Application of Measure of Noncompactness for the System of Functional Integral Equations

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1 Filomat 3:11 (216), DOI /FIL161163A Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: Application of Measure of Noncompactness for the System of Functional Integral Equations Reza Arab a a Department of Mathematics, Sari Branch, Islamic Azad University, Sari, Iran. Abstract. In this paper we introduce the notion of the generalized Darbo fixed point theorem and prove some fixed and coupled fixed point theorems in Banch space via the measure of non-compactness, which generalize the result of Aghajani et al.[6]. Our results generalize, extend, and unify several well-known comparable results in the literature. As an application, we study the existence of solutions for the system of integral equations. 1. Introduction The integral equation creates a very important and significant part of the mathematical analysis and has various applications into real world problems. On the other hand, measures of noncompactness are very useful tools in the wide area of functional analysis such as the metric fixed point theory and the theory of operator equations in Banach spaces. They are also used in the studies of functional equations, ordinary and partial differential equations, fractional partial differential equations, integral and integro-differential equations, optimal control theory, etc., see [1 5, 12, 15 18]. In our investigations, we apply the method associated with the technique of measures of noncompactness in order to generalize the Darbo fixed point theorem [1] and to extend some recent results of Aghajani et al.[6], and also we are going to study the existence of solutions for the following system of integral equations x(t, s) =a(t, s) + f (t, s, x(t, s), y(t, s)) + (t, s, x(t, s), y(t, s)) y(t, s) =a(t, s) + f (t, s, y(t, s), x(t, s)) + (t, s, y(t, s), x(t, s)) k(t, s, u, v, x(u, v), y(u, v))dudv k(t, s, u, v, y(u, v), x(u, v))dudv, for t, s R +, x, y E = BC(R + R + ). We show that Eq. (1) has solutions that belong to E E, where E = BC(R + R + ). (1) 21 Mathematics Subject Classification. 47H8; 47H1, 45B5 Keywords. Measure of noncompactness; System of two variables integral equations; fixed point; coupled fixed point Received: 28 August 214; Accepted: 1 February 215 Communicated by Dragan S. Djordjević address: mathreza.arab@iausari.ac.ir (Reza Arab)

2 R. Arab / Filomat 3:11 (216), Preliminaries In this section, we recall some notations, definitions and theorems to obtain all the results of this work. Denote by R the set of real numbers and put R + = [, + ). Let (E, ) be a real Banach space with zero element. Let B(x, r) denote the closed ball centered at x with radius r. The symbol B r stands for the ball B(, r). For X, a nonempty subset of E, we denote by X and ConvX the closure and the convex closure of X, respectively. Moreover, let us denote by M E the family of nonempty bounded subsets of E and by N E its subfamily consisting of all relatively compact sets. We use the following definition of the measure of noncompactness given in [1]. Definition 2.1. A mapping µ : M E R + is said to be a measure of noncompactness in E if it satisfies the following conditions: (1 ) The family kerµ = {X M E : µ(x) = } is nonempty and kerµ N E, (2 ) X Y µ(x) µ(y), (3 ) µ( X) = µ(x), (4 ) µ(convx) = µ(x), (5 ) µ(λx + (1 λ)y) λµ(x) + (1 λ)µ(y) for λ [, 1], (6 ) If (X n ) is a sequence of closed sets from m E such that X n+1 X n (n = 1, 2,...) and if lim µ(x n) =, then the n set X = n=1 X n is nonempty. The family kerµ defined in axiom (1 ) is called the kernel of the measure of noncompactness µ. One of the properties of the measure of noncompactness is X kerµ. Indeed, from the inequality µ(x ) µ(x n ) for n = 1, 2, 3,..., we infer that µ(x ) =. Further facts concerning measures of noncompactness and their properties may be found in [9, 1, 12]. Darbo s fixed point theorem is a very important generalization of Schauder s fixed point theorem, and includes the existence part of Banach s fixed point theorem. Theorem 2.2. (Schauder [2]) Let C is a closed, convex and bounded subset of a Banach space E. Then every compact, continuous map T : C C has at least one fixed point. In the following we state a fixed-point theorem of Darbo type proved by Banaś and Goebel [1]. Theorem 2.3. Let C be a nonempty, closed, bounded, and convex subset of the Banach space E and F : C C be a continuous mapping. Assume that there exist a constant k [, 1) such that µ(fx) kµ(x) for any nonempty subset of C. Then F has a fixed-point in the set C. Definition 2.4. [14] Let S denote the class of those functions α : [, ) [, 1) which satisfies the condition α(t n ) 1 implies t n. Recently, Aghajani et al. [7] obtained following fixed point theorem which in turn extends Theorem 2.3 and the corresponding result in [1]. Theorem 2.5. Let C be a nonempty, bounded, closed, and convex subset of a Banach space E and T : C C be a continuous function satisfying µ(t(x)) α(µ(x))µ(x), for each X C, where µ is an arbitrary measure of noncompactness and α S. Then T has at least one fixed point in C.

