Sokhuma and Kaewkhao (2011) introduced an iteration scheme to compute a common fixed point of a single-valued nonexpansive mapping and a multivalued nonexpansive mapping on a uniformly convex Banach space. In this paper, we extend the above result of Sokhuma and Kaewkhao from a single-valued mapping to a countable number of mappings and, at the same time, we extend the underlying spaces to strictly convex Banach spaces. The corresponding results are also obtained for the space setting.
MSC: 47H09, 47H10.
Keywords:common fixed point; nonexpansive mapping; strictly convex Banach space; space
Let X be a complete metric space, and E a nonempty subset of X. We will denote by the family of nonempty subsets of E and by the family of nonempty bounded closed subsets of E. Let be theHausdorff distance on , that is,
respectively. If , we call x a fixed point of a single-valued mapping t. Moreover, if , we call x a fixed point of a multivalued mapping T. We use the notation to stand for the set of fixed points of a mapping S. Thus is the set of common fixed points of t and T, i.e., if and only if .
Following , a bounded closed and convex subset E of a Banach space X has the fixed point property for nonexpansive mappings (FPP) (respectively, for multivalued nonexpansive mappings (MFPP)) if every nonexpansive mapping of E into E has a fixed point (respectively, every nonexpansive mapping of E into with compact convex values has a fixed point). For a bounded closed and convex subset E of a Banach space X, a mapping is said to satisfy the conditional fixed point property (CFP) if either t has no fixed points, or t has a fixed point in each nonempty bounded closed convex set that leaves t invariant. A set E is said to have the conditional fixed point property for nonexpansive mappings (CFPP) if every nonexpansive satisfies (CFP). For commuting family of nonexpansive mappings, the following is a remarkable common fixed point property due to Bruck .
Theorem 1.1 ()
LetXbe a Banach space andEa nonempty closed convex subset ofX. IfEhas both the (FPP) and the (CFPP) for nonexpansive mappings, then for any commuting familyof nonexpansive mappings ofEintoE, there is a common fixed point for.
Theorem 1.1 was proved by Belluce and Kirk  when is finite and E is weakly compact and has a normal structure; by Belluce and Kirk  when E is weakly compact and has a complete normal structure; by Browder  when X is uniformly convex and E is bounded; by Lau and Holmes  when is left reversible and E is compact; and finally, by Lim  when is left reversible and E is weakly compact and has a normal structure.
Open Problem (Bruck ). Can commutativity of be replaced by left reversibility?
The answer to this Problem is not known even when the semigroup is left amenable (see  for more details).
In 2011, Sokhuma and Kaewkhao  introduced a new iteration method for approximating a common fixed point of a pair of a single-valued and a multivalued nonexpansive mappings and proved the following strong convergence theorem:
Theorem 1.2 (, Theorem 3.5])
LetEbe a nonempty compact convex subset of a uniformly convex Banach spaceX, and letandbe a single-valued and a multivalued nonexpansive mappings respectively, andsatisfyingfor all. Letbe the sequence of the modified Ishikawa iteration defined by
For a single-valued nonexpansive mapping with , where E is a convex nonexpansive retract of a real uniformly smooth Banach space, Reich and Shemen , Theorem 3.4] obtained a strong convergence to a fixed point of t of a sequence of the form
In 2003, Suzuki  proved the following result.
Theorem 1.3 (, Theorem 2])
LetEbe a compact convex subset of a strictly convex Banach spaceX. Letbe a sequence of nonexpansive mappings onEwith. Letbe a sequence of positive numbers such that, and letbe a sequence of subsets ofsatisfyingforand. Define a sequenceinEbyand
The purpose of this paper is to extend Theorem 1.2 to countably many numbers of single-valued nonexpansive mappings on strictly convex Banach spaces, thereby the result in Theorem 1.3 is covered. The results for spaces are also derived. Our main discoveries are Theorem 3.2 and Theorem 3.6.
We recall that the graph of a multivalued mapping is . The following theorem is essentially proved by Dozo .
Theorem 2.1 (, Theorem 3.1])
LetXbe a Banach space which satisfies Opial’s condition, Ebe a weakly compact convex subset ofX. Let, whereis a family of nonempty compact subsets ofX. Then the graph ofis closed in, whereIdenotes the identity onX, the weak topology andthe norm (or strong) topology.
Let be a family of nonexpansive mappings from E to E. The following lemma proved by Bruck  plays a very important role to our proof of the main result.
Lemma 2.2 (, Lemma 3])
LetEbe a nonempty closed convex subset of a strictly convex Banach spaceX, letbe a family of single-valued nonexpansive mappings onE. Supposeis nonempty. Givena sequence of positive numbers with. Then a mappingtonEdefined by
The following results show examples when the required condition on the nonemptiness of the common fixed point set always satisfies:
Theorem 2.3 (, Theorem 3.1])
LetEbe a weakly compact convex subset of a Banach spaceX. SupposeEhas (MFPP) and (CFPP). Letbe any commuting family of nonexpansive self-mappings ofE. Ifis a multivalued nonexpansive mapping which commutes with every member of, whereis the family of nonempty compact convex subsets ofE. Thenwhere.
Theorem 2.4 (, Theorem 3.2])
LetXbe a Banach space satisfying the Kirk-Massa condition, i.e., the asymptotic center of each bounded sequence ofXin each bounded closed and convex subset is nonempty and compact. LetEbe a weakly compact convex subset ofXand letbe any commuting family of nonexpansive self-mappings ofE. Supposeis a multivalued mapping satisfying conditionfor somewhich commutes with every member of. IfTis upper semi-continuous, then.
