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Existence of Tripled Fixed Points for a Class of Condensing Operators in Banach Spaces
We give some results concerning the existence of tripled fixed points for a class of condensing operators in Banach spaces. Further, as an application, we study the existence of solutions for a general system of nonlinear integral equations.
1. Introduction and Preliminaries
Measures of noncompactness are very useful tools in functional analysis, for instance, in metric fixed point theory and in the theory of operator equations in Banach spaces. The first measure of noncompactness, denoted by , was defined and studied by Kuratowski  in 1930. In 1955, Banaś and Goebel  used the function to prove his fixed point theorem. Darbo’s fixed point theorem  is a very important generalization of Schauder’s fixed point theorem  and several authors had used this concept for the resolution of nonlinear equations, some of whom are Aghajani et al. [4, 5], Banaś , Banaś and Rzepka , Mursaleen and Mohiuddine , and many others. Recently in , Aghajani et al. give a generalization of Darbo’s fixed point theorem. Moreover, they present some results on the existence of coupled fixed points for class of condensing operators. In this paper, we generalize these results to obtain the existence of tripled fixed points for the same class of operators.
Throughout this paper, is assumed to be a Banach space and is the space of all real functions defined, bounded and continuous on . The family of bounded subset, closure, and closed convex hull of are denoted by , , and , respectively.
Definition 1 (see ). Let be a Banach space and the family of bounded subset of . A map is called measure of noncompactness defined on if it satisfies the following.(1) is a precompact set.(2). (3), . (4). (5), for .(6)Let be a sequence of closed sets from such that , , and . Then, the intersection set is nonempty and is precompact.
Theorem 2 (see ). Let be a nonempty closed, bounded, and convex subset of . If is a continuous mapping then has a fixed point.
Theorem 3 (see ). Let be a nonempty closed, bounded, and convex subset of and a continuous mapping such that for any subset of where ; that is, implies . Then, has at least one fixed point.
The following result is a corollary of the previous theorem.
Corollary 4 (see ). Let be a nonempty closed, bounded, and convex subset of and a continuous mapping such that for any subset of where is a nondecreasing and upper semicontinuous functions; that is, for every , . Then, has at least one fixed point.
Definition 5 (see ). A coupled fixed point of a mapping is an element such that and .
Theorem 6 (see ). Let be measures of noncompactness in Banach spaces , (respectively).
Then, the function defines a measure of noncompactness in , where is the natural projection of on , for , and is a convex function defined by such that
Remark 7. Aghajani and Sabzali  illustrated the previous theorem by the following example. Let the mapping be as follows: They showed that or defines a measure of noncompactness in the space , where, for , are measure of noncompactness in and , are the natural projections of on .
Theorem 8 (see ). Let be a nonempty, bounded, closed, and convex subset of a Banach space and let . Assume that is a nondecreasing and upper semicontinuous function. Let be a continuous operator satisfying for any measure of noncompactness . Then, has at least a coupled fixed point.
2. Main Results
Definition 9 (see ). A tripled of a mapping is called a tripled fixed point if
Remark 10. We can notice that by taking or satisfies the conditions of Theorem 6. Thus, for a measure of noncompactness , we have that or defines a measure of noncompactness in the space where , are the natural projections of on .
So, we obtain the following theorem.
Theorem 11. Let be a nonempty, bounded, closed, and convex subset of a Banach space and let be a nondecreasing and upper semicontinuous function such that for all . Then, for any measure of noncompactness , the continuous operator satifying has at least a tripled fixed point.
Proof. To prove this theorem, let us define the measure of noncompactness by and the mapping Since and is a measure of noncompactness, we get By Corollary 4, we obtain that has at least a tripled fixed point.
We can see an application of Theorem 11 in the study of existence of solutions for systems of integral equations defined on the Banach space endowed with the norm The measure of noncompactness on for a positive fixed on is defined as follows: such that Before defining , we need first to introduce the modulus of continuity.
Let and ; is the modulus of continuity of on and let
Assume that(i) are continuous and as ;(ii)the function is continuous and there exist positive , such that for any ;(iii) is continuous and there exists a nondecareasing continuous function with , so that (iv)the function defined by is bounded on ; that is, (v) is a continuous function and there exists a positive solution of the inequality where is positive constant defined by the equality uniformly with respect to .
Theorem 12. Suppose that (i)–(v) hold; then the system of integral equations has at least one solution in the space .
Proof. Let be an operator defined by
For , let
We can easily prove that the solution of (32) in is equivalent to the tripled fixed point of .
Obviously, is continuous for any . Hence, we have Then, by (29) and (30), we get So, we obtain Now, we prove that is continuous. Let , such that, for , Then, Using condition (iii) and (29), there exists such that if , then for any ,. We notice two cases.
Case 1. If , then from (39) and (40)
Case 2. Similarly, for , we have where , and Since is continuous on , we have and . Thus, using (iii), we get Finally, from (42) and (41), we conclude that is a continuous function from into .
Now, we show that the map satisfies all the conditions of Theorem 11. To do this, for an arbitrary fixed and , assume that , , and are nonempty chosen subsets of and , , with . Without loss of generality, let For an arbitrary , where we obtain Further, by the uniform continuity of and on the compact sets and , respectively, we get and as .
Moreover, is a nondecreasing continuous function with and , and we obtain By (48), we get Taking the limit in (50), we obtain Then, for arbitrary , , and , we have Since , , and are arbitrary in (52), Taking again in (53), we obtain We conclude from (51) and (54) that Since is a concave function, (55) implies Finally, since is defined by we get Hence, by Theorem 11, has at least a tripled fixed point in .
We consider the following system of integral equations
We notice that by taking
we get the system integral equations (32).
To solve this system, we need to verify conditions (i)–(v).
Obviously, are continuous and as . Further, the function is continuous for , and we have for any . Conditions (i) and (ii) hold.
Now, let Then, (iii) also holds.
Moreover, then, (iv) is valid.
Let us verify the last condition (v). First, Hence,