3 R. Arab / Filomat 3:11 (216), The following concept of O( f ;.) and its examples was given by Altun and Turkoglu [8]. Let F([, ) be class of all function f : [, ) [, ] and let Θ be class of all operators O( ;.) : F([, )) F([, )), f O( f ;.) satisfying the following conditions: (i) O( f ; t) > for t > and O( f ; ) =, (ii) O( f ; t) O( f ; s) for t s, (iii) lim n O( f ; t n ) = O( f ; lim n t n ), (iv) O( f ; max{t, s}) = max{o( f ; t), O( f ; s)} for some f F([, ). Example 2.6. If f : [, ) [, ) is a Lebesque integrable mapping which is finite integral on each compact subset of [, ), non-negative and such that for each t >, t f (s)ds >, then the operator defined by satisfies the above conditions. O( f ; t) = t f (s)ds Example 2.7. If f : [, ) [, ) is non-decreasing, continuous function such that f () = and f (t) > for t > then the operator defined by f (t) O( f ; t) = 1 + f (t) satisfies the above conditions. 3. Main Results This section is devoted to prove a few generalizations of Darbo fixed point theorem(cf. Theorem 2.3). Theorem 3.1. Let C be a nonempty, bounded, closed, and convex subset of a Banach space E and T : C C be a continuous operator such that O( f ; µ(t(x))) + ϕ(µ(tx)) α(µ(x))[o( f ; µ(x)) + ϕ(µ(x))], (2) for each X C, α S and O( ;.) Θ and ϕ : R + R + is a continuous function, where µ is an arbitrary measure of noncompactness. Then T has at least one fixed point in C. Proof. We define by induction the sequence {C n }, where C = C and C n+1 = Conv(TC n ), for n, such that C C 1... C n C n+1... If there exists a natural number N such that µ(c N ) =, then C N is compact. In this case theorem 2.2 implies that T has a fixed point. So we assume that µ(c n ) for n =, 1, 2,... Also by (2) we have O( f ; µ(c n+1 )) + ϕ(µ(c n+1 )) = O( f ; µ(conv(tc n ))) + ϕ(µ(conv(tc n ))) = O( f ; µ(tc n )) + ϕ(µ(tc n )) α(µ(c n ))[O( f ; µ(c n )) + ϕ(µ(c n ))] < O( f ; µ(c n )) + ϕ(µ(c n )). (3) Since the sequence {O( f ; µ(c n )) + ϕ(µ(c n ))} is nonincreasing and nonnegative, we infer that lim [O( f ; µ(c n)) + ϕ(µ(c n ))] = r. n

4 R. Arab / Filomat 3:11 (216), We show that r =. Suppose, to the contrary, that r >. Then from (3) we have O( f ; µ(c n+1 )) + ϕ(µ(c n+1 )) O( f ; µ(c n )) + ϕ(µ(c n )) α(µ(c n )) < 1, which implies Since α S, we get r =. Hence α(µ(c n )) 1 as n. therefore, O( f ; lim n µ(c n )) + ϕ( lim n µ(c n )) =, lim µ(c n) =, n Since C n C n+1 and TC n C n for all n = 1, 2,..., then from (6 ), X = set, invariant under T and belongs to Kerµ. Therefore Theorem 2.2 completes the proof. Remark 3.2. It is clear that Theorem 3.1 is a generalization of Theorem 2.5, in fact µ(t(x)) = O( f ; µ(t(x))) α(µ(x))o( f ; µ(x)) = α(µ(x))µ(x). X n is a nonempty convex closed Corollary 3.3. Let C be a nonempty, bounded, closed, and convex subset of a Banach space E and T : C C be a continuous operator such that µ(t(x)) + ϕ(µ(tx)) α(µ(x))[µ(x) + ϕ(µ(x))], (4) for each X C and ϕ : R + R + is a continuous function, where µ is an arbitrary measure of noncompactness and α S. Then T has at least one fixed point in C. The following corollary gives us a fixed point theorem with a contractive condition of integral type. Corollary 3.4. Let C be a nonempty, bounded, closed, and convex subset of a Banach space E, k (, 1) and T : C C be a continuous operator such that for any X C one has n=1 µ(t(x)) f (s) ds k