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

Séria I: PRACE MATEMATYCZNE XXX (1990)

**J****a n u s z**** J****a 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.*

**D****e 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^}.*

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

In what follows we shall need the following lemmas:

**L****emma** *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}.*

**L****emma**** 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

*K**p**+ t iix è ^ i t M p + x i)-*

*i = 1 * *i = 1*

**L****emma** *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].

**T****heorem**** 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: D*3*->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.*

**T****heorem**** 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.*

**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 X**e**R, 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 e**B , 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, B**e**M e.*

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

**J. J a n us z e ws k i**

**L****emma**** 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 e**V, te T } .*

*t e T*

**T****heorem**** 3. ***Let p fulfil conditions ***\ ° ~ 6 ° ***and pit)**e**E^ 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 e**T**l** = Dt r\ DE, *
*where Dt = [di-i, df\ (i = 1, ..., n). Set Ц = {u(s): **u e**V, **s e**D(}. 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

**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 *^{e}*M 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 .*

**T****heorem**** 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 e**D.*

*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.*

**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).*

**T****heorem**** 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.

**T****heorem** *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.*

**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}^{D}^{e}*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.

**T****heorem**** 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.

**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