Some generalizations of Darbo s theorem and applications to fractional integral equations

Transcription

1 Jleli et al. Fixed Point Theory and Applications (216) 216:11 DOI /s RESEARCH Open Access Some generalizations of Darbo s theorem and applications to fractional integral equations Mohamed Jleli1, Erdal Karapinar2,3, Donal O Regan4 and Bessem Samet1* * Correspondence: bsamet@ksu.edu.sa 1 Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia Full list of author information is available at the end of the article Abstract In this paper, some generalizations of Darbo s fixed point theorem are presented. An existence result for a class of fractional integral equations is given as an application of the obtained results. MSC: 47H1; 26A33; 45G1 Keywor: Darbo s theorem; measure of noncompactness; fractional integral equation 1 Introduction and preliminaries Let E be a Banach space over R (or C) with respect to a certain norm. For any subsets X and Y of E, we have the following notations: X denotes the closure of X; conv(x) denotes the convex hull of X; P(X) denotes the set of nonempty subsets of X; X + Y and λx (λ R) stand for algebraic operations on sets X and Y. We denote by BE the family of all nonempty bounded subsets of E. Finally, if X is a nonempty subset of E and T : X X is a given operator, we denote by Fix(T) the set of fixed points of T, that is, Fix(T) = {x X : Tx = x}. Banaś and Goebel [ ] introduced the following axiomatic definition of the concept of a measure of noncompactness. Definition. Let σ : BE [, ) be a given mapping. We say that σ is a BG-measure of noncompactness (in the sense of Banaś and Gobel) on E if the following conditions are satisfied: (i) For every X BE, σ (X) = iff X is precompact. (ii) For every pair (X, Y ) BE BE, we have X Y σ (X) σ (Y ). 216 Jleli et al. This article is distributed under the terms of the Creative Commons Attribution 4. International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Jleli et al. Fixed Point Theory and Applications (216) 216:11 Page 2 of 17 (iii) For every X B E,wehave σ (X)=σ (X)=σ ( conv(x) ). (iv) For every pair (X, Y ) B E B R and λ (, 1),wehave σ ( λx +(1 λ)y ) λσ (X)+(1 λ)σ (Y ). (v) If {X n } B E is a decreasing sequence (w.r.t. )ofcloseetssuchthatσ (X n ) as n,thenx := n=1 X n is nonempty. Let X be a nonempty, bounded, closed, and convex subset of the Banach space E. We denote by D X the set of self-mappings D : X X satisfying the following conditions: (i) D is a continuous mapping. (ii) There exist σ : B E [, ), a BG-measure of noncompactness on E,anda constant k (, 1) such that σ (DW ) kσ (W), W P(X ). The following result is known as Darbo s fixed point theorem (see [1, 2]). Theorem 1.2 Let D : X X be a mapping that belongs to D X. Then D has at least one fixed point. Moreover, the set Fix(D) is precompact. Many generalizations and extensions of Darbo s fixed point theorem can be found in the literature (see, for example, [3 9] and the references therein). Using the BG-measure of noncompactness, Aghajani et al. [4] obtained the following generalization of Darbo s theorem. Let F X be the set of self-mappings D : X X satisfying the following conditions: (i) D is a continuous mapping. (ii) There exists σ : B E [, ), a BG-measure of noncompactness on E,suchthatfor all ε >,thereexistssomeδ ε >for which W P(X ), ε σ (W)<ε + δ ε σ (DW )<ε. Theorem 1.3 (Aghajani et al. [4]) Let D : X X be a mapping that belongs to F X. Then D has at least one fixed point. Observe that D X F X.Infact,letD : X X be a given mapping that belongs to D X. Let ε >. From the definition of D X,thereissomek (, 1) such that σ (DW ) kσ (W), for any nonempty subset W of X.Letδ ε =( 1 1)ε. Then for any nonempty subset W of X, k we have ε σ (W)<ε + δ ε = ε k σ (DW ) kσ (W)<ε, so D F X.

