Mann Type Implicit Iteration Approximation for Multivalued Mappings in Banach Spaces
© Huimin He et al. 2010
Received: 16 March 2010
Accepted: 5 July 2010
Published: 25 July 2010
Let be a nonempty compact convex subset of a uniformly convex Banach space and let be a multivalued nonexpansive mapping. For the implicit iterates , , , . We proved that converges strongly to a fixed point of under some suitable conditions. Our results extended corresponding ones and revised a gap in the work of Panyanak (2007).
The fixed point theory of multivalued nonexpansive mappings is much more complicated and difficult than the corresponding theory of single-valued nonexpansive mappings. However, some classical fixed point theorems for single-valued nonexpansive mappings have already been extended to multivalued mappings.
In 1968, Markin  firstly established the nonexpansive multivalued convergence results in Hilbert space. Banach's Contraction Principle was extended to a multivalued contraction in 1969. (Below is stated in a Banach space setting.)
Theorem 1.1 (see ).
In 1974, one breakthrough was achieved by Lim using Edelstein's method of asymptotic centers .
Theorem 1.2 (see Lim ).
In 1990, Kirk and Massa  obtained another important result for multivalued nonexpansive mappings.
Theorem 1.3 (see Kirk and Massa ).
Let be a nonempty closed bounded convex subset of a Banach space and a multivalued nonexpansive mapping. Suppose that the asymptotic center in E of each bounded sequence of X is nonempty and compact. Then T has a fixed point.
where and . He proved that converges strongly to some fixed points of . Xu  extended Theorem 1.3 to a multivalued nonexpansive nonself mapping and obtained the fixed theorem in 2001. The recent fixed point results for nonexpansive mappings can be found in [7–12] and references therein.
and we prove some strong convergence theorems of the sequence defined by (1.6) for nonexpansive multivalued mappings in Banach spaces. The results presented in this paper establish a new type iteration convergence theorems for multivalued nonexpansive mappings in Banach spaces and extend the corresponding results of Panyanak .
where denotes the dual space of and denotes the generalized duality pair. It is well known that if is strictly convex, then is single valued. And if Banach space admits sequentially continuous duality mapping from weak topology to weak star topology, then, by [14, Lemma 1], we know that the duality mapping is also single valued. In this case, the duality mapping is also said to be weakly sequentially continuous; that is, if is a subject of with , then . By Theorem 1 of , we know that if admits a weakly sequentially continuous duality mapping, then satisfies Opial's condition, and is smooth; for the details, see . In the sequel, we will denote the single-valued duality mapping by .
Throughout this paper, we write (resp., ) to indicate that the sequence weakly (resp., weak *) converges to , as usual will symbolize strong convergence. In order to show our main results, the following concepts and lemmas are needed.
Lemma 2.1 (see ).
Example of mappings that satisfy Condition can be founded in .
3. Main Results
Now, we prove our results.
Part (i) is trivial. And part (ii) remains to be proved.
So the desired conclusion follows.
The proof is completed.
Clearly, if and , then the above inequalities cannot be assured. Indeed, from the monotone decreasing sequence of in the proof of (Theorem 3.1 ), we cannot obtain that is a decreasing sequence. Hence, the conclusion of Theorem 3.1 in  cannot be achieved.
Let be a Banach space satisfying Opial's condition and let be a nonempty weakly compact convex subset of . Suppose that is a multivalued nonexpansive mapping, where and , the sequence is generated by (1.6).
Part (i) is trivial. Now we prove part (ii).
This is a contradiction by satisfying Opial's condition.
Which is a contradiction, so the conclusion of the theorem follows.
The proof is completed.
Let be a reflexive Banach space which admits a weakly sequentially continuous duality mapping from to , and let be a nonempty weakly compact convex subset of . Suppose that is a multivalued nonexpansive mapping, where and , the sequence is generated by (1.6).
The proof is completed.
Let be a nonempty compact convex subset of a uniformly convex Banach space and let be a multivalued nonexpansive mapping satisfying Condition , where and , then the sequence generated by (1.6) converges strongly to a fixed point.
The proof of remained part is omitted because it is similar to the proof of Theorem 3.8 in .
The work was supported by the Fundamental Research Funds for the Central Universities, No. JY10000970006, and National Nature Science Foundation, No. 60974082.
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