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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 206759, 5 pages
http://dx.doi.org/10.1155/2013/206759
Research Article

Common Fixed Points for Suzuki-Generalized Nonexpansive Maps

1Department of Mathematics, Science Faculty for Girls, King Abdulaziz University, P.O. Box 4087, Jeddah 21491, Saudi Arabia
2Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
3Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21859, Saudi Arabia

Received 17 May 2013; Revised 27 July 2013; Accepted 27 July 2013

Academic Editor: E. Karapinar

Copyright © 2013 Maryam A. Alghamdi 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

A common fixed point theorem for a pair of maps satisfying condition (C) is proved under certain conditions. We extend the well-knownDeMarr's fixed point theorem to the case of noncommuting family of maps satisfying condition (C). As for an application, an invariant approximation theorem is also derived.

1. Introduction

Jungck [1] initiated the systematic study of finding a common fixed point of a pair of commuting maps. This problem of finding a common fixed point has been of significant interest in the area of fixed point theory and has been studied by many authors such as in [26]. At the first time, the commutativity for two maps was always assumed to find a common fixed point. Later, it was found that the two maps were not necessarily commutative at each point, and then weaker forms of commutativity were defined to obtain a common fixed point for maps on a metric space. For example, the notions of weakly commutative maps [2], compatible maps (weakly compatible maps) [7], biased maps [8], -subweakly commuting maps [4], and occasionally weakly compatible [9] have been introduced and used to find common fixed points of maps.

Recently, Chen and Li [5] introduced the class of Banach operator pairs and, in [10], they investigated the common fixed point problem for nonexpansive maps where is a Banach operator pair. Also, they extended the well-known De Marr's fixed point theorem to the noncommuting case.

More recently, Suzuki [11] introduced a condition on maps, called condition (C) (maps satisfying condition (C) are also known as Suzuki-generalized nonexpansive maps), and obtained some fixed point theorems and convergence theorems for such maps. Dhompongsa et al. [12] and Dhompongsa and Kaewcharoen [13] made significant contribution to fixed point theory for maps satisfying condition (C). For more results see [14].

In this paper, we discuss a common fixed point problem for a Banach operator pair satisfying condition (C). A family of maps satisfying condition (C) is also investigated. As for an application, an invariant approximation theorem is obtained.

2. Preliminaries

Let be a Banach space. is said to be(i)strictly convex if for all , with and ,(ii)uniformly convex in every direction (UCED) if, for and with , there exists such that for all , with and .

It is obvious that being UCED implies strict convexity.

Let be a nonempty subset of and let be a self-map of . We denote by the set of fixed points of ; that is, . Also, if and are self-maps of , we denote by the set of common fixed points of and ; that is, . If is a nonempty family of self-maps of , a point is called common fixed point of if it is the fixed point of each member of .

The map is called(i)nonexpansive if (ii)quasi-nonexpansive if and

Suzuki [11] introduced a condition on maps, called condition (C), which is weaker than nonexpansiveness and stronger than quasi-nonexpansiveness.

Definition 1 (see [11]). A self-map of is said to satisfy condition if for all .

Example 2 (see [13]). Define a map on by Then is a continuous map satisfying condition () and is not nonexpansive.

Proposition 3 (see [11]). Let be a nonempty subset of a Banach space . Assume that is a nonexpansive map. Then satisfies condition .

Proposition 4 (see [11]). Let be a nonempty subset of a Banach space . Assume that a map satisfies condition and has a fixed point. Then is a quasi-nonexpansive map.

Chen and Li [5] introduced the class of Banach operator pairs.

Definition 5 (see [5]). Let be a metric space; the pair of two self-maps and of is called a Banach operator pair if the set of fixed points of is invariant; that is, .

A Banach operator pair depends on the order of and ; that is, if is a Banach operator pair, need not be such a pair. It is well known that for two self-maps and of a metric space , the pair is a Banach pair if and only if and commute on the set [5].