3 Jleli et al. Fixed Point Theory and Applications (216) 216:11 Page 3 of 17 In [6], Dhage introduced the following axiomatic definition of the measure of noncompactness. Definition 1.4 Let σ : B E [, ) be a given mapping. We say that σ is a D-measure of noncompactness (in the sense of Dhage) on E if the following conditions are satisfied: (i) For every X B E, σ (X)=iff X is precompact. (ii) For every pair (X, Y ) B E B E,wehave X Y σ (X) σ (Y ). (iii) For every X B E,wehave σ (X)=σ (X)=σ ( conv(x) ). (iv) If {X n } B E is a decreasing sequence (w.r.t. )suchthatσ (X n ) as n, then the X := n=1 X n is nonempty. Observe that if σ : B E [, ) is a BG-measure of noncompactess on E, thenσ is a D-measure of noncompactess on E. In this paper, using the axiomatic definition of the measure of noncompactness given by Dhage, we obtain new generalizations of Theorem 1.2. Finally, an existence result for a certain class of fractional integral equations will be given as an application. 2 Main results Let X be a nonempty, bounded, closed, and convex subset of a Banach space E.Wecontinue to use the same notations presented in the previous section of this paper. Let F X be the set of self-mappings D : X X satisfying the following conditions: (i) D is a continuous mapping. (ii) There exists σ : B E [, ), a D-measure of noncompactness on E,suchthatfor all ε >,thereexistssomeδ ε >for which W P(X ), ε σ (W)<ε + δ ε σ (DW )<ε. We have the following result. Theorem 2.1 Let D : X X be a mapping that belongs to F X. Then D has at least one fixed point. The result of Theorem 2.1 can be obtained using the same arguments of the proof of Theorem 1.3 in [4]. By Theorem 2.1, we want just to mention that Theorem 1.3 is still valid for any D-measure of noncompactness. Let G X be the set of mappings D : X X such that (i) D is continuous. (ii) There exists a function ω :[, ) [, ) such that (ω 1 ) ω(t)=iff t =; (ω 2 ) ω is nondecreasing and right continuous;

4 Jleli et al. Fixed Point Theory and Applications (216) 216:11 Page 4 of 17 (ω 3 ) for every ε >,thereexistsγ ε >such that W P(X ), ε ω ( σ (W) ) < ε + γ ε ω ( σ (DW ) ) < ε, where σ : B E [, ) is a D-measure of noncompactness. The following lemma can be proved using a similar argument as in the proof of Theorem 2.6 in [4]. Lemma 2.2 We have G X F X. Using Theorem 2.1and Lemma2.2, weobtain the followingresult. Corollary 2.3 Let D : X X be a mapping that belongs to G X. Then D has at least one fixed point. Let be the set of functions ϕ :[, ) [, ) satisfying the conditions: ( 1 ) ϕ L 1 loc [, ); ( 2 ) for every ξ >,wehave ξ ϕ(s) >. Let H X be the set of mappings D : X X such that (H 1 ) D is continuous; (H 2 ) for every ε >,thereexistssomeγ ε >such that W P(X ), ε σ (W) ϕ(s) < ε + γ ε σ (DW ) ϕ(s) < ε, where ϕ and σ : B E [, ) is a D-measure of noncompactness. Lemma 2.4 We have H X G X. Proof Take ω(t)= t ϕ(s), t, we obtain the desired result. Using Corollary 2.3 and Lemma 2.4, we obtain the following result. Corollary 2.5 Let D : X X be a mapping that belongs to H X. Then D has at least one fixed point.

