Iterative Algorithms for a Finite Family of Multivalued Quasi-Nonexpansive Mappings

Abstract

Let be a nonempty closed and convex subset of a uniformly convex real Banach space and let be multivalued quasi-nonexpansive mappings. A new iterative algorithm is constructed and the corresponding sequence is proved to be an approximating fixed point sequence of each ; that is, . Then, convergence theorems are proved under appropriate additional conditions. Our results extend and improve some important recent results (e.g., Abbas et al. (2011)).

1. Introduction

Let be a metric space, a nonempty subset of , and a multivalued mapping. An element is called a fixed point of if . For single valued mapping, this reduces to . The fixed point set of is denoted by .

For several years, the study of fixed point theory for multivalued nonlinear mappings has attracted, and continues to attract, the interest of several well known mathematicians (see, e.g., Brouwer [1], Kakutani [2], Nash [3, 4], Geanakoplos [5], Nadler Jr. [6], and Downing and Kirk [7]).

Interest in the study of fixed point theory for multivalued nonlinear mappings stems, perhaps, mainly from its usefulness in real-world applications such as Game Theory and Nonsmooth Differential Equations.

Game Theory. Nash showed the existence of equilibria for noncooperative static games as a direct consequence of multivalued Brouwer or Kakutani fixed point theorem. More precisely, under some regularity conditions, given a game, there always exists a multivalued mapping whose fixed points coincide with the equilibrium points of the game. This, among other things, made Nash a recipient of Nobel Prize in Economic Sciences in 1994. However, it has been remarked that the applications of this theory to equilibrium are mostly static: they enhance understanding conditions under which equilibrium may be achieved but do not indicate how to construct a process starting from a nonequilibrium point and convergent to equilibrium solution. This is part of the problem that is being addressed by iterative methods for fixed point of multivalued mappings.

Nonsmooth Differential Equations. A large number of problems from mechanics and electrical engineering lead to differential inclusions and differential equations with discontinuous right-hand sides, for example, a dry friction force of some electronic devices. Below are two models: where and are fixed in . These types of differential equations do not have solutions in the classical sense. A generalized notion of solution is what is called a solution in the sense of Fillipov.

Consider the following multivalued initial value problem: Under some conditions, the solutions set of (1) and (2) coincides with the fixed point set of some multivalued mappings.

Let be a nonempty subset of a normed linear space . The set is called proximinal (see, e.g., [8–10]) if for each , there exists such that where for all . Every nonempty, closed, and convex subset of a real Hilbert space is proximinal. We denote by the families of nonempty closed and bounded subsets of , the families of nonempty compact subsets of , and the families of nonempty compact convex subsets of . The Hausdorff metric on is defined by for all . A multivalued mapping is called Lipschitzian if there exists such that When in (5), we say that is a contraction, and is called nonexpansive if . Finally, A multivalued mapping is said to be quasi-nonexpansive if and for all and .

Several papers deal with the problem of approximating fixed points of multivalued nonexpansive mappings (see, e.g., [8–12] and the references therein) and their generalizations (see, e.g., [13–15]).

On the other hand, Abbas et al. [11] introduced a new one-step iterative process for approximating a common fixed point of two multivalued nonexpansive mappings in a real uniformly convex Banach space and established weak and strong convergence theorems for the proposed process under some basic boundary conditions. Let be two multivalued nonexpansive mappings. They introduced the following iterative scheme: where and are such that and , , and are real sequences in satisfying .

The following lemma is a consequence of the definition of Hausdorff metric, as remarked by Nadler Jr. [6].

Lemma 1. Let and . For every , there exists such that

Following the work of Abbas et al. [11], Rashwan and Altwqi [16] introduced a new scheme for approximation of a common fixed point of three multivalued nonexpansive mappings in uniformly convex Banach space. Let be three multivalued nonexpansive mappings. They employed the following iterative process: where , , and are such that and , , and are real sequences in satisfying .

Remark 2. Note that if , , and are known, then the existence of , , and satisfying (10) is guaranteed by Lemma 1.

Before we state the result of Rashwan and Altwqi [16], we need the following definition.

The mappings are said to satisfy condition if , for and , or , for and , or , for and .

Let be the set of all common fixed points of the mappings , , and .

Theorem RA (Rashwan and Altwqi [16]). Let be a uniformly convex Banach space and a nonempty closed and convex subset of . Let be multivalued nonexpansive mappings satisfying condition and the sequence defined by (9) and (10). If and for any , then

Recently, Bunyawat and Suantai [17] introduced a one-step iterative scheme for finding a common fixed point of a finite family of multivalued quasi-nonexpansive mappings in a uniformly convex real Banach space. They proposed the following algorithm: let be a nonempty closed and convex subset of a uniformly convex real Banach space and multivalued quasi-nonexpansive mappings. Let be the sequence defined iteratively from by where the sequence satisfies    and with .

Then, they proved the following theorem.

Theorem BS (Bunyawat and Suantai [17]). Let be a real Banach space and a closed convex subset of . Let be a finite family of multivalued quasi-nonexpansive mappings from into with . Then the sequence defined by (12) converges strongly to a common fixed point of if and only if .

It is our purpose in this paper to construct a new iterative algorithm and prove strong convergence theorems for approximating a common fixed point of a finite family of multivalued quasi-nonexpansive mappings in uniformly convex real Banach spaces. The class of mappings used in our theorems is much more larger than that of multivalued nonexpansive mappings. Our theorems generalize and extend those of Abbas et al. [11], Rashwan and Altwqi [16], and Bunyawat and Suantai [17] and many other important results.

