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Common fixed-point results in uniformly convex Banach spaces
Fixed Point Theory and Applications volume 2012, Article number: 171 (2012)
We introduce a condition on mappings, namely condition . In a uniformly convex Banach space, the condition is weaker than quasi-nonexpansiveness and weaker than asymptotic nonexpansiveness. We also present the existence theorem of common fixed points for a commuting pair consisting of a mapping satisfying condition and a multivalued mapping satisfying conditions and for some .
In 1978, Itoh and Takahashi  established the existence of common fixed points of a quasi-nonexpansive mapping and a multivalued nonexpansive mapping by an elementary constructive method in a Hilbert space. In 2006, Dhompongsa et al.  obtained a common fixed point result for a commuting pair of single-valued and multivalued nonexpansive mappings in uniformly convex Banach spaces. The analogy result in CAT(0) spaces was also proved by Dhompongsa et al. . Since then, Shahzad and Markin  studied an invariant approximation problem and provided sufficient conditions for the existence of such that and , where , t and T are commuting nonexpansive mappings on E. In 2009, Shahzad  also obtained a common fixed point and invariant approximation result in a CAT(0) space in which t and T are weakly commuting.
Motivated by Suzuki’s result , Garcia-Falset et al.  introduced two kinds of generalizations for condition , namely conditions and and studied both the existence of fixed points and their asymptotic behavior. Recently, Abkar and Eslamian  proved that if E is a nonempty closed convex and bounded subset of a complete CAT(0) space X, is a single-valued quasi-nonexpansive mapping, and is a multivalued mapping satisfying conditions and for some such that t and T are weakly commuting, then there exists a point such that . This result was extended to the general setting of uniformly convex metric spaces by Laowang and Panyanak .
In this paper, we first introduce the following condition.
Definition 1.1 Let t be a mapping on a subset E of a Banach space X. Then t is said to satisfy condition if
the fixed point set is nonempty closed and convex, and
for every , a closed convex subset A with , and such that , we have .
We show that, in a uniformly convex Banach space, condition is weaker than quasi-nonexpansiveness and weaker than asymptotic nonexpansiveness. We also present the existence theorem of common fixed points for a commuting pair consisting of a mapping satisfying condition and a multivalued mapping satisfying conditions and for some . Consequently, such a theorem extends many results in the literature.
In this section, we give some preliminaries.
A mapping t on a subset E of a Banach space X is called an asymptotically nonexpansive mapping if for each , there exists a positive constant with such that
If for all , then t is called a nonexpansive mapping. We denote by the set of fixed points of t, i.e., .
We shall denote by the family of nonempty bounded closed subsets of E and by the family of nonempty compact convex subsets of E. Let be the Hausdorff distance on , i.e.,
where is the distance from the point a to the subset B.
A multivalued mapping is said to be nonexpansive if
Definition 2.1 A multivalued mapping is said to satisfy condition provided that
We say that T satisfies condition whenever T satisfies for some .
Definition 2.2 A multivalued mapping is said to satisfy condition for some provided that
A point x is called a fixed point for a multivalued mapping T if . A single valued mapping and a multivalued mapping are said to be commute if for all .
A Banach space X is said to be strictly convex if
for all with and . We recall that a Banach space X is called uniformly convex (Clarkson ) if, for each , there is a such that if then
It is obvious that uniform convexity implies strict convexity.
In 1991, Xu  proved the characterization of uniform convexity as follows.
Theorem 2.3 
A Banach space X is uniformly convex if and only if for each fixed number , there exists a continuous function , , such that
for all and all such that and .
Let E be a nonempty closed and convex subset of a Banach space X and be a bounded sequence in X. For , define the asymptotic radius of at x as the number
The number r and the set A are, respectively, called the asymptotic radius and asymptotic center of relative to E. It is known that is as nonempty, weakly compact and convex as E is . The sequence is called regular relative to E if for each subsequence of .
Lemma 2.4 Let be a bounded sequence in X, and let E be a nonempty closed convex subset of X. Then has a subsequence which is regular relative to E.
The following result was proved by Goebel and Kirk .
Lemma 2.5 Let and be bounded sequences in a Banach space X, and let . If, for every natural number n, we have and , then .
