An approximation of a common fixed point of nonexpansive mappings on convex metric spaces
© Anakkamatee and Dhompongsa; licensee Springer 2012
Received: 17 March 2012
Accepted: 28 June 2012
Published: 19 July 2012
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.
where is the distance from the point a to the subset B.
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 ()
Let X be a Banach space and E a nonempty closed convex subset of X. If E has both the (FPP) and the (CFPP) for nonexpansive mappings, then for any commuting familyof nonexpansive mappings of E into E, 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])
where, and, . Thenconverges strongly to a common fixed point of t and T.
where is a retraction on the subset E and the sequences satisfy conditions: (i) , (ii) and . Clearly, conditions (i) and (ii) on the sequences are different from the ones in Theorem 1.2.
In 2003, Suzuki  proved the following result.
Theorem 1.3 (, Theorem 2])
for. Thenconverges strongly to a common fixed point of.
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])
Let X be a Banach space which satisfies Opial’s condition, E be a weakly compact convex subset of X. Let, whereis a family of nonempty compact subsets of X. Then the graph ofis closed in, where I denotes the identity on X, the weak topology andthe norm (or strong) topology.
We will use the theorem in the following form: If is a sequence in E such that converges weakly to some and converges to 0, then .
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])
for allis well defined, nonexpansive and.
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])
Let E be a weakly compact convex subset of a Banach space X. Suppose E has (MFPP) and (CFPP). Letbe any commuting family of nonexpansive self-mappings of E. Ifis a multivalued nonexpansive mapping which commutes with every member of, whereis the family of nonempty compact convex subsets of E. Thenwhere.
Theorem 2.4 (, Theorem 3.2])
Let X be a Banach space satisfying the Kirk-Massa condition, i.e., the asymptotic center of each bounded sequence of X in each bounded closed and convex subset is nonempty and compact. Let E be a weakly compact convex subset of X and letbe any commuting family of nonexpansive self-mappings of E. Supposeis a multivalued mapping satisfying conditionfor somewhich commutes with every member of. If T is upper semi-continuous, then.
Note that strictly convex Banach spaces satisfy the condition in the above theorems.
It is an open problem to find a sufficient condition to assure that the condition (2.1) is satisfied.
for each .
X is a space.
- (ii)X satisfies the (CN) inequality: If and is the midpoint of and , then
In particular, (2.2) holds inspaces.
By (2.6), , it is seen that . Thus the limit x is independent of the choice of .
Lemma 2.7 (, Lemma 3.8])
Let C be a nonempty closed convex subset of a completespace X, letbe a family of single-valued nonexpansive mappings on C. Supposeis nonempty. Definebyfor allwherewithandas . Then t is nonexpansive and.
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 . □
where , and . Put .
Theorem 3.2 Let E be a nonempty compact convex subset of a strictly convex Banach space X, letbe a family of single-valued nonexpansive mappings on E, 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.
Step 1. exists for all :
Therefore, is a bounded decreasing sequence in , and hence exists.
Step 2. :
Step 3. :
Thus . By Lemma 3.1, since , .
Step 4. :
From Step 2 and (3.3), we obtain .
Step 5. :
It follows that . Since exists by Step 1, . □
The following example shows that the condition ‘ for all ’ in Theorem 3.2 is necessary.
We will show that does not converge to a common fixed point of T and .
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.
, and .
Note also that the above result is equivalent to:
, and . Then the sequence converges strongly to some .
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 3.5 Let X be a strictly convex Banach space satisfying the Opial’s condition, E be a weakly compact convex subset of X, letbe a family of single-valued nonexpansive mappings on E, and letbe a multivalued nonexpansive mapping. Supposeandfor all. Given a sequence of positive numberswithandwith. Then the sequencedefined by (3.1) converges weakly to some.
Proof In the proof of Theorem 3.2, by applying the Opial’s condition, it follows from a standard argument that converges weakly to some . Then Theorem 2.1 implies that v is a point in F. □
where , , and . Put .
Theorem 3.6 Let E be a compact convex subset of a completespace X. Letbe a family of single-valued nonexpansive mappings on E, 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.
Step 2. :
and hence .
Step 3. :
This also implies that .
Step 4. , where :
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.
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