5 Jleli et al. Fixed Point Theory and Applications (216) 216:11 Page 5 of 17 Let I X be the set of mappings D : X X such that (I 1 ) D is continuous; (I 2 ) thereexistssomeϕ such that σ (DW ) ϕ(s) k σ (W) ϕ(s), W P(X ), where k (, 1) is a constant and σ : B E [, ) is a D-measure of noncompactness. Lemma 2.6 We have I X H X. Proof Let D : X X be a mapping that belongs to I X.Letε >befixed.letγ ε =( 1 k 1)ε. Take W P(X )suchthat ε σ (W) From (I 2 ), we obtain ϕ(s) < ε + γ ε = ε k. σ (DW ) ϕ(s) k σ (W) ϕ(s) < k ε k = ε, so D H X. Using Corollary 2.5 and Lemma 2.6, we obtain the following result. Corollary 2.7 Let D : X X be a mapping that belongs to I X. Then D has at least one fixed point. Remark 2.8 Take ϕ(t)=1,t in Corollary 2.7, we obtain Theorem 1.2. Let J X be the set of mappings D : X X such that (J 1 ) D is continuous; (J 2 ) there exists a function η :(, ) R such that (η 1 ) foreachsequence{ n } (, ),wehave lim η( n)= lim n =; n n (η 2 ) thereexistsτ >such that W P(X ), σ (W)σ (DW )> τ + η ( σ (DW ) ) η ( σ (W) ), where σ : B E [, ) is a D-measure of noncompactness. Theorem 2.9 Let D : X X be a mapping that belongs to J X. Then D has at least one fixed point.

6 Jleli et al. Fixed Point Theory and Applications (216) 216:11 Page 6 of 17 Proof Consider the sequence {X n } of subsets of E defined by { X := X, X n+1 := conv(dx n ), n =,1,2,... (2.1) By induction, we observe easily that X n+1 X n, n =,1,2,... (2.2) If for some N,wehaveσ (X N ) =, then by the property (i) of the D-measure of noncompactness, X N is compact. Since D(X N ) X N (from (2.2)), Schauder s fixed point theorem applied to the self-mapping D : X N X N gives the desired result. So, without loss of the generality, we may assume that σ (X n )>, n =,1,2,... For n =,sinceσ (X )>andσ (DX )=σ (X 1 )>,fromtheproperty(η 2 )wehave τ + η ( σ (DX ) ) η ( σ (X ) ), which yiel η ( σ (X 1 ) ) η ( σ (X ) ) τ. Similarly, for n =1, wehave η ( σ (X 2 ) ) η ( σ (X 1 ) ) τ η ( σ (X ) ) 2τ. By induction, we obtain η ( σ (X n ) ) η ( σ (X ) ) nτ, n =,1,2,... Since lim n η( σ (X ) ) nτ =, we deduce that lim n η( σ (X n ) ) =, so from the property (η 1 )wehave lim σ (X n)=. (2.3) n From the property (iv) of the D-measure of noncompactness, the set M := n=1 X n is nonempty. Moreover, for every p =,1,2,..., we have M X p, (2.4)

7 Jleli et al. Fixed Point Theory and Applications (216) 216:11 Page 7 of 17 which implies from (2.2)that DM DX p X p+1 X p, p =,1,2,... Then D : M M is well defined. On the other hand, from (2.4) and the property (ii) of the D-measure of noncompactness, we have σ (M) σ (X p ), p =,1,2,... Passing to the limit as p and using (2.3), we obtain σ (M)=, which implies from the property (i) of the D-measure of noncompactness that M = M is compact. Applying Schauder s fixed point theorem to the mapping D : M M,weobtain the desired result. Remark 2.1 Observe that D X J X.Infact,ifD : X X belongs to D X,thatis, σ (DW ) kσ (W), W P(X ), then W P(X), σ (W)σ (DW )> ln σ (DW ) ln k ln σ (W). Then D J X with η(t)=ln t, t >. Therefore, Theorem 2.9 is a generalization of Theorem 1.2. Let K X be the set of mappings D : X X such that (K 1 ) D is continuous; (K 2 ) there exists a function θ :(, ) (1, ) such that (θ 1 ) foreachsequence{u n } (, ),wehave lim θ(u n)=1 lim u n =; n n (θ 2 ) thereexist k (, 1) and a D-measure of noncompactness σ : B E [, ) such that W P(X ), σ (W)σ (DW )> θ ( σ (DW ) ) [ θ ( σ (W) )] k. We have the following result. Theorem 2.11 Let D : X X be a mapping that belongs to K X. Then D has at least one fixed point.

