Abstract and Applied Analysis

Volume 2011 (2011), Article ID 435686, 15 pages

http://dx.doi.org/10.1155/2011/435686

## The Fixed Point Theory for Some Generalized Nonexpansive Mappings

^{1}Departamento de Análisis Matemático, Facultad de Matemáticas, Universitat de Valencia, 46100 Burjassot, Valéncia, Spain^{2}Departamento de Matemáticas, Ciencias Naturales y Ciencias Sociales Aplicadas a la Educación, Universidad Católica de Valencia San Vicente Mártir, 46100 Godella, Valéncia, Spain

Received 30 September 2010; Accepted 18 January 2011

Academic Editor: Martin D. Schechter

Copyright © 2011 Enrique Llorens Fuster and Elena Moreno Gálvez. 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 study some aspects of the fixed point theory for a class of generalized nonexpansive mappings, which among others contain the class of generalized nonexpansive mappings recently defined by Suzuki in 2008.

#### 1. Introduction

Nonexpansive mappings are those which have Lipschitz’s constant equal to 1. A Banach space is said to have the fixed point property for nonexpansive mappings (FPP in short) provided that every nonexpansive self-mapping of every nonempty, closed, convex, bounded subset of has a fixed point.

Since 1965 considerable effort has been aimed to study the fixed point theory for nonexpansive mappings in the setting of both reflexive and nonreflexive Banach spaces. It turns out that property (FPP) closely depends upon geometric characteristics of the Banach space under consideration. Even when is a weakly compact convex subset of , a nonexpansive self-mapping of needs not have fixed points. Nevertheless, if the norm of has suitable geometric properties (as e.g., uniform convexity, among many others), every nonexpansive self-mapping of every weakly compact convex subset of has a fixed point. In this case, is said to have the weak fixed point property, WFPP in short.

Although nonexpansive mappings are perhaps one of the most important topic in the so-called metric fixed point theory, one can find in the literature considerable amount of research about more general classes of mappings than the nonexpansive ones. Among many others, mappings such that for every ,
where are nonnegative constants with were studied since the late sixties of the last century. In many instances (see [1, 2]), this class of mappings are called *generalized nonexpansive* mappings. Despite its perhaps questionable usefulness, this type of condition appears to be quite natural from a geometric point of view. Of course, one easily verifies that the above condition (1.1) is, equivalent to the existence of nonnegative constants , , with such that for all
Particular cases of mappings satisfying condition (1.1) have been studied by various authors independently [2–5].

On the other hand, in a recent paper, Suzuki [6] defined a class of generalized nonexpansive mappings as follows.

Let be a nonempty subset of a Banach space . We say that a mapping satisfies * condition * on if for all ,

Of course, every nonexpansive mapping satisfies condition on , but in [6] some examples are given of noncontinuous mappings satisfying condition .

The aim of this paper is to study a class of mappings which properly contains those satisfying either condition (1.3) or (1.2) in many cases. We will show that one of the most important geometric conditions which implies the (WFPP) for nonexpansive mappings, namely the so-called normal structure, also allows us to derive fixed point results for our class of mappings. In particular, for mappings satisfying Suzuki's condition , this result is more general than the ones included in [6, 7].

#### 2. Preliminaries

We will assume throughout this paper that is a Banach space and is a nonempty, closed, convex, bounded subset of . For a given mapping , the (possibly empty) set of all fixed points of will be denoted by . In the same way, a sequence in is called an *almost fixed point sequence for * (a.f.p.s. in short) provided that . It is well known that every nonexpansive mapping has a.f.p. sequences. The same holds if satisfies Suzuki's condition on , (see [6, Lemma 6]).

We now recall further concepts which will be useful in the forthcoming sections.

We begin with some classes of mappings. Definitions (1) and (2) are given in [8], and the first one is a generalization of condition given by Suzuki in [6]. (1)For , we say that a mapping satisfies *condition * on if for all with one has that . Of course, the original Suzuki condition is just . (2)For , a mapping is said to satisfy *condition ** on * if, for all ,
We say that satisfies condition on if satisfies on for some . In [6] is shown that if a mapping satisfies Suzuki's condition , then it satisfies condition .(3)A mapping is said to be *quasi-nonexpansive on * provided that it has at least one fixed point and for every , and for all ,
This concept is essentially due to Díaz and Metcalf [9] and Dotson [10]. The mapping defined by
is quasi-nonexpansive on but not nonexpansive. (4)Finally, a mapping is said to be *asymptotically regular* on if, for each , it is the case that .