3 Main results
We recall that a mapping t on a subset E of a Banach space X is called quasi-nonexpansive  if for all and . From the definition, we can see that nonexpansive mappings with a fixed point are quasi-nonexpansive.
We first show that a quasi-nonexpansive mapping defined on a strictly convex Banach space X satisfies condition .
Proposition 3.1 Let E be a strictly convex subset of a Banach space. If is a quasi-nonexpansive mapping, then t satisfies condition .
Proof It is known that is nonempty closed and convex [, Theorem 1]. Let and A be a closed convex subset with . Let be such that . Quasi-nonexpansiveness of t implies that
Since E is strictly convex and , it must be the case that . Therefore, t satisfies condition . □
An asymptotically nonexpansive mapping defined on a uniformly convex Banach space also satisfies condition .
Proposition 3.2 Let X be a uniformly convex Banach space and E be a nonempty subset of X. If is an asymptotically nonexpansive mapping with , then t satisfies condition .
Proof The set is closed and convex [, Theorem 2]. Let , A be a closed convex subset of E with , and be such that . By Theorem 2.3, there exists a continuous function ψ such that for all integers ,
Since and A is convex, we have
Since t is asymptotically nonexpansive, the right-hand side of the inequality tends to zero as l, m tend to infinity. Hence, is a Cauchy sequence. Let . We have
By letting , we can conclude that , that is . Now, letting in (3.1) yields
Since and , thus . Therefore, . □
The following example shows that the class of mappings satisfying condition is strictly wider than the class of quasi-nonexpansive mappings and asymptotically nonexpansive mappings.
Example 3.3 Let f be a function on defined by
Then f is neither quasi-nonexpansive nor asymptotically nonexpansive. However, f satisfies condition .
We are now in a position to state our main theorem.
Theorem 3.4 Let E be a nonempty bounded closed convex subset of a uniformly convex Banach space X. Let be a mapping satisfying condition , and let be a multivalued mapping satisfying conditions and for some . If t and T are commute, then they have a common fixed point, that is, there exists a point such that .
Proof Commutative property of t and T implies that for all . Then we have for all since t satisfies condition .
Now, we find an approximate fixed point sequence in for T. Take . Since , we choose . Define
Since is convex, we have . Let such that . We get since t satisfies condition . Put
Again, choose such that . Similarly, we get . We have a sequence such that
where and . For every natural number , we have
It follows that
Since T satisfies condition (), we have
Hence, for each , we have
We now apply Lemma 2.5 to conclude that . That is, as ,
Passing through a subsequence, if necessary, we can assume that is regular. Let . For each , we choose such that . Since t satisfies condition , we have . The compactness of implies that the sequence has a convergent subsequence with the limit point . We also obtain since is closed. By the condition of T, we have for some ,
Since is regular, and an asymptotic center of a bounded sequence in a uniformly convex Banach space is a singleton set, these show that . Hence, . □
As a consequence of Proposition 3.1, Proposition 3.2, and Theorem 3.4, we obtain the following corollaries.
Corollary 3.5 Let E be a nonempty bounded closed convex subset of a uniformly convex Banach space X. Let be a quasi-nonexpansive mapping, and let be a multivalued mapping satisfying conditions and for some . If t and T are commute, then they have a common fixed point, that is, there exists a point such that .
Corollary 3.6 Let E be a nonempty bounded closed convex subset of a uniformly convex Banach space X. Let be an asymptotically nonexpansive mapping, and let be a multivalued mapping satisfying conditions and for some . If t and T are commute, then they have a common fixed point, that is, there exists a point such that .
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The first author would like to thank the Office of the Higher Education Commission, Thailand, for supporting grant fund under the program of Strategic Scholarships for Frontier Research Network for the PhD Program, Thai Doctoral degree, for this research.
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Akkasriworn, N., Kaewkhao, A., Keawkhao, A. et al. Common fixed-point results in uniformly convex Banach spaces. Fixed Point Theory Appl 2012, 171 (2012). https://doi.org/10.1186/1687-1812-2012-171
- common fixed point
- quasi-nonexpansive mapping
- generalized nonexpansive mapping
- uniformly convex Banach space