8 Jleli et al. Fixed Point Theory and Applications (216) 216:11 Page 8 of 17 Proof Consider the sequence {X n } of subsets of E defined by (2.1). As in the proof of Theorem 2.9, without loss of the generality, we may assume that σ (X n )>, n =,1,2,... For n =,sinceσ (X )>andσ (DX )=σ (X 1 )>,wehave θ ( σ (DX ) ) [ θ ( σ (X ) )] k, that is, θ ( σ (X 1 ) ) [ θ ( σ (X ) )] k. Again, for n =1,sinceσ (X 1 )>andσ (DX 1 )=σ (X 2 )>,wehave θ ( σ (DX 1 ) ) [ θ ( σ (X 1 ) )] k, that is, θ ( σ (X 2 ) ) [ θ ( σ (X 1 ) )] k, so θ ( σ (X 2 ) ) [ θ ( σ (X ) )] k 2. Therefore, by induction, we get 1<θ ( σ (X n ) ) [ θ ( σ (X ) )] k n, n =,1,2,... Passing to the limit as n,weobtain lim n θ( σ (X n ) ) =1, so from the property (θ 1 )wehave lim σ (X n)=. n The rest of the proof is similar to that in the proof of Theorem 2.9. Corollary 2.12 Let D : X X be a continuous mapping. Suppose that there exist a constant k (, 1) and a D-measure of noncompactness σ : B E [, ) such that 2 2 ( ) π arctan 1 [2 2π ( )] arctan 1 k, σ (DW ) σ (W) for any W P(X ) with σ (W)σ (DW )>.Then D has at least one fixed point.

9 Jleli et al. Fixed Point Theory and Applications (216) 216:11 Page 9 of 17 Proof Taking θ(t)=2 2 π arctan ( 1 t ), t >, in Theorem 2.11,weobtainthedesiredresult. Remark 2.13 Observe that D X K X.Infact,ifD : X X belongs to D X,thatis, σ (DW ) kσ (W), W P(X ), then W P(X ), σ (W)σ (DW )> e σ (DW ) [ e σ (W)] k. Therefore D K X with θ(t)=e t. Let L X be the set of mappings D : X X such that (L 1 ) D is continuous; (L 2 ) there exists a function ζ :[, ) [, ) R such that (ζ 1 ) ζ (z 1, z 2 )<z 2 z 1, for all z 1, z 2 >; (ζ 2 ) if {u n } and {v n } are two sequences in (, ) such that lim n u n = lim n v n = l >,then (ζ 3 ) lim sup ζ (u n, v n )<, n ζ ( σ (DW ), σ (W) ), W P(X ), where σ : B E [, ) is a D-measure of noncompactness. Theorem 2.14 Let D : X X be a mapping that belongs to L X. Then D has at least one fixed point. Proof Consider the sequence {X n } of subsets of E defined by (2.1). From the property (ζ 3 ), we have ζ ( σ (X n+1 ), σ (X n ) ), n =,1,2,... (2.5) As before, without loss of the generality, we may assume that σ (X n )>, n =,1,2,... (2.6) From the property (ζ 1 ), (2.5)and(2.6), we get σ (X n ) σ (X n+1 ), n =,1,2,...