µ(x) f (s) ds, where µ is an arbitrary measure of noncompactness and f : [, ) [, ) is a Lebesgue-integrable mapping which is summable (i.e. with finite integral) on each compact subset of [, ), non-negative and such that for each ɛ >, ɛ f (s) ds >. Then T has at least one fixed point in C. Definition 3.5. [13] An element (x, y) X X is called a coupled fixed point of a mapping T : X X X if T(x, y) = x and T(y, x) = y. Theorem 3.6. [1] Suppose µ 1, µ 2,..., µ n be the measures in E 1, E 2,..., E n respectively. Moreover assume that the function F : [, ) n [, ) is convex and F(x 1, x 2,..., x n ) = if and only if x i = for i = 1, 2,..., n. Then µ(x) = F(µ 1 (X 1 ), µ 2 (X 2 ),..., µ n (X n )) defines a measure of noncompactness in E 1 E 2... E n where X i denotes the natural projection of X into E i for i = 1, 2,..., n.

5 R. Arab / Filomat 3:11 (216), Now, as results from Theorem 3.6, we present the following example. Example 3.7. [1] Let µ be a measure of noncompactness. We define F(x, y) = x + y for any x, y [, ). Then F has all the properties mentioned in Theorem 3.6. Hence µ(x) = µ(x 1 ) + µ(x 2 ) is a measure of noncompactness in the space E E where X i, i = 1, 2 denote the natural projections of X. Theorem 3.8. Let C be a nonempty, bounded, closed, and convex subset of a Banach space E and T : C C C be a continuous function such that O( f ; µ(t(x 1 X 2 ))) + ϕ(µ(t(x 1 X 2 ))) 1 2 α(µ(x 1) + µ(x 2 ))[O( f ; µ(x 1 ) + µ(x 2 )) + ϕ(µ(x 1 ) + µ(x 2 ))], (5) for any subset X 1, X 2 of C, where µ is an arbitrary measure of noncompactness and ϕ : [, ) [, ) is a nondecreasing, continuous and ϕ(t + s) ϕ(t) + ϕ(s) for all t, s and α S. Also O( ;.) Θ and O( f ; t + s) O( f ; t) + O( f ; s) for all t, s. Then T has at least a coupled fixed point. Proof. First note that, from Example 3.7, µ(x) = µ(x 1 ) + µ(x 2 ) for any bounded subset X E E defines a measure of noncompactness on E E where X 1 and X 2 denote the natural projections of X. we define a mapping T : C C C C by T(x, y) = (T(x, y), T(y, x)). It is obvious that T is continuous. Now we claim that T satisfies all the conditions of Theorem 3.1. To prove this, let X C C be any nonempty subset. Then by (2 ), (5) and (ii) we obtain O( f ; µ( T(X))) + ϕ( µ( T(X))) O( f ; µ(t(x 1 X 2 ) T(X 2 X 1 ))) + ϕ( µ(t(x 1 X 2 )) T(X 2 X 1 ))) = O( f ; µ(t(x 1 X 2 )) + µ(t(x 2 X 1 ))) + ϕ(µ(t(x 1 X 2 )) + µ(t(x 2 X 1 ))) = O( f ; µ(t(x 1 X 2 ))) + O( f ; µ(t(x 2 X 1 ))) + ϕ(µ(t(x 1 X 2 ))) + ϕ(µ(t(x 2 X 1 ))) = O( f ; µ(t(x 1 X 2 ))) + ϕ(µ(t(x 1 X 2 ))) + O( f ; µ(t(x 2 X 1 ))) + ϕ(µ(t(x 2 X 1 ))) 1 2 α(µ(x 1) + µ(x 2 ))[O( f ; µ(x 1 ) + µ(x 2 )) + ϕ(µ(x 1 ) + µ(x 2 ))] α(µ(x 2) + µ(x 1 ))[O( f ; µ(x 2 ) + µ(x 1 )) + ϕ(µ(x 2 ) + µ(x 1 ))] = α(µ(x 1 ) + µ(x 2 ))[O( f ; µ(x 1 ) + µ(x 2 )) + ϕ(µ(x 1 ) + µ(x 2 ))] = α( µ(x))[o( f ; µ(x)) + ϕ( µ(x))]. Hence, from Theorem 3.1, T has at least one fixed point in C C. Now the conclusion of theorem follows from the fact that every fixed point of T is a coupled fixed point of T. Corollary 3.9. Let C be a nonempty, bounded, closed, and convex subset of a Banach space E and T : C C C be a continuous function. Assume that there exists a constant k [, 1) such that µ(t(x 1 X 2 )) k 2 (µ(x 1) + µ(x 2 )), for any subset X 1, X 2 of C, where µ is an arbitrary measure of noncompactness. Then T has at least a coupled fixed point.

