that ||/(t, s, x)

—/ (t, s, x)|| ^ dh(x, t, s) (t, t, s e D , ||xII ^ h) and limt_t §Ddh(T, t, s)ds = 0.

Theorem 2. Assume, in addition to I and II, that the function (s, x)-*f(t, s, x) is uniformly continuous and р(7)е£д for teD. Moreover, assume that there exists an integrable function к: D2 —► R + such that- (4) p({p(t)} u (p(t) +f{t, s, X))) ^ k{t, s)p{X)

for any s, te D and for any bounded subset X of E. Then there exists q > 0 such that for any Я, 0 ^ Я < q, the equation (1) has at least one continuous solution.

(3)

Existence o f solutions o f nonlinear integral equations 87

P roof. Denote by C the Banach space of all continuous functions и: D-+E with the usual supremum norm || • ||c. Let r(K) be the spectral radius of the integral operator К defined by

Put

Ku(t) = J k(t, s)u(s)ds (u e C , teD).

D

q = min sup---- ’h~\\p\\

,й > О Ф ) r{K)’ mes D J For fixed XeR, 0 X < q, choose b > 0 in such a way that

(5) \\p\\c + Xa(b) ^ b.

Put В = {xeC: ||x||c < b}. Consider the operator G defined by G(x)(t) = p(t) + Xj f ( t , s, x(s))ds (x eB , t ED).

D

The assumptions I, II and (5) imply that G is a continuous mapping B-+B and the set G(B) is equicontinuous (see [7] or [9]).

Let F be a subset of В such that F c conv(G(F) u {p}). Obviously V(t) c: conv(G(F)(£) и {p(0}) f°r tED.

Let us fix teD . In virtue of the uniform continuity of/ the transformation (Jx)(s) = f(t , s, x(s)) maps equicontinuous sets into equicontinuous sets. Hence

p(F(0) ^ p(conv(G(F)(0 u {p(t)})) = p(G(V)(t) u {p(t)})

= p({p(t)} u (p(t) + X j (fV)(s) ds))

D

= p(p{t) + x \ (/V u {9})(s)ds),

D

by Lemma 3 (F c G (В) is equicontinuous) it follows that p(p{t) + X\${JVv{0}){s)ds) < X\ p(p{t) + {JVu {9})(s))ds

D D

= Я f n({p{t)} U (p(t)+f(t, s, V(s))))ds.

Thus, by (4), we obtain D

p(V{t)) ^ X J k(t, s)p(V(s))ds.

D

Because this inequality holds for every î e D and Xr(K) < 1, by applying the theorem on integral inequalities, we conclude that n(V(tj) = 0 for tED.

Hence Ascoli’s theorem proves that the set F is relatively compact.

Consequently, by Theorem 1, G has a fixed point.

R em ark. If the measure p has the property: p{{a) u A) = p(A) for any then the assumption (4) has the form p(p(t)+f(t, s, X)) < k(t, s)p(X).

In the next theorem we shall assume additionally that the measure p has the maximum property:

6° p(A u B) = max(p(A), p(B)) for A, BeM e.

Then arguing similarly to [1], we can prove the following Ambrosetti-type

(4)

J. J a n us z e ws k i

Lemma 4. Let V be a bounded equicontinuous subset of C(D, E). Then for each compact subset T of D

p(F(T)) = supp(V(t)), where V(T) = {x(t): x eV, te T } .

t e T

Theorem 3. Let p fulfil conditions \ ° ~ 6 ° and pit)eE^ for tED. Assume, in addition, that there exists an integrable function k : D2-+R+ such that for every tED, e > 0 and for every bounded subset X of E there exists a closed subset De of D such that mes (D\De) < e and

(6) p(p(t)+f(t, T x X ) ) ^ sup/c(L s)g(X)

s e T

for any compact subset T of De.

Then there exists g > 0 such that for each a, 0 ^ Я < g, there exists at least one continuous solution of (1).

P ro o f. Choose g, b, B, G as in the preceding proof. Let F be a subset of В such that F c conv(G(F) u {p}). Since F is equicontinuous, the function t-*v(t) = p(F(f)) is continuous on D.