Let be a bounded sequence on . One defines the *asymptotic radius of ** at * as the number

In the same way, the *asymptotic radius of ** in * is the number

and the *asymptotic center* of in as the (possibly empty) set

It is well known, (e.g., see [11]), that whenever is weakly compact and that if is convex, then is convex.

Finally, we recall also some geometric properties of normed spaces that will appear in the remainder of this paper. (1)A normed space is said to satisfy the *Opial condition* if for any sequence in such that it happens that , ,
It can be readily established, on the extraction of appropriate subsequences, that the lower limits can be replaced with upper limits in the above definition. (2)A Banach space is said to have *normal structure* if for each bounded, convex, subset of with diam there exists a *nondiametral* point , that is a point such that
This property was introduced in 1948 by Brodskii and Milman. Since 1965, it has been widely studied due to its relevance in fixed point theory for nonexpansive mappings. For more information see, for instance [11].

#### 3. Condition

Next we introduce a class of nonlinear mappings.

*Definition 3.1. *A mapping satisfies condition , (or it is an -type mapping), on provided that it fulfills the following two conditions. (1)If a set is nonempty, closed, convex and -invariant, (i.e., ), then there exists an a.f.p.s. for in . (2)For any a.f.p.s. of in and each

From now on, if not specified, a mapping is said to satisfy condition , whenever it satisfies it on its domain.

Assumption (1) of this definition is automatically satisfied by several classes of nonlinear mappings. For instance, it is a well-known property of nonexpansive mappings. Thus, if a mapping is nonexpansive with respect to any equivalent renorming of , then satisfies (1). Asymptotically regular mappings automatically satisfy (1) too.

We point out that if satisfies condition and the set is nonempty, then is quasi-nonexpansive. Indeed, if , then the sequence with is, of course, an a.f.p.s. for , and from assumption (2),

This means, among other things, that we have some information about the set when this set is nonempty and satisfies condition . Indeed, according to Theorem 1 of Dotson's paper [10], if is strictly convex, then is closed and convex, and is continuous on .

However, quasi-nonexpansive mappings are not relevant concerning the existence of fixed points (because this existence of fixed points is assumed by definition). Nevertheless, as we will see below, there are fixed point free mappings satisfying condition and, moreover it is possible to give some fixed point results for such -type mappings.

Next, we will see that there are quasi-nonexpansive mappings which fail to be -type mappings.

*Example 3.2. *Let , and consider the compact convex set . Let be the mapping given by
It is straightforward to check that
This set is nonconvex, but this does not contradict the above-mentioned Dotson result, because is not strictly convex.

First, we will see that is quasi-nonexpansive. For and ,
On the other hand, if and ,
Thus, for every , and , , that is, is quasi-nonexpansive.

Moreover, taking and , it is easy to check that is not nonexpansive on .

Finally, let us see that fails to satisfy condition on . Define the sequence
Of course , and then is an a.f.p.s. for on . But, for , one has
while

Next, we will give some examples of mappings satisfying condition . Let be a mapping.

Proposition 3.3. *If is nonexpansive, then it satisfies condition .*

*Proof. *Let be a nonexpansive mapping. It is well-known that if is a closed convex -invariant subset of , then has a.f.p. sequences in . Moreover, since is nonexpansive, for every a.f.p.s. for and every ,
Then, satisfies condition .

As a direct consequence, the well known examples of fixed point free nonexpansive self-mappings of weakly compact convex subsets, are instances of -type fixed point free self-mappings of such class of sets.

Proposition 3.4. *If satisfies Suzuki’s condition , then it satisfies condition .*

*Proof. *Recall that if is a type mapping, and is a closed, convex, -invariant subset of , then there exist a.f.p. sequences for in (see [6, Lemma 6]). Moreover, in [6, Lemma 7], it was shown that, for every ,
Hence, if is an a.f.p.s. for and ,
Hence, such mappings satisfy condition .

The above proof can be easily adapted for mappings which satisfy condition .

Proposition 3.5. *If satisfies condition for some . Then, satisfies condition provided that it satisfies assumption (1) of Definition 3.1.*

*Proof. *Replace 3 with in the above proof.

We will study further relationships between the class of mappings and those which satisfy condition in Theorem 4.7 in the next section.

In [6, 8] are given some examples of noncontinuous mappings satisfying conditions and .