Note that strictly convex Banach spaces satisfy the condition in the above theorems.
Remark 2.5 In our main theorems (Theorem 3.2 and Theorem 3.6), we assume the following conditions:
It is an open problem to find a sufficient condition to assure that the condition (2.1) is satisfied.
Let be a metric space. A geodesic joining to is a mapping c from a closed interval to X such that and for all . Thus c is an isometry and . The image of c is called a geodesic (or metric) segment joining x and y. We denote for this geodesic if it is unique. Write for . The space X is said to be a geodesic space if every two points of X are joined by a geodesic. It is said to be of hyperbolic type if it satisfies:
for all . Let and with . It had been defined, by induction, in  that
Following , a metric space X is said to be a space if it is geodesically connected and if every geodesic triangle in X is at least as thin as its comparison triangle in the Euclidean plane . In fact (cf.  p.163), the following are equivalent for a geodesic space X:
Lemma 2.6 (, Proposition 2.2]) LetXbe aspace. Then for eachand
In  the element has been defined. Let be a given sequence in such that , let be a bounded sequence in X, and let be an arbitrary point in X. Let and assume that as . Set
Thus, by (2.3),
We know that is a Cauchy sequence (see ). Thus as for some . Write
Lemma 2.7 (, Lemma 3.8])
3 Main results
3.1 Strictly convex Banach spaces
The following result is a generalization of the result of , Lemma 1.3].
Proof We suppose on the contrary that . Since E and are compact, there exist subsequences of , of and of such that , , for some with and for some . From (i) and (ii) we have and . Using the strict convexity of X and (iii), we have , a contradiction. Hence . □
Now we introduce a new iteration method for a family of single-valued nonexpansive mappings and a multivalued nonexpansive mapping. Let E be a nonempty bounded closed convex subset of a Banach space X, let be a family of single-valued nonexpansive mappings on E, and let be a multivalued nonexpansive mapping. Given a sequence of positive numbers with . The sequence of the modified Ishikawa iteration is defined by , and
Theorem 3.2LetEbe a nonempty compact convex subset of a strictly convex Banach spaceX, letbe a family of single-valued nonexpansive mappings onE, and letbe a multivalued nonexpansive mapping. Supposeandfor all. Given a sequence of positive numberswithandwith. Then the sequencedefined by (3.1) converges strongly to some.
Proof We follow the proof of , Theorem 3.6] and split the proof into five steps.
Consider the following estimates:
We also have
From (3.1), we can see that
We note from Step 3 that
Example 3.3 We consider the space X of Example 3.9 in . Let X be the Hilbert space with the usual norm, and let be a continuous strictly concave function such that and for all . Let be defined by and be defined by
Proof Clearly, is a divergent sequence. We note that and for each with , we have for all i. If we put , then for all n. Since , we must have as . Suppose converges to z for some . Thus also converges to z, a contradiction. □
It is noticed that F is not convex. Thus it is not a nonexpansive retract of any convex set. It can be also observed that if we redefine the mapping T as we can easily verify that T is nonexpansive and the condition (2.1) is satisfied.
Note also that the above result is equivalent to:
Thus Theorem 3.2 contains Theorem 1.3.
With the application of the demiclosedness principle (Theorem 2.1), a weak convergence version of Theorem 3.2 also holds:
Theorem 3.5LetXbe a strictly convex Banach space satisfying the Opial’s condition, Ebe a weakly compact convex subset ofX, letbe a family of single-valued nonexpansive mappings onE, and letbe a multivalued nonexpansive mapping. Supposeandfor all. Given a sequence of positive numberswithandwith. Then the sequencedefined by (3.1) converges weakly to some.
Let E be a nonempty bounded closed convex subset of a complete space X, let be a family of single-valued nonexpansive mappings on E, and be a multivalued nonexpansive mapping. Given a sequence of positive numbers with and as where . The sequence of the modified Ishikawa iteration is defined by
Theorem 3.6LetEbe a compact convex subset of a completespaceX. Letbe a family of single-valued nonexpansive mappings onE, and letbe a multivalued nonexpansive mapping. Supposeandfor all. Givena sequence of positive numbers withandaswhere. If, then the sequencedefined by (3.5) converges strongly to some.
As before, we consider the proof in 5 steps. Because of the same details in some cases, we only present proofs for Step 2 to Step 4.
By (3.6) and using (CN) inequality,
This implies that
Using (3.6) and (CN) inequality, we have
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
The authors are grateful to the referees for their valuable comments and suggestions. They also would like to thank the Junior Science Talent Project (JSTP) under Thailand’s National Science and Technology Development Agency for financial support.
Dhompongsa, S, Kaewkhao, A, Panyanak, B: On Kirk’s strong convergence theorem for multivalued nonexpansive mappings on spaces. Nonlinear Anal.. 75, 459–468 (2012). Publisher Full Text
Holmes, RD, Lau, AT-M: Non-expansive actions of topological semigroups and fixed points. J. Lond. Math. Soc.. 5, 330–336 (1972). Publisher Full Text
Lim, TC: Characterizations of normal structure. Proc. Am. Math. Soc.. 43, 313–319 (1974). Publisher Full Text
Schu, J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc.. 43, 153–159 (1991). Publisher Full Text