Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 237858, 8 pages

http://dx.doi.org/10.1155/2013/237858

## Generalized Virtually Stable Maps and Their Associated Sequences

^{1}Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand^{2}Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok 10400, Thailand

Received 18 April 2013; Accepted 8 June 2013

Academic Editor: Janusz Matkowski

Copyright © 2013 P. Chaoha 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 extend the concept of virtual stability of continuous self-maps to arbitrary selfmaps and investigate the structure of sequences associated with uniformly virtually stable selfmaps. We also obtain a necessary and sufficient condition for a uniformly virtually stable selfmap to have the largest possible associated sequence. Examples of a uniformly virtually stable selfmap having the prescribed largest sequence and a uniformly virtually stable selfmap having no largest sequence are given.

#### 1. Introduction

Virtual stability of a self-map was first introduced in [1], where it was proved to unify various (continuous) types of nonexpansiveness in fixed-point theory. The main feature of a virtually stable self-map is that its fixed-point set is always a retract of its convergence set, and this fact immediately allows us to connect topological structures (e.g., connectedness and contractibility) of the fixed-point set to those of the convergence set. Although the original definition of virtual stability requires the continuity of a self-map, it seems that the connection between the fixed-point set and the convergence set also remains valid to some degree in discontinuous settings as hinted in [2, Corollary 2.6] and the work on mean-type mappings in [3–5] (see Example 4 below). A careful investigation of this fact will definitely result in a more powerful notion of virtual stability that enables us to study fixed-point sets of interesting discontinuous self-maps (e.g., Suzuki generalized nonexpansive maps [6]). Moreover, since all well-known self-maps in metric fixed-point theory are always uniformly virtually stable with respect to the same sequence of all natural numbers, it is quite natural to ask whether this also holds in general for a uniformly virtually stable self-map. It turns out that the answer to this simple question is highly nontrivial and interestingly involves algebraic properties of a certain object. Therefore, the purpose of this work is to revise the concept of virtual stability as well as to take care of the previos question. We will first redefine the notion of virtual stability to include discontinuous self-maps and investigate some properties of these newly defined virtually stable self-maps and prove the expected connection between the convergence set and the fixed-point set. Then we will show that the sequence associated with a uniformly virtually stable self-map can be made, in some sense, largest possible provided that the set of all associated sequences has a certain structure. We will also see that when the largest sequence exists, it may not be unless the map is continuous. For the sake of completeness, since the existence of the largest possible sequence associated with a given self-map cannot be guaranteed in general, we will provide two constructive examples of uniformly virtually stable self-maps: one having the prescribed largest sequence and the other having no largest sequence.

#### 2. Virtual Stability without Continuity

Let be a (nonempty) Hausdorff space and a self-map (that may not be continuous). We will also let and denote the sets of (open) neighborhoods of in and in , respectively. Recall that the fixed-point set and the convergence set of are defined, respectively, to be and where denotes the -th iterate of . We also define the map by for all . Notice that may not be continuous, and since we do not assume the continuity of , we only have . However, in this work, we will always assume that This condition is quite natural in the sense that it is automatically satisfied whenever is continuous with , and it is also required for the iterative sequence to be a scheme according to [2, Definition 2.1]. With the above assumption in mind, we are able to previous define the concept of virtual stability just as in [1, Definition 2.1].

*Definition 1. *A fixed point of is said to be *virtually **-stable* if for each , there exist together with an increasing sequence of positive integers such that for all . We simply call *virtually stable* if every fixed point of is virtually stable. Moreover, we will call a fixed point of *uniformly virtually **-stable* with respect to an increasing sequence of positive integers if for each , there exists such that for all . When every fixed point of is uniformly virtually -stable with respect to the same sequence, we will simply call *uniformly virtually stable*.

Clearly, a uniformly virtually stable self-map is always virtually stable, and a continuously (uniformly) virtually stable self-map as defined in [1] is also (uniformly) virtually stable according to our new definition. Moreover, if is uniformly virtually stable with respect to , it is immediately uniformly virtually stable with respect to any subsequence of .