Proposition 3.6. *Let be a generalized nonexpansive selfmap of . If any of the following conditions holds, then satisfies condition : *(1)*, *(2)* and , *(3)* and , *(4)* and , *(5)* and , which implies that .*

*Proof. *We first need to verify that there exist almost fixed point sequences for in any nonempty, closed, convex, and -invariant subset of . We need to split the proof in cases according to the above list.

(1) If , it is proved in [12, Theorem 4], that a generalized nonexpansive mapping with coefficients , , satisfying from a nonempty, closed, bounded, convex set into itself has a fixed point . Then, if such is -invariant, then the sequence given by , is obviously an a.f.p.s. for in .

(2) If and , in the proof of Theorem 1 of [5], it is seen that for each , the orbit is an a.f.p.s. Thus, has a.f.p. sequences on each invariant closed convex subset of .

(3) If and , in the proof of Theorem 1.1 in [13], it is shown that . Thus, has again a.f.p. sequences on each -invariant closed convex subset of .

(4) If and , then is asymptotically regular on , that is, any orbit of is an a.f.p.s. (see [1, pages 83–85]).

(5) If and , then and therefore, is nonexpansive. Thus, has a.f.p. sequences on each -invariant, closed, convex subset of .

Next, we will prove the second condition, that is, given an a.f.p.s. for on , for each the following inequality holds:
It is well known that (see the proof of Lemma 3.1 of [2]) if is a generalized nonexpansive mapping and , then
Let be an a.f.p.s. on and . Then,
Thus,
and this leads to

As , then and hence .

Thus,
Consequently,
as desired.

The inclusions which follow from Propositions 3.3, 3.6, and 3.4 are strict, as the following easy example shows.

*Example 3.7. *Let given by . Let us see that satisfies condition on , but it fails to be generalized nonexpansive and to satisfy Suzuki's condition .* satisfies condition ** on *.

Let us observe that if is an a.f.p.s. for , then since is continuous, for any convergent subsequence one has that . Thus, is convergent to 0.

The only invariant subsets of are the intervals with , and it is clear that in has a.f.p. sequences, namely all those which are convergent to 0.

On the other hand, if is an a.f.p.s. for , and , then* fails Suzuki's condition ** on **.*

Take and . One has
while
* fails to be generalized nonexpansive on **.*

Suppose for a contradiction that there exist positive constants ,, with such that for every ,
Then, if we take and , we obtain
which implies that
Therefore, which implies that and . But, in this case, would be nonexpansive, which is impossible because nonexpansive mappings satisfy Suzuki's condition.

*Remark 3.8. *If and (which implies that ), then the above proposition also holds whenever the space satisfies suitable geometrical conditions. (1)From [14], the following fact is well known: every weakly compact convex subset of has close-to-normal structure if and only if every map of the above type on a weakly compact convex subset of has a unique fixed point, and hence it has a.f.p. sequences in every -invariant, closed, convex subset of .(2)If has uniformly normal structure (see [15]), from [4, Theorem 1], we know that has a unique fixed point and hence it has a.f.p. sequences. The same conclusion is true if is Lipschitzian on , without the assumption of uniformly normal structure for (see [4, Theorem 4]).

*Remark 3.9. *Condition (1) and (2) in the definition of type mappings are independent as it is shown in the following two examples.

*Example 3.10 (see [16]). *Let be the standard orthonormal basis of and let
For each , let
where . Define the mapping by

In [16], it is shown that is fixed point free and asymptotically regular, that is, for every , , and hence any orbit of is an a.f.p.s. for . Therefore, if is a nonempty invariant subset of , then for every , is an a.f.p.s. in . Thus, satisfies condition (1) of Definition 3.1.

Consider now the orbit of the mapping which starts at the point . We claim that
Suppose that for a positive integer our claim holds. Then,
Since is asymptotically regular in , is an a.f.p.s. for . Considering this a.f.p.s. and the point , we will see that fails condition (2). Indeed,

*Example 3.11. *Let be the mapping given by . It is easy to see that the only closed convex, -invariant subsets of are and the intervals whenever . Since 0 is a fixed point for , then in , we have the (trivial) a.f.p.s. given by . In the same way, since 1 is a fixed point for , then in , we have the (trivial) a.f.p.s. given by . Thus, fulfills condition (1) of Definition 3.1.

On the other hand, for the sequence given by one has that, if ,
Thus, fails to satisfy condition (2) of the definition of -type mappings.

#### 4. Fixed Point Theorems

*Remark 4.1. *Let be a nonempty closed and convex subset of a Banach space , and a mapping which satisfies condition . If is an a.f.p.s. for , then, for every ,
that is, asymptotic centers of a.f.p. sequences are invariant under mappings satisfying condition .