6 4. Existence of Solutions for a System of Integral Equations R. Arab / Filomat 3:11 (216), In what follows we will work in the classical Banach space BC(R + R + ) consisting of all real functions defined, bounded and continuous on R + R + equipped with the standard norm x = sup{ x(t, s) : t, s }. Now, we present the definition of a special measure of noncompactness in BC(R + R + ) which will be used in the sequel, a measure that was introduced and studied in [1]. To do this, let X be fix a nonempty and bounded subset of BC(R + R + ) and fix a positive number T. For x X and ɛ >, denote by ω T (x, ɛ) the modulus of the continuity of function x on the interval [, T], i.e., ω T (x, ɛ) = sup{ x(t, s) x(u, v) : t, s, u, v [, T], t u ɛ, s v ɛ}. Further, let us put ω T (X, ɛ) = sup{ω T (x, ɛ) : x X}, and ω T (X) = lim ɛ ωt (X, ɛ) ω (X) = lim T ω T (X). Moreover, for two fixed numbers t, s R + let us define the function µ on the family M BC(R+ R + ) by the following formula µ(x) = ω (X) + α(x), where α(x) = lim sup diamx(t, s), X(t, s) = {x(t, s) : x X} and diamx(t, s) = sup{ x(t, s) y(t, s) : x, y X}. t,s Similar to [1] (cf. also [11]), it can be shown that the function µ is the measure of noncompactness in the space BC(R + R + ). As an application of our results we are going to study the existence of solutions for the system of integral equations (1). Consider the following assumptions (A 1 ) α i : R + R + are continuous, nondecreasing and lim t α i (t) =, i = 1, 2. (A 2 ) The function a : R + R + R + is continuous and bounded. (A 3 ) k : R + R + R + R + R R R is continuous and there exists a positive constant M such that M = sup Morever, lim t,s { k(t, s, u, v, x(u, v), y(u, v)) dudv : t, s R +, x, y BC(R + R + ) }. (6) [k(t, s, u, v, x 2 (u, v), y 2 (u, v)) k(t, s, u, v, x 1 (u, v), y 1 (u, v))]dudv =, (7) uniformly respect to x 1, y 1, x 2, y 2 BC(R + R + ). (A 4 ) The functions f, : R + R + R R R are continuous and there exist two bounded functions a 1, a 2 : R + R + R with bound K = max{ sup a 1 (t, s), sup a 2 (t, s)}. (t,s) R + R + (t,s) R + R +

7 R. Arab / Filomat 3:11 (216), Also there exist two positive constant D and λ < 1 such that and f (t, s, x 2, y 2 ) f (t, s, x 1, y 1 ) a 1(t, s)λ( x 2 x 1 + y 2 y 1 ) D + λ( x 2 x 1 + y 2 y 1 ), (t, s, x 2, y 2 ) (t, s, x 1, y 1 ) a 2(t, s)λ( x 2 x 1 + y 2 y 1 ) D + λ( x 2 x 1 + y 2 y 1 ), for all t, s R + and x 1, y 1, x 2, y 2 R. Moreover, we assume that 2K(1 + M) D. (A 5 ) The functions H 1, H 2 : R + R + R + defined by H 1 (t, s) = f (t, s,, ) and H 2 (t, s) = (t, s,, ) are bounded on R + R + with H = max{ sup H 1 (t, s), sup H 2 (t, s)}. (t,s) R + R + (t,s) R + R + Theorem 4.1. If the assumptions (A 1 ) (A 5 ) are satisfied, then the system of equation (1) has at least one solution (x, y) E E. Proof. Define the operator T : E E E associated with the integral equation (1) by where, T(x, y)(t, s) = a(t, s) + f (t, s, x(t, s), y(t, s)) + (t, s, x(t, s), y(t, s))[f(x, y)(t, s)], (8) F(x, y)(t, s) = k(t, s, u, v, x(u, v), y(u, v))dudv. (9) Solving Eq.