Fix tED and e > 0. By (6) and the Lusin theorem there exists a compact subset Ds of D such that mes(D\De) < s and p ( p ( t ) + f ( t , T x X ) ) ^ supseT k{t, s)p(X) for any compact subset T of Ds, while the function s-*/c(t, s) is continuous.

We divide the interval D = [0, d] into n parts 0 = d0 < dx < ... < dn — d in such a way that \k(t, s)v{r) — k(t, u)v(z)\ < г for s, r, u, z eTl = Dt r\ DE, where Dt = [di-i, df\ (i = 1, ..., n). Set Ц = {u(s): u eV, s eD(}. By Lemma 1 (7) p(p(t) + X \$f(t, s, V(s))ds) ^ p(p(t) + X j f{t, s, V{s))ds)

D De

+ || J f ( t , s , V(s))ds\\p(K(B(D), 1)).

Let us observe that DS'De

Я J f ( t , s, V(s))ds c= £ Я j f(t , s, V(s))ds -

Dc i = l . Ti

Hence

Я £ mes 7]conv/(f, 7] x V f i= 1

n

р(р(0 + Я j f(t , s, V(s))ds) ^ р(р(0 + Я £ mes f com f i t , 7) x Vf)

De i = l

n

## < я

£ mes 7>(p(f) + conv/(t, f x Vf)

i = 1

## < я

£ mes 7]p(conv {pit) + fit, Tt xVf)) i = 1

^ Я £ mes 7] sup ^it, s)p(^)

i = 1 s e T i n

= Я £ mes T f i t, q M s f i= 1

(5)

Existence o f solutions of nonlinear integral equations 89

where qte 7], s.eD,. Moreover, as \k(t, s)v{s) — k{t, < e for seTJ, we have

mes Ttk(t, g , ) ^ ) ^ J k(t, s)v(s)ds + s mes 7].

Ti

Thus Л

р(Я J f(t, s, V(s))ds + p(t)) ^ Я £ (J k(t, s)v(s)ds + ernes 7])

DE i = l Ti

= Я J /c(r, s)t>(s)<is + A£mesD£.

As г is arbitrarily small, from this and (7) we deduce that р(р(0 + Я{ /(C s, F(s))ds) ^ Я J /c(r, s)v(s)ds,

i> D

and therefore fi{V{t)) ^ y.JD/c(t, 5 )15 (5 ) ds. Further we argue as in the final part of the proof of Th. 2.

Now we assume additionally that the measure p has the property 7° if Х я eM E, X n = X„, X n +1 c for n = 1, 2, ... and if ü m , ^ p{Xn) = 0, then X x = O?=1X n * 0 .

Theorem 4. Let p fulfil conditions l°-5° and 7°, let the function (s, x)-» f(t, s, x) be uniformly continuous and pitfeE^for any teD. Assume, in addition, that there exists an integrable function к : D2-+R+ such that for any bounded subset X of E

(8) p(p(t)+f(t, s, X)) ^ k(t, s)p(X).

Then there exists q > 0 such that for every Я, 0 < Я < q, the equation (1) has at least one continuous solution x(t) such that x(t)eЕц for teD.

P roof. Choose q, b, B, G as in the proof of Th. 2. Denote by X 0 the set of all functions x e B such that

IIx(t) x(s)|| ^ ||p(r)-p(s)|| + j d b(s, t, T)di for t, s eD.

D

The set X 0 is bounded, closed, convex and equicontinuous. Set X n+ i = conv GXn (n = 1 ,2 ,...). Since G{B) cz B, therefore all these sets are of the same type as X 0 and X n+1c : X n. Put un(t) = p(Xn(t)). Obviously 0 ^ un + 1(t) ^ un(t) (n — 0, 1, ...). The functions щ are continuous. Therefore the sequence un(t) converges uniformly to a function u^f). We have

*V+i(0 = p(con\{GXn)(tj) = p(p(t) + X J ( f X n)(s)ds)

D

< Я j p(p(t) + { f X n)(s})ds ^ Я j k(t, s)un(s)ds,

D D

which implies u^it) ^ ЯJDk(t, s)uo0{s)ds for tED. Since Xr(K) < 1, we get uoo(0 = 0 for teD.