Let us recall that, when is a metric space, a self-map is called (1)*virtually nonexpansive* [7] if is continuous and the family of all iterates of is equicontinuous on (equivalently, on ), (2)*uniformly quasi-Lipschitzian* if there is such that
for all .

Notice that a virtually nonexpansive self-map is always continuous and uniformly virtually stable with respect to the sequence of all natural numbers, while a uniformly quasi-lipschitzian self-map may not be continuous in general.

Proposition 2. *Suppose is a self-map on a metric space . *(1) *If ** is continuous and uniformly virtually stable with respect to **, then it is virtually nonexpansive*. (2) *If ** is uniformly quasi-lipschitzian, then it is uniformly virtually stable with respect to **.*

*Proof. *Suppose is a self-map on a metric space . (1) Let and . Then there is such that for all . Thus, for whose , we have for all . This implies that is equicontinuous on and hence is virtually nonexpansive. (2) Suppose there is such that for all . It follows that for all , and . Thus, is uniformly virtually stable with respect to .

It is not difficult to see that the class of uniformly quasi-lipschitzian self-maps includes various important (possibly discontinuous) self-maps in metric fixed-point theory such as nonexpansive maps, Kannan maps [8], Suzuki generalized nonexpansive maps [6], quasi-nonexpansive maps, and even asymptotically quasi-nonexpansive maps. Hence, the previous proposition immediately implies that those maps are all uniformly virtually stable with respect to . The following theorem directly extends [1, Theorem 2.6] to include discontinuous maps.

Theorem 3. *Suppose that is a regular space. If is virtually stable and is continuous for some , then is continuous and hence is a retract of .*

*Proof. *Let and . Since is regular, so is and then there is such that . By virtual stability of , for all , for some sequence and . Also there is, by the fact that , such that for all . Then and for each ,
Thus, is continuous and hence is a retract of .

*Example 4. *Let and , and let be an interval. Consider the subspace of equipped with the maximum norm . Since the maximum norm on also induces the usual topology, is a regular space. Recall that a self-map is called *mean-type* if , where each is a *mean* on ; that is, for each ,
Notice that may not be continuous in general. However, if satisfies
for all , and , it is always quasi-nonexpansive (with respect to the maximum norm), and hence uniformly virtually stable (with respect to the sequence ). For if and , one has

In particular, the mean-type mapping defined by , satisfies , and hence it is uniformly virtually stable. Since is also continuous, is a retract of by the previous theorem. In fact, it follows from [4] that , , and for . Therefore, is clearly a retract of .

*Example 5. *Consider that equipped with the maximum norm, and a discontinuous self-map given by
It is straightforward to verify that is quasi-nonexpansive (hence, uniformly virtually stable) and is continuous. Therefore, by the previous theorem, is continuous and hence is a retract of . Notice also that is not convex.

The following examples show that if we drop the continuity assumption of some iterate in Theorem 3, the map may or may not be continuous.

*Example 6. *Let be defined by
Observe that is uniformly virtually stable and is not continuous for all . Moreover, and . Hence is not continuous.

*Example 7. *Consider that equipped with the supremum norm. Define by
It is straightforward to verify that is quasi-nonexpansive (hence, uniformly virtually stable) and is not continuous for all . However, is continuous because for all .

The next theorem gives a condition allowing us to enlarge the sequence associated with a given uniformly virtually stable self-map. As a result, we immediately obtain a refinement and a generalization of [1, Theorem 2.17].

*Definition 8. *An increasing sequence of natural numbers is said to satisfy the *sup-finite condition* if .

Theorem 9. *If, for some , is continuous on and is uniformly virtually stable with respect to where satisfies the sup-finite condition, then is uniformly virtually stable with respect to .*

*Proof. *Let , , and . Because is uniformly virtually stable with respect to , there is such that for all . By continuity at of , , there exists for which for all . Now let . If , , whereas if , there is such that for some and so

Corollary 10. *Suppose that is a metric space. If is continuous and uniformly virtually stable with respect to , where satisfies the sup-finite condition, then is virtually nonexpansive. In particular, is a -set when is complete.*

*Proof. *By setting in the previous theorem, is uniformly virtually stable with respect to and hence virtually nonexpansive by Proposition 2 (1). For, if is complete, by [9, Theorem 1.2], is then a -set.