Example 6 (see [5]). Let and be two self-maps of defined by for . Directly, we have The following assertions can be verified: (i), and hence is a Banach operator pair on ; equivalently, and commute on the set .(ii) is not a Banach operator pair, since for is not in .

The following proposition for Banach operator pairs can be found in [10].

Proposition 7. If is a star shaped set (i.e., for any and with , then is a Banach operator pair if and only if the pairs s are Banach operator pairs for all , where .

Definition 8 (see [10]). Let and be two self-maps of a metric space . The pair is called symmetric Banach operator pair if both and are Banach operator pairs; that is, and .

The pair is a symmetric Banach operator pair if and only if and are commuting on . It is easy to see that a Banach operator pair may not be a symmetric Banach operator pair; see [10].

Definition 9 (see [10]). Let be a nonempty family of self-maps of a metric space .   is called a Banach operator family if for all ,, is a symmetrical Banach operator pair.

In 1963, DeMarr [15] stated the following well-known fixed point theorem for a family of commuting nonexpansive maps.

Theorem 10 (DeMarr [15]). If is a nonempty compact convex subset of a Banach space and is a nonempty family of commuting nonexpansive maps of into itself, then the family has a common fixed point in .

Recently Chen and Li [10] extended DeMarr's theorem to the noncommuting case.

Theorem 11 (see [10]). Let be a nonempty closed convex subset of a normed space and let be a nonempty family of nonexpansive maps of into itself. If is a Banach operator family and there exists a such that is compact, then has a common fixed point in .

We now collect some results about condition (C) which will be used in the sequel.

Lemma 12 (see [11]). Let be a nonempty closed subset of a Banach space . Assume that satisfies condition (C). Then is closed. Moreover, if is strictly convex and is convex, then is also convex.

Theorem 13 (see [14]). Let be a closed bounded convex subset of a Banach space . Assume that is a map satisfying condition and that is compact. Then has a fixed point.

Lemma 14 (see [11]). Let be a nonempty subset of a Banach space . Assume that is a map satisfying condition (C). Then for , the following hold: (i), (ii)either or holds,(iii)either or holds.

3. Main Results

Lemma 15 (see [16] or [15]). Let be a nonempty compact subset of a Banach space . Let be the diameter of . If , then there exists an element such that where is the smallest closed convex set containing .

Following [15], we are able to prove the following lemma.

Lemma 16. Let be a nonempty closed convex subset of a Banach space . Suppose that satisfies condition such that there exists a compact set not reduced to a point. Then there exists a nonempty closed convex set such that (1) and ,(2).

Proof. Let be the diameter of . Since is not reduced to a point, we have . According to Lemma 15, there is such that For each , define Since for each , it follows that It is easy to check that is closed and convex. Let . Then is not empty since . For any and any , we have ; that is, . Since we obtain that That is, . This is true for any ; thus . This shows that for all . Recalling that is compact, therefore, there exist such that . Thus, ; that is, .

Theorem 17. Let be a nonempty closed bounded convex subset of a Banach space . Suppose that and are two self-maps on satisfying condition . If is a Banach operator pair, is nonexpansive, and is compact, then .

Proof. Let be the set of all nonempty closed bounded convex subsets of such that and and is compact. Since , then is nonempty. Define a partial order “≤” by set inclusion on the set ; that is, whenever .
Let be any total ordering subset of and . Since is closed, we have , and since is compact, it follows that It is clear that . By Zorn’s lemma, has a minimal set .
Since satisfies condition (C) and is compact, then, by Theorem 13, has a nonempty fixed point set . It follows that is a closed subset of and thus is compact. On the other hand, we have , and since is a Banach operator pair, it implies that . Using Zorn's lemma again, there exists a minimal nonempty compact subset of which satisfies and ( is not necessarily convex).
Next, we show . If , then we have , and is compact because is continuous. Also, we have since . This contradicts the minimality of .
If has only one point, the proof is finished. Suppose that has at least two points. By Lemma 16 there exists a set satisfying and . Since is nonexpansive and , it follows that which implies that is a proper subset of and this contradicts the minimality of . This completes the proof.