Theorem 4.2. *Let be a nonempty compact convex subset of a Banach space and a mapping satisfying condition . Then, has a fixed point.*

*Proof. *Since is nonempty, closed, bounded and convex, and -invariant, there exists an a.f.p.s. for , say , in . Since is compact, there exists a subsequence of such that converges to some . By assumption (2) of Definition 3.1,
and by unicity of the limit, .

Corollary 4.3. *Suppose that the asymptotic center in of each sequence in is nonempty and compact. Then, has a fixed point.*

*Proof. *Since satisfies condition , there exists an a.f.p.s. for in , say . Let be the asymptotic center of relative to . By our assumption, is nonempty and compact and, by Remark 4.1, is -invariant. Then, from Theorem 4.2, has a fixed point on .

Notice that whenever the asymptotic center in of a bounded sequence consists of just one point, this point has to be a fixed point for mappings that leave asymptotic centers invariant, which is the case for the mappings satisfying condition studied here.

Some geometrical regularity conditions of the space or the set force the asymptotic centers to be “nice”. For example, it is known that in uniformly convex spaces asymptotic centers are singletons. Even more, it is also known that the asymptotic centers of bounded sequences wit respect to weakly compact convex sets are compact on -uniformly convex Banach spaces. (see [17, page 77].)

However, Banach spaces which satisfy either Opial condition or uniform convexity have normal structure and then they fall into the scope of the following theorem.

Let be weakly compact convex set and an arbitrary self-map of . The standard Zorn's Lemma argument gives that contains a closed, convex, -invariant, minimal subset .

Theorem 4.4. *Let be a Banach space with normal structure. Let be a nonempty, weakly compact and convex subset of . Let be a mapping satisfying condition . Then, has a fixed point.*

*Proof. *Let be a minimal subset of . Since satisfies condition , there exists an a.f.p.s. for in . This sequence is either constant, and hence it consists of a fixed point of , or it is nonconstant. In this case, since has normal structure, from Corollary 1 of [5], the real function given by
is not constant in . Then, takes at least two values. If is an intermediate value, then the set
is nonempty, convex, and closed and . From condition , is also -invariant which contradicts the minimality of .

*Remark 4.5. *In the above proof, we have shown that if is a minimal subset of , then the functions are constant on . It is unclear if this constant value coincides with the diameter of , as in the nonexpansive case. In the affirmative, we would obtain a Karlovitz-like result for type mappings.

Theorem 4.6. *Let be a Banach space which satisfies the Opial condition. Let be a mapping satisfying condition . Then, if is an a.f.p.s. for such that it converges weakly to , then is a fixed point of .*

*Proof. *Since is an a.f.p.s. for , and satisfies condition ,
Given that , if , from the Opial condition, we obtain
which is a contradiction.

We finish with a result which establishes an alternative for mappings satisfying some condition .

Theorem 4.7. *Let be a closed, convex, bounded subset of a Banach space and let be a continuous mapping satisfying condition on for some . Then, at least one of the following statements is true: *(1)* has a fixed point, *(2)* satisfies condition .*

*Proof. *It is shown in [8] that mappings satisfying some condition have a.f.p. sequences on each closed bounded convex -invariant subset of . Let be an a.f.p.s. for in .

If for some , there is a subsequence of converging to , since is an a.f.p.s., then the sequence has the same limit as , and therefore by the continuity of , , and statement 1 holds.

Suppose now that for every , the sequence does not have any subsequence converging to . We claim that there exists some such that, for all
Otherwise, for every , there would exist such that
Then, since does not have any subsequence converging to ,
a contradiction which proves our claim.

From (4.7), bearing in mind that satisfies condition , we have
and then
that is, satisfies condition .

For continuous mappings, a more general result than Theorem 8 of [8] can be obtained by combining the above result with Theorem 4.4.

Corollary 4.8. *Let be a Banach space with normal structure and a convex weakly compact subset of . Let be a continuous mapping satisfying some condition on . Then, has a fixed point.*

*Proof. *If does not have a fixed point in , then from the above theorem, it satisfies condition . Consequently, from Theorem 4.4, we get a contradiction.

#### Acknowledgments

E. Moreno Gálvez was partially supported by the Oficina de Transferencia de Resultados de Investigación of the Universidad Católica de Valencia (Spain). E. Llorens Fuster was partially supported by the Ministerio de Educación y Ciencia of Spain (Grant no. MTM2009-10696-C02-02).

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