(6)

90 J. J a n u s z e w s k i

Set цс(Х) — max {p(X(t)): teD} for X equicontinuous. Then pc is a mea­

sure of noncompactness (cf. [2], p. 79). Since lim max{n„(t): teD} = 0 , n-> 00

therefore pc(Xn)-> 0 as n-> oo.

Hence the set X œ = P)“=1 X n is nonempty, convex, closed and Х ж c= £ д.

Now we can apply the Schauder fixed point theorem to the mapping G|Xoo, which yields the existence of x e X ^ such that x = G(x).

R em ark. The proofs of Theorems 3 and 4 are suggested by the corresponding proofs from [10] and [6] for differential equations.

3. The Volterra integral equation. Consider now the integral equation (2) assuming that p and / satisfy I and II. Choose b > 0 in such a way that b > 2sup(6D ||p(f)||. From II(ii) it follows that there is a number a1, 0 < ^ d, such that jo mb(t, s)ds ^ b/2 for 0 < t < a^.

Let J = [0, a], where a = min(al5 1). Put В = {u e C ( J , E): ||w||c ^ b} and t

F(x)(t) = p(t) + J / ( t , s, x{s))ds for x e B , te J . о

Similarly to the Urysohn integral equation, we can show that F is a continuous mapping B-+B and the set F(B) is equiuniformly continuous.

Further, let P = {(t , s, z) eR3: 0 ^ s ^ t ^ l, |z| < c}, where l > a, c > 2b.

Assume that a nonnegative real-valued function (t, s, z)-+h(t, s, z) defined on P is a Kamke function, i.e. h satisfies the Carathéodory conditions II(i)—(iii) and

(iv) for each fixed t, s the function z->h(t, s, z) is nondecreasing, (v) for each q, 0 < q ^ /, the zero function is the unique continuous solution of the equation z(t) = ÿ0h(t, s, z(s))ds defined on [0, q).

Theorem 5. Assume that for each t e J the function (s, x )-> /(t, s, x) is uniformly continuous on {(5, x): 0 ^ s < t, ||x|| ^ b}, p i f e E ^ for t e J and (9) p({p{t)} u(p(t)+f{t, s, X)))^ h(t, s, p{X))

for 0 ^ s ^ t ^ a and for each bounded subset X of E. Then the equation (2) has at least one continuous solution on J.

The proof is similar to that of Th. 2.

Theorem 6. Let p fulfil conditions l°-6°. Assume, in addition, that for any e > 0, bounded X <= E and t e J there exists a closed subset IE o /[ 0, f] such that mes([0, t ] \ / £) < e and

(10) p(p{t)+f{t, T x X)) ^ sup h(t, s, p(X))

s e T

for each closed subset T of IE. Then the equation (2) has at least one continuous solution on J.

(7)

Existence o f solutions o f nonlinear integral equations 91

P roof. Let F be a subset of В such that V a conv(F(F)u {p}). Let us fix teJ, s > 0.

By the Scorza Dragoni theorem there exists a closed subset De of J such that mes(J\Ds) < e and the function h is uniformly continuous on Ds x [0, bf\, where = bp{K(B(J), 1)). As Fis equicontinuous, the function t-^v(t) = p(V(t)) is continuous on J . Choose Ô > 0 such that \h(t, s, r) — h(t, u, z)\ < г for s, u e De and r, z e[0 , bf\ satisfying \s — u\ < ô, \r — z\ < Ô, and choose r] such that 0 < q < Ô and |u(s) — t>(w)| < <5 for s , w e J such that \s — w| < ц.

We divide the interval [0, r] into n parts 0 = t0 < tl < ... < t n = t in such a way that |t£ — tr- il < for i = l , . . . , w. Let = [tt, ti-f\ n DE and Vt = {u(s): ueV, s e D,}.

By (10) we may choose a closed subset Je of J such that m e s(J\J£) < e and p(p(t)+f(t, T x V f ) ^ sup h(t, s, p{Vt))

s e T

for any compact subset T of JE and z = 1, 2, . . n. Put P = [0, t] n Den J e, Q = [0, t~\\P and 7] = Dt n J E. We see that

j / ( t , s, V(s))dsa j f(t , s, F(s))ds + j / ( t , s, F(s))ds.