10 Jleli et al. Fixed Point Theory and Applications (216) 216:11 Page 1 of 17 Then there is some r suchthat lim n σ (X n)=r. If r >, then from the property (ζ 2 ), we have lim sup ζ ( σ (X n+1 ), σ (X n ) ) <, n which contradicts (2.5). As consequence, we have lim n σ (X n)=. The rest of the proof is similar to the proof of Theorem 2.9. Remark 2.15 Taking ζ (z 1, z 2 )=kz 2 z 1, where k (, 1) is a constant, we obtain Theorem 1.2. Corollary 2.16 Let D : X X be a continuous mapping such that σ (DW ) σ (W) ( σ (W) ), W P(X ), where :[, ) [, ) is a lower semi-continuous function with 1 () = {} and σ : B E [, ) is a D-measure of noncompactness. Then D has at least one fixed point. Proof Taking ζ (z 1, z 2 )=z 2 (z 2 ) z 1 in Theorem 2.14,weobtainthedesiredresult. Corollary 2.17 Let D : X X be a continuous mapping such that σ (DW ) ψ ( σ (W) ), W P(X ), where ψ :[, ) [, ) is an upper semi-continuous function with ψ(t)<tforallt> and σ : B E [, ) is a D-measure of noncompactness. Then D has at least one fixed point. Proof Taking ζ (z 1, z 2 )=ψ(z 2 ) z 1 in Theorem 2.14,weobtainthedesiredresult.

11 Jleli et al. Fixed Point Theory and Applications (216) 216:11 Page 11 of 17 3 An existence result for a fractional integral equation The measure of noncompactness argument is a useful tool in Nonlinear Analysis. In particular, such argument can be used to obtain existence results for various classes of integral equations. For more details on the applications of the measure of noncompactness concept, we refer the reader to [1, 3, 4, 6, 8 15] and the references therein. In this section, we discuss the existence of solutions to the fractional integral equation y(t)= f (t, y(t)) Ɣ() t, (g(t) g(s)) t [, T], (3.1) where T >, (, 1), u, f :[,T] R R and g :[,T] R. We suppose that the following conditions are satisfied. (i) The function f :[,T] R R is continuous. (ii) There exists an upper semi-continuous function ψ :[, ) [, ) such that ψ() =, ψ(t)<t for all t >, ψ is nondecreasing, and f (t, x) f (t, y) ψ ( x y ), (t, x, y) [, T] R R. (iii) The function u :[, ) [, ) is continuous and there exists a nondecreasing function ω :[, ) [, ) such that u(t, z) ω ( z ), (t, z) [, T] R. (iv) The function g :[,T] R is C 1 and nondecreasing. (v) There exists r >such that ( ψ(r )+F ) ω(r ) ( g(t) g() ) r and ω(r ) ( ) g(t) g() 1, where F = max{ f (t,):t [, T]}. Let E = C([, T]; R) be the set of real continuous functions defined in [, T]. The set E endowed with the norm z = max { z(t) : t [, T] }, z E, is a Banach space. Let W be a nonempty and bounded subset of E. Let us define the mapping γ : W [, ) [, )by γ (z, ρ)=sup { z(a) z(b) : a, b [, T], a b ρ }, z W, ρ. Set γ (W, ρ)=sup { γ (z, ρ):z W }, ρ.