Corollary 11. *Suppose that is a complete metric space. If, for some , is uniformly virtually stable with respect to and is continuous, then is a -set.*

*Proof. *Let . Then is uniformly virtually stable with respect to . Thus, is a -set by the previous corollary.

#### 3. Sequences Associated with Uniformly Virtually Stable Self-Maps

In view of Proposition 2 (2) and Theorem 9, the sequence associated with a given uniformly virtually stable self-map seems to have an implicit structure. In fact, some questions may naturally arise. (i)Can we replace with the larger (in some sense) sequence so that is still uniformly virtually stable with respect to ? (ii)If can be enlarged to , can we make largest possible? and how does it look like? (iii)Does the largest possible sequence satisfy the sup-finite condition? (iv)Is the largest possible sequence always ?

As we will see later on in this section, it turns out that the answers to these questions are interestingly nontrivial and require some careful considerations in both continuous and discontinuous settings.

Throughout this section, is a Hausdorff space and is a self-map of . Every sequence is assumed to be an increasing sequence of natural numbers, so it is uniquely determined by its image. We then always identify a sequence with the infinite set , and with this identification in mind, the following notations become natural: (i) if is a subsequence of ; (ii) represents the sequence whose image is ; (iii) represents the sequence whose image is ; (iv) represents the sequence whose image is , and so represents the sequence whose image is ; (v) represents the sequence whose image is , and .

For a nonempty subset of , the natural number satisfying (1) for all and (2)if for all , then

will be called *the greatest common divisor* of and denoted by . Also, we let denote the set of all finite sums of elements in ; that is,
Notice that and if , we have . As usual, will represent , and will represent at the same time both the set and the sequence whose image is . We may simply write for .

Moreover, for a uniformly virtually stable self-map , we let
Clearly, is partially ordered by , and a maximal element in will be called a *maximal sequence associated with *. The following are some basic properties of .

Proposition 12. *Suppose is uniformly virtually stable with respect to . one has the following: *(1)*If **, then **. *(2)*If **, then **. *(3)* for any **. *(4)* for any **.*

*Proof. *Let and .(1)It is straightforward from the definition of virtual stability.(2)There exist such that and for all . Hence for all where .(3)By induction, it suffices to show that . There are, by virtual stability of , such that and for all . This implies that for all .(4)By letting , we have for some , and hence, and by (1) and (2).

*Remark 13. *From (2) in the previous proposition, if and are maximal elements in , then . This implies that a maximal sequence associated with , if exists, is always unique.

Before considering the existence, we will first investigate the structure of the maximal sequence associated with a given self-map.

Lemma 14. *Suppose is uniformly virtually stable. For a sequence , one has *(1)* for some , *(2)*there are and such that for all , *(3)*if and satisfies the sup-finite condition, then . *

*Proof. *(1) The fact that follows directly from the definition. Now, let . Then, for some . If for some , then , a contradiction. Also, for any satisfying for all , we have for all , and hence, . This implies .

(2) If , the proof is clear. So, we assume that . By (1), for some . Then, for some partition of , where . Set and . Notice that because , and for all . Then for each and ,
Since for , we have for some and ,
Therefore, the result follows by letting .

(3) Assume that and . By (2), there is for which for all . By Proposition 12, we have
and hence . The proof will immediately follow by Proposition 12 (1) once we can show that . To see this, let . The case of is clear. Otherwise, for some . Since , we have for some . This implies that .

Theorem 15. *Suppose is uniformly virtually stable and is the maximal sequence associated with . One has*(1)*, *(2)* for some finite subset of , *(3)*if is continuous, then . *

*Proof. *(1) It suffices to show that . Let . By Proposition 12, , and hence, by maximality of .

(2) From (1), we have . Then by Lemma 14 (2), there exist such that for all . By setting , we clearly have . On the other hand, for each , notice that (i)if , then , and (ii)if , then for some .

Therefore, .