(1) is equivalent to finding a coupled fixed point of the operator T defined on the space E E. For better readability, we break the proof into a sequence of cases. Case 1: T transforms the space E E into E. By considering conditions of theorem we infer that T(x, y) is continuous on R + R +. Now we prove that T(x, y) E for any (x, y) E E. For arbitrarily fixed (t, s) R + R + we have Indeed, (T(x, y))(t, s) a(t, s) + f (t, s, x(t, s), y(t, s)) + (t, s, x(t, s), y(t, s)) F(x, y)(t, s) a(t, s) + Kλ( x(t, s) + y(t, s) ) D + λ( x(t, s) + y(t, s) ) + H + f (t, s, x(t, s), y(t, s)) f (t, s, x(t, s), y(t, s)) f (t, s,, ) + f (t, s,, ) a 1(t, s)λ( x(t, s) + y(t, s) ) + H 1 (t, s) D + λ( x(t, s) + y(t, s) ) Kλ( x(t, s) + y(t, s) ) D + λ( x(t, s) + y(t, s) ) + H, (t, s, x(t, s), y(t, s)) (t, s, x(t, s), y(t, s)) (t, s,, ) + (t, s,, ) a 2(t, s)λ( x(t, s) + y(t, s) ) + H 2 (t, s) D + λ( x(t, s) + y(t, s) ) Kλ( x(t, s) + y(t, s) ) D + λ( x(t, s) + y(t, s) ) + H, (F(x, y)(t, s) = k(t, s, u, v, x(u, v), y(u, v))dudv k(t, s, u, v, x(u, v), y(u, v)) dudv M. [ Kλ( x(t, s) + y(t, s) ) D + λ( x(t, s) + y(t, s) ) + H ] M. (1)

8 By assumption (A 4 ), we get T(x, y) a + R. Arab / Filomat 3:11 (216), ( ) Kλ( x + y ) D + λ( x + y ) + H (1 + M) a + (K + H )(1 + M). (11) Thus T maps the space E E into E. More precisely, from (11) we obtain that T(B r B r ) B r,where r = a + (K + H )(1 + M). Case 2: we show that map T : B r B r B r is continuous. To do this, let us fix arbitrarily ɛ > and take (x, y), (z, w) B r B r such that (x, y) (z, w) ɛ. Then (T(x, y)(t, s)) (T(z, w)(t, s)) = f (t, s, x(t, s), y(t, s)) + (t, s, x(t, s), y(t, s))[f(x, y)(t, s)] f (t, s, z(t, s), w(t, s)) (t, s, z(t, s), w(t, s))[f(z, w)(t, s)] f (t, s, x(t, s), y(t, s)) f (t, s, z(t, s), w(t, s)) + (t, s, x(t, s), y(t, s)) (F(x, y))(t, s) (F(z, w))(t, s) + (t, s, x(t, s), y(t, s) (ts, z(t, s), w(t, s)) (F(z, w))(t, s) K(1 + M)λ( x z + y w ) D + λ( x z + y w ) [ ] Kλ( x + y ) + D + λ( x + y ) + H (F(x, y))(t, s) (F(z, w))(t, s), (12) Furthermore, with due attention to the condition (A 2 ) there exist N > such that for t > N we have (F(x, y))(t, s) (F(z, w))(t, s) = [k(t, s, u, v, x(u, v), y(u, v)) k(t, s, u, v, z(u, v), w(u, v))]dudv < ɛ. (13) Suppose that t, s > N. It follows (12) and (13) that T(x, y)(t, s) T(z, w)(t, s) < ɛ. (14) If t, s [, N], then we obtain (F(x, y))(t, s) (F(z, w))(t, s) α 2 N ω 1(k, ɛ), (15) where we denoted α N = sup{α i (t) : t [, N], i = 1, 2}, and ω 1 (k, ɛ) = sup{ k(t, s, u, v, x, y) k(t, s, u, v, z, w) : t, s [, N], u, v [, α N ], x, y, z, w [ r, r], (x, y) (z, w) ɛ}. By using the continuity of k on [, N] [, N] [, α N ] [, α N ] [ r, r] [ r, r], we have ω 1 (k, ɛ) as ɛ. Now, linking the inequalities (12) and (15) we deduce that T(x, y)(t, s) T(z, w)(t, s) ɛ + [K + H ]α 2 N ω 1(k, ɛ). (16) The above established facts we conclude that T is continuous on B r B r. Case 3: In the sequel, we show that for any nonempty set X 1, X 2 B r, µ(t(x 1 X 2 )) λ 2 (µ(x 1) + µ(x 2 )).