12 Jleli et al. Fixed Point Theory and Applications (216) 216:11 Page 12 of 17 Let B E be the set of all nonempty bounded subsets of E. Then the mapping σ : B E [, ) defined by σ (W)= lim ρ + γ (W, ρ), W B E, is a BG-measure of noncompactness (then it is a D-measure of noncompactness) on the space E (see [1]). We have the following existence result. Theorem 3.1 Under the assumptions (i)-(v), equation (3.1) has at least one solution y E. Moreover, we have y r. Proof Let us consider the operator D defined on E by (Dy)(t)= f (t, y(t)) Ɣ() t, (g(t) g(s)) (y, t) E [, T]. (3.2) Atfirst, weshowthattheoperator D maps E into itself. Set t (Hy)(t)=, (y, t) E [, T]. (3.3) (g(t) g(s)) Fromtheassumption(i),wehavejusttoshowthatH maps E into itself. In order to prove this fact, let us fix some y E.ObservethatHy :[,T] R is a well-defined function. In fact, using the assumptions (iii) and (iv), for all t [,T] we have that is, t (Hy)(t) g (s) u(s, y(s)) (g(t) g(s)) ω ( y ) t g (s) (g(t) g(s)) = ω( y ) ( ), g(t) g() (Hy)(t) ω( y ) ( ) g(t) g() <, t [, T]. (3.4) Let us prove the continuity of Hy at. To do this, let {t n } be a sequence in [, T]suchthat t n + as n.from(3.4), for all n we have (Hy)(tn ) ω( y ) ( g(tn ) g() ). Passing to the limit as n and using the continuity of g at, we obtain lim n (Hy)(t n)==(hy)(). Then Hy is continuous at.

13 Jleli et al. Fixed Point Theory and Applications (216) 216:11 Page 13 of 17 Now, let t (, T] befixedand{t n } be a sequence in (, T] suchthatt n t as n. Without restriction of the generality, we may assume that t n t for n large enough. For every n,wehave (Hy)(t n ) (Hy)(t) tn = g (s)u(s, y(s)) t (g(t n ) g(s)) (g(t) g(s)). For n large enough, we can write (Hy)(t n ) (Hy)(t) t ( g ) (s)u(s, y(s)) (g(t n ) g(s)) g (s)u(s, y(s)) dτ (g(t) g(s)) tn + t (g(t n ) g(s)) ω ( y ) t( g (s) (g(t) g(s)) g ) (s) (g(t n ) g(s)) + ω ( y ) t n Since g is continuous in [, T], we have t g (s) (g(t n ) g(s)) = ω( y ) (( ) ( g(t) g() + g(tn ) g(t) ) ( g(tn ) g() ) ) + ω( y ) ( g(tn ) g(t) ). lim n ω( y ) (( ) ( g(t) g() + g(tn ) g(t) ) ( g(tn ) g() ) ) ω( y ) ( + g(tn ) g(t) ) =, which yiel lim n (Hy)(t n ) (Hy)(t) =.ThenHy is continuous at t.asconsequence, Hy E, for all y E,andD : E E is well defined. On the other hand, using the assumptions (ii) and (iii), for an arbitrarily fixed y E and t [, T], we have (Dy)(t) f (t, y(t)) t g (s) u(s, y(s)) Ɣ() (g(t) g(s)) Then f (t, y(t)) f (t,) + f (t,) Ɣ() t (ψ( y(t) )+F)ω( y ) ( ) g(t) g() (ψ( y )+F)ω( y ) ( ). g(t) g() Dy (ψ( y )+F)ω( y ) ( ), g(t) g() y E. g (s)ω( y(s) ) (g(t) g(s)) Using the above inequality, the fact that the functions ψ, ω :[, ) [, ) are nondecreasing, and the assumption (v), we infer that the operator D maps B(, r )intoitself,