2. Preliminaries

Lemma 3 (Chang et al. [18]). Let be a uniformly convex real Banach space. For arbitrary , let . Then, for any given subset and for any positive real numbers with , there exists a continuous, strictly increasing, and convex function with such that, for any , with ,

3. Main Results

In this paper, we propose the following iterative algorithm.

Let and a nonempty closed convex subset of a uniformly convex real Banach space . Let be multivalued quasi-nonexpansive mappings. Let be a sequence defined iteratively as follows: where , , , , with . In the sequel, we will write for the set of common fixed points of the mappings , .

Lemma 4. Let be a nonempty, closed, and convex subset of a real Banach space . For , let be a finite family of multivalued quasi-nonexpansive mappings from into with and such that and . Let be the sequence defined by (14). Then, for all , exits.

Proof. Let . By (14) and the quasi-nonexpansiveness of we have Equation (15) implies that the sequence is monotonically decreasing. Since it is bounded from below by , we conclude that exists.

Next, we prove the following theorem.

Theorem 5. Let be a nonempty, closed, and convex subset of a real uniformly convex Banach space . For let be a multivalued quasi-nonexpansive mapping. Suppose that and that for all . Let be the sequence defined by (14). Then, for all ,

Proof. Using Lemma 4 and the quasi-nonexpansiveness of , there exists some positive real such that This implies that, for each , , Next, let ; using inequality (17) and Lemma 3, we have Therefore This implies that It follows that Since , we have This completes the proof.

We now approximate common fixed points of through strong convergence of the sequence defined by (14). We start with the following definition.

Definition 6. A family is said to satisfy Condition , if there exists a strictly increasing function with , for all and , , such that

Theorem 7. Let be a nonempty, closed, and convex subset of a uniformly convex real Banach space . For , let be a multivalued quasi-nonexpansive mapping. Assume that satisfies condition . If and for all , then the sequence converges strongly to a common fixed point of .

Proof. From Theorem 5, we have for all . Using the fact that satisfies condition , it follows that there exists some such that . Thus there exist a subsequence of and a sequence such that By setting in place of and following the same arguments as in the proof of Lemma 4, we obtain from inequality (15) that We now show that is a Cauchy sequence in . Notice that This shows that is a Cauchy sequence in and thus converges strongly to some . Using the fact that is quasi-nonexpansive and , we have so that and thus . Therefore, and converges strongly to . Setting in place of in the proof of Lemma 4, it follows from inequality (15) that exists. So, converges strongly to . This completes the proof.

Definition 8. A mapping is called hemicompact if, for any sequence in such that as , there exists a subsequence of such that . One notes that if is compact, then every multivalued mapping is hemicompact.

Theorem 9. Let be a nonempty, closed, and convex subset of a uniformly convex real Banach space . For let be a multivalued continuous and quasi-nonexpansive mapping. Assume that is hemicompact for some . If and for all , then the sequence defined by (14) converges strongly to a common fixed point of .

Proof. From Theorem 5, we have that . Since is hemicompact, there exists a subsequence of such that for some . Since for each , is continuous, we have . Therefore, and so for all . Setting in the proof of Theorem 5, it follows from inequality (15) that the sequence is decreasing and bounded from bellow. Therefore, exists. So, converges strongly to . This completes the proof.

The following result gives a necessary and sufficient condition for the strong convergence of the sequence in (14) to a common fixed point of a finite family of multivalued nonexpansive maps , .

Theorem 10. Let be a nonempty, closed, and convex subset of a real Banach space . For let be a multivalued nonexpansive mapping. Assume that and that for all . Then the sequence defined by (14) converges strongly to a common fixed point of if and only if .

Proof. The necessity is obvious. Conversely, suppose that . Let . By (15), we have . This gives . Hence, exists. By hypothesis, so we must have . Next, we show that is a Cauchy sequence in . Let be given and since , there exists such that, for all , we have In particular, , so there exists such that Now for , we have Hence, is a Cauchy sequence in a closed subset of a Banach space and therefore, it converges in . Let . Now, for each , we have By letting , it follows that , , which implies that . Consequently, .

Corollary 11 (Abbas et al. [11]). Let be a real Banach space and a nonempty, closed, and convex subset of . Let be multivalued nonexpansive mappings of into such that . Let , , and be sequences in satisfying . Let be a sequence defined iteratively by where , are such that and whenever is a fixed point of any one of the mappings and . Then, converges strongly to a common fixed point of if and only if .

Corollary 12 (Bunyawat and Suantai [17]). Let be a real Banach space and a closed convex subset of . Let be a finite family of multivalued quasi-nonexpansive mappings from into with . Then the sequence defined by (12) converges strongly to a common fixed point of if and only if .

Remark 13. The recursion formula (14) used in our theorems is easier to use than the recursion formula (6) of Abbas et al., the one of Rashwan and Altwqi (9), and the one of Bunyawat and Suantai (12) in the following sense: in our algorithm, for and do not have to satisfy the restrictive conditions: (7) in the recursion formula (6), (10) in the recursion formulas (9), and similar restrictive conditions in the recursion formula (12).

Conflict of Interests

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

Acknowledgment

The authors thank the referee for his work and his valuable suggestion that helped to improve the presentation of this paper.