(3) Suppose that is continuous. It follows that , and by maximality of , . Therefore, by (1), we have .

The next theorem gives a necessary and sufficient condition for the existence of the maximal sequence.

Theorem 16. *Suppose is uniformly virtually stable. Then the maximal sequence associated with exists if and only if there is satisfying the sup-finite condition.*

*Proof. * If is maximal, then and hence
Assume that for some . By Lemma 14 (3), . If , we are done. Suppose that . For , let
and if , otherwise . Then . By Proposition 12 (2), and again by Lemma 14 (3), since where , . Notice that is maximal because whenever for some , there is such that .

Towards the end of this section, we will construct some interesting examples including a uniformly virtually stable self-map whose maximal sequence is prescribed as well as a uniformly virtually stable self-map having no maximal sequence.

*Example 17. *For , let . Then for all . Consider the sequence . Set and for any . Define by
Then, can be illustrated in Figure 1 and it is uniformly virtually stable with respect to the maximal sequence .

To see this, note that , for all , and for each and ,
To show the virtual stability of with respect to , it suffices to prove that for all . Let . For each ,
for some , and . Thus, we get the claim. Moreover is maximal because whenever , either or for some (and so for some ) holds, which the first case implies that for all , while the latter implies that
for all ; that is, . Thus for all , which induces the maximality of as desired.

Lemma 18. *Let and for ,
**
Then forms a partition of , and is infinite for each .*

*Proof. *Following from the fact that for all , if , then for all . Also for each , since , we have for some . Hence, forms a partition of . For the next implication, notice that is clearly infinite. Now, assume that . We first claim that . Since , we have for some . It follows that and for if , then and , and hence . Next, we claim that for all . Let . By the previous claim, we have . Hence, for some ; that is, . Since , we must have for some . Thus, as claimed. Finally, since , then is infinite as desired.

*Example 19. *For each , let be defined as in the previous lemma, and for each , let be the smallest prime number such that . For examples, and . Clearly, whenever . Now, let be a subspace of equipped with the maximum norm, and let denote the open ball .

The goal of this example is to find a uniformly virtually stable self-map having no maximal sequence. This can be done by constructing a uniformly virtually stable self-map with respect to but not . For if is the maximal sequence associated with such an , we must have and hence , but this contradicts the property that is not uniformly virtually stable with respect to .

To obtain the desired self-map, it suffices to require the map to satisfy the following three conditions: (C1) ,(C2) for each , (C3) For each , there exists such that .

Notice that (C1) and (C2) imply the virtual stability of with respect to , while (C1) and (C3) imply that, for each , there exists such that , and hence is not uniformly virtually stable with respect to .

We are now ready to give the explicit description of such an . Consider the following subsets of : , and for some , and for some , and either or for some .

Observe that (O1) whenever . All cases are trivial except for the case of , where it follows from the fact that forms a partition of ;(O2) for all , because for each with , we have and hence .

Define by

Clearly, is well-defined by (O1). The following properties are also satisfied.(P1), and . So, is -invariant. Moreover, since ’s are all distinct primes, we have if and only if , and hence is injective, where denotes the second-coordinate projection . (P2)For each , one can verify that In particular, we have the following. (i) since and . (ii), where , since and . (P3)For each with and , we have , , by (P2), and for some . Again by (P2), it follows that

Clearly, (C1) is implied by (P1). To prove (C2), let , and it suffices to consider the following two cases.(1)If , then , and hence (2)If , then for some , and . If , we have , a contradiction. Thus , and by (P3), there is some such that From above cases, it follows that .

Finally, for (C3), let and . Then by (P2), we have .

*Remark 20. *From the previous example, we can easily check that is uniformly virtually stable with respect to for any , just by showing that for any . Hence, Proposition 12 (2) does not hold in general for an arbitrary union of sequences in .

#### Acknowledgments

The first author is (partially) supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The third author wishes to thank the H. M. King Bhumibol Adulyadej’s 72nd Birthday Anniversary Scholarship. The authors are grateful to Professor Pimpen Vejjajiva and the anonymous referee(s) for their valuable comments and suggestions for improving this paper.

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