14 Jleli et al. Fixed Point Theory and Applications (216) 216:11 Page 14 of 17 where B(, r )= { z E : z r }. Now, we claim that the operator D : B(, r ) B(, r ) is continuous. From (3.2), we can write D in the form where Dy = 1 Gy Hy, y E, Ɣ() (Gy)(t)=f ( t, y(t) ), (y, t) E [, T], and Hy is defined by (3.3). In order to prove our claim, it is sufficient to show that the operators G and H are continuous on B(, r ). First of all, we show that G is a continuous operator on B(, r ). To do this, we take a sequence {y n } B(, r )andy B(, r )such that y n y asn, and we have to prove that Gy n Gy asn.in fact, for all t [, T], using the condition (ii), we have (Gyn )(t) (Gy)(t) ( = f t, yn (t) ) f ( t, y(t) ) ψ ( yn (t) y(t) ) ψ ( y n y ) y n y. Thus we have Gy n Gy y n y, for all n. Passing to the limit as n in the above inequality, we obtain lim n Gy n Gy =. This proves that G is a continuous operator on B(, r ). Next, we show that H is a continuous operator on B(, r ). To do this, we fix a real number ε > and we take arbitrary functions x, y B(, r )suchthat x y < ε. For all t [, T], we have where (Hx)(t) (Hy)(t) t u(r, ε) g (s) u(s, x(s)) u(s, y(s)) (g(t) g(s)) t g (s) (g(t) g(s)) u(r, ε) ( ), g(t) g() u(r, ε)=sup { u(τ, v) u(τ, w) : τ [, T], v, w [ r, r ], v w < ε }.

15 Jleli et al. Fixed Point Theory and Applications (216) 216:11 Page 15 of 17 Therefore, Hx Hy u(r, ε) ( ). g(t) g() Since u is uniformly continuous on the compact [, T] [ r, r ], we have u(r, ε) as ε + and, therefore, the last inequality gives us lim Hx Hy =. ε + Then H is continuous on B(, r )anddmaps continuously the set B(, r )intoitself. Further, let W be a nonempty subset of B(, r ). Let ρ >befixed,y W,andt 1, t 2 [, T]besuchthat t 1 t 2 ρ. Without restriction of the generality, we may assume that t 1 t 2.Wehave (Dy)(t1 ) (Dy)(t 2 ) f (t 1, y(t 1 )) t1 Ɣ() (g(t 1 ) g(s)) f (t 2, y(t 2 )) t2 Ɣ() (g(t 2 ) g(s)) f (t 1, y(t 1 )) t1 Ɣ() (g(t 1 ) g(s)) f (t 2, y(t 2 )) t1 Ɣ() (g(t 1 ) g(s)) + f (t 2, y(t 2 )) t1 g (s)u(s, y(s)) t2 Ɣ() (g(t 1 ) g(s)) (g(t 2 ) g(s)) f (t 1, y(t 1 )) f (t 1, y(t 2 )) + f (t 1, y(t 2 )) f (t 2, y(t 2 )) t1 Ɣ() + f (t 2, y(t 2 )) f (t 2,) + f (t 2,) t1 Ɣ() (g(t 1 ) g(s)) t2 (g(t 1 ) g(s)) + f (t 2, y(t 2 )) f (t 2,) + f (t 2,) t2 Ɣ() (g(t 1 ) g(s)) t2 (g(t 2 ) g(s)) ψ( y(t 1) y(t 2 ) )+ω f (r, ρ) ω(r ) ( g(t 1 ) g() ) + (ψ(r )+F)ω(r ) Ɣ() t1 t 2 g (s) (g(t 1 ) g(s)) g (s) u(s, y(s)) (g(t 1 ) g(s)) + (ψ(r )+F)ω(r )(( g(t2 ) g() ) ( + g(t1 ) g(t 2 ) ) ( g(t1 ) g() ) ) ψ(γ (y, ρ)) + ω f (r, ρ) ψ(γ (y, ρ)) + ω f (r, ρ) ω(r ) ( g(t) g() ) 2(ψ(r )+F)ω(r )( + g(t1 ) g(t 2 ) ) ω(r ) ( g(t) g() ) 2(ψ(r )+F)ω(r )[ ], + γ (g, ρ)

