An approximation of a common fixed point of nonexpansive mappings on convex metric spaces

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.

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,

where is the distance from the point a to the subset B.

A mapping and a multivalued mapping are said to be nonexpansive if for each ,

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 [8], 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 [6].

Theorem 1.1 ([6])

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 [1] when is finite and E is weakly compact and has a normal structure; by Belluce and Kirk [2] when E is weakly compact and has a complete normal structure; by Browder [4] when X is uniformly convex and E is bounded; by Lau and Holmes [11] when is left reversible and E is compact; and finally, by Lim [14] when is left reversible and E is weakly compact and has a normal structure.

Open Problem (Bruck [6]). Can commutativity of be replaced by left reversibility?

The answer to this Problem is not known even when the semigroup is left amenable (see [13] for more details).

In 2011, Sokhuma and Kaewkhao [17] 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 ([17], 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

where, and, . Thenconverges strongly to a common fixed point oftandT.

For a single-valued nonexpansive mapping with , where E is a convex nonexpansive retract of a real uniformly smooth Banach space, Reich and Shemen [15], Theorem 3.4] obtained a strong convergence to a fixed point of t of a sequence of the form

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 [18] proved the following result.

Theorem 1.3 ([18], 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

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 [10].

Theorem 2.1 ([10], 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.

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 [5] plays a very important role to our proof of the main result.

Lemma 2.2 ([5], 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

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 ([8], 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 ([8], 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[12] if it satisfies:

for all . Let and with . It had been defined, by induction, in [7] that

The definition of ⊕ in (2.3) is an ordered one in the sense that it depends on the order of points . Under (2.2) we can see that

for each .

Following [3], 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. [3] p.163), the following are equivalent for a geodesic space X:

(i) X is a space.

(ii) X satisfies the (CN) inequality: If and is the midpoint of and , then

Lemma 2.6 ([3], Proposition 2.2]) LetXbe aspace. Then for eachand

In particular, (2.2) holds inspaces.

In [9] 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),

where and for each

We know that is a Cauchy sequence (see [9]). Thus as for some . Write

By (2.6), , it is seen that . Thus the limit x is independent of the choice of .

Lemma 2.7 ([9], Lemma 3.8])

LetCbe a nonempty closed convex subset of a completespaceX, letbe a family of single-valued nonexpansive mappings onC. Supposeis nonempty. Definebyfor allwherewithandas . Thentis nonexpansive and.

3.1 Strictly convex Banach spaces

The following result is a generalization of the result of [16], Lemma 1.3].

Lemma 3.1LetEbe a compact subset of a strictly convex Banach spaceX, letbe a sequence of real numbers such thatfor all, and let, be sequences ofEsatisfying, for some,

(i) ,

(ii) and

(iii) .

Then, .

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

where , and . Put .

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 [17], Theorem 3.6] and split the proof into five steps.

Step 1. exists for all :

We first note that, since ,

Consider the following estimates:

Therefore, is a bounded decreasing sequence in , and hence exists.

Step 2. :

From Step 1, suppose . We have

Thus

We also have

By Lemma 3.1, since , .

Step 3. :

From (3.1), we can see that

and hence . Therefore, and by (3.2) we obtain

Thus . By Lemma 3.1, since , .

Step 4. :

We note from Step 3 that

and

for all . Therefore,

From Step 2 and (3.3), we obtain .

Step 5. :

Define a mapping by

for any . By Lemma 2.2, t is well defined, nonexpansive and . Since E is compact, there exists a subsequence of which converges to v for some . Using Step 3 and Step 4, we have

and

It follows that . Since exists by Step 1, . □

The following example shows that the condition ‘ for all ’ in Theorem 3.2 is necessary.

Example 3.3 We consider the space X of Example 3.9 in [8]. 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

It is straightforward showing that T and each are nonexpansive. Set and for a subsequence in with . Let be a sequence in defined as

where

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.

Remark 3.4 With the same proof, Theorem 3.2 is valid when is of the following form: For a permutation π on , define in E by and

, and .

Note also that the above result is equivalent to:

Let be a sequence of subsets of satisfying for and . Define in E by and

, 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 3.5LetXbe 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.

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. □

3.2 spaces

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

where , , and . Put .

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.

Proof The proof follows along the lines with the proof of Theorem 3.2. Recall that and for all . Thus, by (3.5),

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. :

Let , we have for all n. Using the nonexpansiveness of , we see that

By (3.6) and using (CN) inequality,

Let . Since ,

This implies that

and hence .

Step 3. :

Using (3.6) and (CN) inequality, we have

and thus

As before,

This also implies that .

Step 4. , where :

Since E is compact, there exists a subsequence of such that as for some . Using the nonexpansiveness of and t, we have

Therefore, . From Step 2 and Step 3 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.

1. Belluce, LP, Kirk, WA: Fixed-point theorems for families of contraction mappings. Pac. J. Math.. 18, 213–217 (1966)

2. Belluce, LP, Kirk, WA: Nonexpansive mappings and fixed-points in Banach spaces. Ill. J. Math.. 11, 474–479 (1967)

3. Bridson, M, Haefliger, A: Metric Spaces of Non-positive Curvature, Springer, Berlin (1999)

4. Browder, FE: Nonexpansive nonlinear operators in a Banach space. Proc. Natl. Acad. Sci. USA. 54, 1041–1044 (1965). PubMed Abstract | Publisher Full Text | PubMed Central Full Text

5. Bruck, RE: Properties of common fixed-point sets of nonexpansive mappings in Banach spaces. Trans. Am. Math. Soc.. 179, 251–262 (1973)

6. Bruck, RE: A common fixed point theorem for a commuting family of nonexpansive mappings. Pac. J. Math.. 53, 59–71 (1974)

7. Butsan, T, Dhompongsa, S, Fupinwong, W: Schauder’s conjecture and the Kakutani fixed point theorem on convex metric spaces. J. Nonlinear Convex Anal.. 11, 513–526 (2010)

8. Dhompongsa, S, Nanan, N: A common fixed points for commuting family of single valued nonexpansive mappings and a multivalued nonexpansive mapping. Fixed Point Theory Appl.. 2011, (2011)

9. 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

10. Dozo, EL: Multivalued nonexpansive mappings and Opial’s condition. Proc. Am. Math. Soc.. 38, 286–292 (1973)

11. 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

12. Goebel, K, Kirk, WA: Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge (1990)

13. Lau, AT-M: Normal structure and common fixed point properties for semigroups of nonexpansive mappings in Banach spaces. Fixed Point Theory Appl.. 2010, (2010)

14. Lim, TC: Characterizations of normal structure. Proc. Am. Math. Soc.. 43, 313–319 (1974). Publisher Full Text

15. Reich, S, Sheman, L: Two algorithms for nonexpansive mappings. Fixed Point Theory. 12(2), 443–448 (2011)

16. Schu, J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc.. 43, 153–159 (1991). Publisher Full Text

17. Sokhuma, K, Kaewkhao, A: Ishikawa iterative process for a pair of single-valued and multivalued nonexpansive mappings in Banach spaces. Fixed Point Theory Appl.. 2011, (2011)

18. Suzuki, T: Convergence theorems to common fixed points for infinite families of nonexpansive mappings in strictly convex Banach spaces. Nihonkai Math. J.. 14, 43–54 (2003)