9 R. Arab / Filomat 3:11 (216), Indeed, by virtue of assumptions (A 1 ) (A 5 ), we conclude that for any (x, y), (z, w) X 1 X 2 and t, s R +, where (T(x, y))(t, s) (T(z, w))(t, s) K(1 + M)λ( x(t, s) z(t, s) + y(t, s) w(t, s) ) D + λ( x(t, s) z(t, s) + y(t, s) w(t, s) ) [ ] Kλ( x(t, s) + y(t, s) ) + D + λ( x(t, s) + y(t, s) ) + H β(t, s) ] 1 2 λ( x(t, s) z(t, s) + y(t, s) w(t, s) ) + [ Kλ( x(t, s) + y(t, s) ) D + λ( x(t, s) + y(t, s) ) + H β(t, s) = sup { This estimate allows us to derive the following one β(t, s), } [k(t, s, u, v, x(u, v), y(u, v)) k(t, s, u, v, z(u, v), w(u, v))]dudv : x, y E. diam(t(x 1 X 2 ))(t, s) λ 2 (diamx 1(t, s) + diamx 2 (t, s)) + Consequently, from (17) and assumption (7) that [ ] Kλ( x(t, s) + y(t, s) ) D + λ( x(t, s) + y(t, s) ) + H β(t, s). (17) lim sup diam(t(x 1 X 2 ))(t, s) λ 2 [lim sup diamx 1 (t, s) + lim sup diamx 2 (t, s)]. (18) Next, fix arbitrarily N > and ɛ >. Let us choose t 1, t 2, s 1, s 2 [, N], with t 2 t 1 ɛ, s 2 s 1 ɛ. Without loss of generality, we may assume that t 1 t 2 and s 1 s 2. Then, for (x, y) X 1 X 2 we get and f (t 2, s 2, x(t 2, s 2 ), y(t 2, s 2 )) f (t 1, s 1, x(t 1, s 1 ), y(t 1, s 1 )) f (t 2, s 2, x(t 2, s 2 ), y(t 2, s 2 )) f (t 2, s 2, x(t 1, s 1 ), y(t 1, s 1 )) + f (t 2, s 2, x(t 1, s 1 ), y(t 1, s 1 )) f (t 1, s 1, x(t 1, s 1 ), y(t 1, s 1 )) Kλ( x(t 2, s 2 ) x(t 1, s 1 ) + y(t 2, s 2 ) y(t 1, s 1 ) ) D + λ( x(t 2, s 2 ) x(t 1, s 1 ) + y(t 2, s 2 ) y(t 1, s 1 ) ) + f (t 2, s 2, x(t 1, s 1 ), y(t 1, s 1 )) f (t 1, s 1, x(t 1, s 1 ), y(t 1, s 1 )) λ 2(1 + M) (ωn (x, ɛ) + ω N (y, ɛ)) + ω N ( f, ɛ), (F(x, y)(t 2, s 2 ) (F(x, y)(t 1, s 1 ) + α1 (t 2 ) α2 (s 2 ) α1 (t 2 ) α2 (s 2 ) α 1 (t 1 ) α 2 (s 1 ) α1 (t 2 ) α2 (s 2 ) k(t 2, s 2, u, v, x(u, v), y(u, v)) k(t 1, s 1, u, v, x(u, v), y(u, v)) dudv k(t 1, s 1, u, v, x(u, v), y(u, v)) dudv ω N (k, ɛ)dudv + α1 (t 2 ) α2 (s 2 ) α 1 (t 1 ) α 2 N ωn (k, ɛ) + ω N (α 1, ɛ)ω N (α 2, ɛ)k N, α 2 (s 1 ) K N dudv

10 R. Arab / Filomat 3:11 (216), and Therefore, (t 2, s 2, x(t 2, s 2 ), y(t 2, s 2 ))(F(x, y)(t 2, s 2 ) (t 1, s 1, x(t 1, s 1 ), y(t 1, s 1 ))(F(x, y)(t 1, s 1 ) (t 2, s 2, x(t 2, s 2 ), y(t 2, s 2 ))(F(x, y)(t 2, s 2 ) (t 1, s 1, x(t 1, s 1 ), y(t 1, s 1 ))(F(x, y)(t 2, s 2 ) + (t 1, s 1, x(t 1, s 1 ), y(t 1, s 1 ))(F(x, y)(t 2, s 2 ) (t 1, s 1, x(t 1, s 1 ), y(t 1, s 1 ))(F(x, y)(t 1, s 1 ) Kλ( x(t 2, s 2 ) x(t 1, s 1 ) + y(t 2, s 2 ) y(t 1, s 1 ) ) D + λ( x(t 2, s 2 ) x(t 1, s 1 ) + y(t 2, s 2 ) y(t 1, s 1 ) ) (F(x, y)(t 2, s 2 ) Kλ( x(t 1, s 1 ) + y(t 1, s 1 ) ) + [ D + λ( x(t 1, s 1 ) + y(t 1, s 1 ) ) + H ] (F(x, y)(t 2, s 2 ) (F(x, y)(t 1, s 1 ) Mλ 2(1 + M) (ωn (x, ɛ) + ω N (y, ɛ)) + (K + H )[α 2 N ωn (k, ɛ) + ω N (α 1, ɛ)ω N (α 2, ɛ)k N ]. (T(x, y))(t 2, s 2 ) (T(x, y))(t 2, s 2 ) a(t 2, s 2 ) a(t 1, s 1 ) + f (t 2, s 2, x(t 2, s 2 ), y(t 2, s 2 )) f (t 1, s 1, x(t 1, s 1 ), y(t 1, s 1 )) + (t 2, s 2, x(t 2, s 2 ), y(t 2, s 2 ))(F(x, y)(t 2, s 2 ) (t 1, s 1, x(t 1, s 1 ), y(t 1, s 1 ))(F(x, y)(t 1, s 1 ) ω N λ (a, ɛ) + 2(1 + M) (ωn (x, ɛ) + ω N (y, ɛ)) + ω N ( f, ɛ) Mλ + 2(1 + M) (ωn (x, ɛ) + ω N (y, ɛ)) + (K + H )[α 2 N ωn (k, ɛ) + ω N (α 1, ɛ)ω N (α 2, ɛ)k N ] (19) where we defined ω N ( f, ɛ) = sup{ f (t 2, s 2, x, y) f (t 1, s 1, x, y) : t 1, t 2, s 1, s 2 [, N], t 2 t 1 ɛ, s 2 s 1 ɛ, x, y [ r, r]} ω N (k, ɛ) = sup{ k(t 2, s 2, u, v, x, y) k(t 1, s 1, u, v, x, y) : t 1, t 2, s 1, s 2 [, N], t 2 t 1 ɛ, s 2 s 1 ɛ, u, v [, α N ], x, y [ r, r]} ω N (α i, ɛ) = sup{ α i (t) α i (s) : t, s [, N], t s ɛ, i = 1, 2} ω N (x, ɛ) = sup{ x(t 2, s 2 ) x(t 1, s 1 ) : t 1, t 2, s 1, s 2 [, N], t 2 t 1 ɛ, s 2 s 1 ɛ} K N = sup{ k(t, s, u, v, x, y) : t, s [, N], u, v [, α N ], x, y [ r, r]} ω N (a, ɛ) = sup{ a(t 2, s 2 ) a(t 1, s 1 ) : t 1, t 2, s 1, s 2 [, N], t 2 t 1 ɛ, s 2 s 1 ɛ}, Since (x, y) was an arbitrary element of X 1 X 2, the inequality (2) implies that ω N (T(X 1 X 2 ), ɛ) ω N (a, ɛ) + λ 2 (ωn (X 1, ɛ) + ω N (X 2, ɛ)) + ω N ( f, ɛ) + (K + H )[α 2 N ωn (k, ɛ) + ω N (α 1, ɛ)ω N (α 2, ɛ)k N ], (2) In view of the uniform continuity of the functions a, f and k on [, N] [, N] and [, N] [, N] [ r, r] and [, N] [, N] [, α N ] [, α N ] [ r, r] [ r, r] respectively, we have that ω N (a, ɛ), ω N ( f, ɛ) and ω N (k, ɛ). Moreover, it is obvious that the constant K N is finite and ω N (α 1, ɛ) and ω N (α 2, ɛ) as ɛ. Thus, linking the established facts with the estimate (2) we get ω o (T(X 1 X 2 )) λ 2 (ω o(x 1 ) + ω o (X 2 )). (21)

11 R. Arab / Filomat 3:11 (216), Finally, from (18), (21) and the definition of the measure of noncompactness µ, we obtain µ(t(x 1 X 2 )) = ω (T(X 1 X 2 )) + lim sup diam(t(x 1 X 2 ))(t, s) t,s λ 2 (ω o(x 1 ) + ω o (X 2 )) + λ 2 λ 2 (ω o(x 1 ) + lim sup (lim sup diamx 1 (t, s) + lim sup diamx 2 (t, s)) diamx 1 (t, s) + ω o (X 2 ) + lim sup diamx 2 (t, s)) (22) = λ 2 (µ(x 1) + µ(x 2 )). Finally, applying Corollary 3.9, we obtain the desired result. References [1] R.P. Agarwal, D. O