16 Jleli et al. Fixed Point Theory and Applications (216) 216:11 Page 16 of 17 where ω f (r, ρ)=sup { f (t, u) f (s, u) : u [ r, r ], t, s [, T], t s ρ }. Therefore, γ (DW, ρ) ψ(γ (W, ρ)) + ω f (r, ρ) ω(r ) ( g(t) g() ) 2(ψ(r )+F)ω(r )[ ]. + ω(g, ρ) Passing to the limit superior as ρ + and using the fact that ψ is upper semi-continuous, we obtain σ (DW ) ψ(σ (W)) ω(r ) ( g(t) g() ). Then, from the assumption (v), we obtain σ (DW ) ψ ( σ (W) ). As a consequence, for any nonempty subsets W of B(, r ), we have ζ ( σ (DW ), σ (W) ), where ζ :[, ) [, ) R is defined by ζ (z 1, z 2 )=ψ(z 2 ) z 1, (z 1, z 2 ) [, ) [, ). Under the assumptions on the function ψ, the operator D : B(, r ) B(, r )belongsto the family of operators L X,whereX = B(, r ). Then by Theorem 2.14,wededucethatD has at least one fixed point y B(, r ), which is a solution to equation (3.1). Competing interests The authors declare that they have no competing interests. Authors contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details 1 Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia. 2 Department of Mathematics, Atilim University, Incek, Ankara, 6836, Turkey. 3 Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah, 21589, Saudi Arabia. 4 School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland. Acknowledgements The first and fourth authors would like to extend their sincere appreciations to the Deanship of Scientific Research at King Saud University for its funding of this Prolific Research Group Project No. PRG Received: 17 September 215 Accepted: 4 January 216 References 1. Banaś, J, Goebel, K: Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Appl. Math., vol. 6. Dekker, New York (198) 2. Darbo, G: Punti uniti in trasformazioni a codominio noncompatto. Rend. Semin. Mat. Univ. Padova 24, (1955) 3. Aghajani, A, Allahyari, R, Mursaleen, M: A generalization of Darbo s theorem with application to the solvability of systems of integral equations. J. Comput. Appl. Math. 26, (214) 4. Aghajani, A, Mursaleen, M, Shole Haghighi, A: Fixed point theorems for Meir-Keeler condensing operators via measure of noncompactness. Acta Math. Sci. 35(3), (215)

17 Jleli et al. Fixed Point Theory and Applications (216) 216:11 Page 17 of Cai, L, Liang, J: New generalizations of Darbo s fixed point theorem. Fixed Point Theory Appl. 215, Article ID 156 (215) 6. Dhage, BC: Asymptotic stability of nonlinear functional integral equations via measures of non-compactness. Commun. Appl. Nonlinear Anal. 15(2),89-11 (28) 7. Papageorgiou, SN, Kyritsi-Yiallourou, ST: Handbook of Applied Analysis. Springer Science & Business Media, vol. 19. Springer, Berlin (29) 8. Samadi, A, Ghaemi, MB: An extension of Darbo s theorem and its application. Abstr. Appl. Anal. 214, Article ID (214) 9. Samadi, A, Ghaemi, MB: An extension of Darbo fixed point theorem and its applications to coupled fixed point and integral equations. Filomat 28(4), (214) 1. Banaś, J, Nalepa, R: On a measure of noncompactness in the space of functions with tempered increments. J. Math. Anal. Appl. 435, (216) 11. Banaś, J, Mursaleen, M: Sequences Spaces and Measures of Noncompactness with Applications to Differential and Integral Equations. Springer, New Delhi (214) 12. Banaś, J, Rodriguez, JR, Sadarangani, K: On a class of Urysohn-Stieltjes quadratic integral equations and their applications. J. Comput. Appl. Math. 113, 35-5 (2) 13. Banaś, J, Sadarangani, K: Solvability of Volterra-Stieltjes operator-integral equations and their applications. Comput. Math. Appl. 41, (21) 14. Darwish, MA: On quadratic integral equation of fractional orders. J. Math. Anal. Appl. 311, (25) 15. Darwish, MA, Henderson, J, O Regan, D: Existence and asymptotic stability of solutions of a perturbed fractional functional-integral equation with linear modification of the argument. Bull. Korean Math. Soc. 48, (211)