Theorem 18. Let be a nonempty closed bounded convex subset of a strictly convex space . Suppose that and are two self-maps on satisfying condition . If is a Banach operator pair and is compact, then .

Proof. By Theorem 13 and Lemma 12, is a nonempty closed bounded convex set. It is compact since is compact. Since , again by Theorem 13, has a fixed point in ; that is, .

Corollary 19. Let be a nonempty closed bounded convex subset of an UCED Banach space . Suppose that and are two self-maps on satisfying condition (C). If is a Banach operator pair and is compact, then .

Example 20. Consider with the usual metric and let . Define a map on by and define a map on by . Then is a continuous map satisfying condition and is not nonexpansive (see [13]) and is nonexpansive and hence satisfies condition . Also is a Banach operator pair. Therefore, all conditions of Theorem 17 (and Theorem 18) are satisfied and and have a common fixed point. Note that Theorem 2.1 in Chen and Li [10] is not applicable here.

Next, we show a common fixed point theorem of a countable family of maps satisfying condition (C). We need first the following proposition which shows that for a given map there are a lot of maps such that is a symmetric Banach operator pair.

Proposition 21 (see [10]). Let be a self-map on a convex subset of a normed space and let be a map from to such that the set is invariant; that is, . Define Then is a symmetric Banach operator pair.

Theorem 22. Let be a nonempty closed bounded convex subset of a Banach space . Suppose that is a nonempty family of self-maps on satisfying condition (C). If is a Banach operator family and there exists a such that is compact and every (except ) in the family is nonexpansive, then has a common fixed point in .

Proof. Let and let be the set of all nonempty closed bounded convex subsets of such that , and is compact. On the set , define a partial order by set inclusion; then we can find a minimal set .
As in the proof of Theorem 17, and have a nonempty compact common fixed point set in satisfying and . Since and are Banach operator pairs, we have . Using Zorn's lemma, there exists a minimal nonempty compact subset of which satisfies , and . Using an argument similar to that in Theorem 17, we can show that . If reduces to a point, then . If has at least two different points, then, by Lemma 16, this contradicts the minimality of . Therefore we obtain that is a singleton and .

For any finite maps , we have by induction that . We now let . Thus for any is a nonempty compact set, and for each , we have This implies that the set family has the finite intersect property. Thus, Therefore the family has a common fixed point in .

4. Applications

Let be a subset of the normed space and ; then the distance of a point to the subset is defined by The set of best approximants of a point in is denoted by and defined by It is well known that is always a bounded subset of and is a closed and convex set if is so. Also, if is compact, then is nonempty. For more details, we refer to [17].

Let denote the class of closed convex subsets of containing . For and , let It is clear that .

The following result provides a partial solution of an existence problem of approximation theory in the following result (see also [14]).

Theorem 23. Let be a Banach space and let be a self-map of with such that . Assume that satisfies condition (C) on and is compact. Then the set of best approximations is nonempty.

Proof. Without loss of generality we may assume that . If , then As a result Since is compact, we can find such that and so by Lemma 14, for all . Hence and thus .

The following is an application of Theorem 17 to invariant approximations for convex sets.

Theorem 24. Let be a Banach space, and self-maps of with , and with and . If is a Banach operator pair on , both and are maps satisfying condition (C) on is nonexpansive, and is compact, then .

Proof. By Theorem 23, is a nonempty. Since is closed and convex, then is a closed convex set. We now show that is invariant. Let . Then dist. Since satisfies condition (C) on , by Lemma 14, we obtain that and so This implies that . Consequently, we have , and, similarly, we can prove that . Since and is compact, we have that is compact. Now, Theorem 17 guarantees that .

Acknowledgments

The authors are grateful to the referees for their valuable suggestions. This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The first and third authors acknowledge with thanks DSR for financial support. The second author is grateful to Chiang Mai University for the financial support.

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