0 P Q

Arguing similarly to the proof of Th. 3, we get

П

p(p(t) + J /(L s, V(s})ds) ^ X mes TiH** Vi’ v(si))

P i = 1

for qie T i, sieDi and further

p(V{t)) ^ /x(p(0 + j / ( t , s, V(s)) ds) ^ J h(t, s, v(s))ds

о 0

for te J .

From the property of Kamke functions and the theorem on integral inequalities, we conclude that p(V(t)) = 0 for t e J . The proof is completed as that of Th. 2.

Theorem 7. Let pt fulfil conditions l°-5° and 7°, and p(t)eE^for any te J.

Assume, in addition, that for teD the function (s, x)-+f(t, s, x) is uniformly continuous on {(s, x): 0 ^ s ^ t, ||x|| < b] and

f(p(0 + /(L s, X)) < h(t, s, fi(X))

for 0 ^ s ^ t ^ a and for each bounded subset X of E. Then the equation (2) has at least one continuous solution x(t) such that x i f e E ^ f o r te J .

The proof is similar to that of Th. 4.

(8)

92 J. Ja n u s z e w s k i

References

[1] A. A m b r o s e t t i, Un teorema di esistenza per le equazioni differenziali negli spazi di Banach, Rend. Sem. Mat. Univ. Padova 39 (1967), 349-360.

[2] J. В an as, Applications o f measures o f noncompactness to various problems, Zeszyty Nauk.

Politech. Rzeszowskiej 34(1987).

[3] J. В an a s and K. G o e b e l, Measures o f Noncompactness in Banach Spaces, Lecture Notes in Pure and Appl. Math. 60 (1980).

[4] J. B an as, A. H a j n o s z and S. W ç d r y c h o w ic z , Some generalization ofSzuflas theorem for ordinary differential equations in Banach space, Bull. Acad. Polon. Math. 29 (1981), 459- 464.

[5] K. D e im lin g , Ordinary Differential Equations in Banach Spaces, Lecture Notes in Math. 596, Springer, 1977.

[6] K. G o e b e l and W. R z y m o w s k i, An existence theorem for the equation x ' = f { t , x ) in Banach space, Bull. Acad. Polon. Sci. Math. 18 (1970), 367-370.

[7] R. K. M ille r , Nonlinear Volterra Equations, Benjamin, California 1971.

[8] B. N. S a d o v s k ii, Limit-compact and compactifying operators, Uspekhi Mat. Nauk 27 (1972), 81-146 (in Russian).

[9] S. S z u fla , On the existence o f solutions o f Volterra integral equations in Banach space, Bull.

Acad. Polon. Sci. Math. 22 (1974), 1209-1213.

[10] —, On the existence o f solutions o f differential equations in Banach spaces, ibid. 30(1982), 507-515.

[11] —, On the application o f measure o f noncompactness to existence theorems, Rend. Sem. Mat.

Univ. Padova 75 (1986), 1-14.

INSTYTUT MATEMATYKI I FIZYKI, AKADEMIA TECHNICZNO-ROLNICZA INSTITUTE O F MATHEMATICS AND PHYSICS, ATR

KALISKIEGO 7, 85-790 BYDGOSZCZ, POLAND

Cytaty

Powiązane dokumenty

In final section, we give our main result concerning with the solvability of the integral equation (1) by applying Darbo fixed point theorem associated with the measure

Abstract: Using the technique associated with measure of non- compactness we prove the existence of monotonic solutions of a class of quadratic integral equation of Volterra type in

The statis- tical model for stochastic processes, defined by (1), is essentially more gen- eral than that considered in Magiera and Wilczy´ nski (1991) (it also contains some models

Keywords: Banach space, difference equation, fixed point, measure of noncompactness, asymptotic behaviour of solutions.. 2000 Mathematics Subject Classification:

In this paper, we present some results concerning the existence and the local asymptotic stability of solutions for a functional integral equation of fractional order, by using

Key words and phrases: existence of solution, measure of noncompactness, nonlinear Fredholm integral equation, Henstock-Kurzweil integral, HL

[r]

T heorem 3.. On the other hand Hille, Yosida and Kato in these situations have proved directly the convergence of corresponding sequences Un{t 1 s), obtaining in this