Regan, Singular Volterra integral equations, Appl. Math. Lett. 13 (2) [2] R.P. Agarwal, D. O Regan, Fixed point theory and applications, Cambridge University Press, 24. [3] R.P. Agarwal, M. Benchohra, D. Seba, On the application of measure of noncompactness to the existence of solutions for fractional differential equations, Results Math. 55 (29) [4] A. Aghajani, J. Banas, Y. Jalilian, Existence of solution for a class of nonlinear Voltrra sigular integral equations, Computer and Mathematics with Applications 62 (211) [5] A. Aghajani, N. Sabzali, A coupled fixed point theorem for condensing operators with application to system of integral equations, Journal of nonlinear and convex Analysis (To appear). [6] A. Aghajani, J. Banas, N. Sabzali, Some generalizations of Darbo fixed point theorem and applications, Bull. Belg. Math. Soc. Simon Stevin 2 (2) (213) [7] A. Aghajani, R. Allahyari, M. Mursaleen, A generalization of Darbo s theorem with application to the solvability of systems of integral equations, Journal of Computational and Applied Mathematics 26 (214) [8] I. Altun, D. Turkoglu, A fixed point theorem for mappings satisfying a general contractive condition of operator type, J. Comput. Anal. Appl. 9 (1) (27) [9] J. Banas, Measures of noncompactness in the space of continuous tempered functions, Demonstratio Math. 14 (1981) [1] J. Banas, K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Dekker, New York 6 (198). [11] J. Banas, R. Rzepka, An application of a measure of noncompactnessin the study of asymptotic stability, Appl. Math. Lett. 16 (23) 1 6. [12] J. Banas, M. Mursaleen, Sequence Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations, Springer, New Delhi, 214. [13] S.S. Chang, Y.J. Huang, Coupled Fixed Point Theorems With Applications, J. Korean Math. Soc. 33 (3) (1996) [14] M. Geraghty, On contractive mappings, Proc. Amer. Math. Soc. 4 (1973) [15] X. Hu, J. Yan, The global attractivity and asymptotic stability of solution of a nonlinear integral equation. J. Math. Anal. Appl. 321 (26) [16] Z. Liu, S.M. Kang, Existence and asymptotic stability of solutions to functional-integral equation, Taiwan J. Math. 11 (27) [17] L. Liu, F. Guo, C. Wu, Y. Wu, Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl. (25) [18] M. Mursaleen, A. Alotaibi, Infinite System of Differential Equations in Some Spaces, Abstract and Applied Analysis (212) doi:1.1155/212/

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