The Scientific World Journal

Volume 2014 (2014), Article ID 541862, 9 pages

http://dx.doi.org/10.1155/2014/541862

Research Article

Existence of Tripled Fixed Points for a Class of Condensing Operators in Banach Spaces

Vatan Karakaya,¹ Nour El Houda Bouzara,² Kadri Doğan,¹ and Yunus Atalan²

¹Department of Mathematical Engineering, Faculty of Chemistry-Metallurgical, Yildiz Technical University, 34210 Istanbul, Turkey

²Department of Mathematics, Faculty of Science and Letters, Yildiz Technical University, 34210 Istanbul, Turkey

Received 28 May 2014; Accepted 23 June 2014; Published 14 September 2014

Academic Editor: S. A. Mohiuddine

Copyright © 2014 Vatan Karakaya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 [1] in 1930. In 1955, Banaś and Goebel [2] used the function to prove his fixed point theorem. Darbo’s fixed point theorem [2] is a very important generalization of Schauder’s fixed point theorem [3] and several authors had used this concept for the resolution of nonlinear equations, some of whom are Aghajani et al. [4, 5], Banaś [6], Banaś and Rzepka [7], Mursaleen and Mohiuddine [8], and many others. Recently in [9], 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 [10]). 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 [2]). Let be a nonempty closed, bounded, and convex subset of . If   is a continuous mapping

then has a fixed point.

Theorem 3 (see [9]). 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 [9]). 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 [11]). A coupled fixed point of a mapping is an element such that and .

Theorem 6 (see [12]). 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 [13] 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 [9]). 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 [14]). 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.

3. Applications

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 .

Example 1. 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,

Furthermore, for any ,

Thus,

It is easy to see that, for any , we have that

holds and condition (v) is valid.

Consequently, the system has at least one solution in .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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