Generalized Mann Iterations for Approximating Fixed Points of a Family of Hemicontractions
© Liang-Gen Hu et al. 2008
Received: 10 January 2008
Accepted: 15 May 2008
Published: 19 May 2008
This paper concerns common fixed points for a finite family of hemicontractions or a finite family of strict pseudocontractions on uniformly convex Banach spaces. By introducing a new iteration process with error term, we obtain sufficient and necessary conditions, as well as sufficient conditions, for the existence of a fixed point. As one will see, we derive these strong convergence theorems in uniformly convex Banach spaces and without any requirement of the compactness on the domain of the mapping. The results given in this paper extend some previous theorems.
It is easy to see that the class of pseudocontractive mappings with fixed points is a subset of the class of hemicontractions.
There are many papers in the literature dealing with the approximation of fixed points for several classes of nonlinear mappings (see, e.g., [1–11], and the reference therein). In these works, there are two iterative methods to be used to find a point in . One is explicit and one is implicit.
The explicit one is the following well-known Mann iteration.
It has been applied to many classes of nonlinear mappings to find a fixed point. However, for hemicontractive mappings and strictly pseudocontractive mappings, the iteration process of convergence is in general not strong (see a counterexample given by Chidume and Mutangadura ). Most recently, Marino and Xu  proved that the Mann iterative sequence converges weakly to a fixed point for strictly pseudocontractive mappings in a Hilbert space, while the real sequence satisfying (i) and (ii) .
In order to get strong convergence for fixed points of hemicontractive mappings and strictly pseudocontractive mappings, the following Mann-type implicit iteration scheme is introduced.
Recently, in the setting of a Hilbert space, Rafiq  proved that the Mann-type implicit iterative sequence converges strongly to a fixed point for hemicontractive mappings, under the assumption that the domain of is a compact convex subset of a Hilbert space, and for some .
In this paper, we will study the strong convergence of the generalized Mann-type iteration scheme (see Definition 2.1) for hemicontractive and, respectively, pseudocontractive mappings. As we will see, our theorems extend the corresponding results in  in four aspects. (1) The space setting is a more general one: uniformly convex Banach space, which could not be a Hilbert space. (2) The requirement of the compactness on the domain of the mapping is dropped. (3) A single mapping is replaced by a family of mappings. (4) The Mann-type implicit iteration is replaced by the generalized Mann iteration. Moreover, we give answers to a question asked in .
2. Preliminaries and Lemmas
Definition 2.1 (generalized Mann iteration).
Definition 2.2 (see ).
Lemma 2.3 (see ).
Lemma 2.6 (see ).
A simple computation shows the conclusion.
3. Main Results
This completes the proof.
The necessity is obvious.
Since satisfies condition , and for each , it follows from the existence of that . Applying the similar arguments as in the proof of Theorem 3.2, we conclude that converges strongly to a common fixed point of . This completes the proof.
As a direct consequence of Theorem 3.3, we get the following result.
Corollary 3.4 (see [12, Theorem 3]).
By , we have . Equation (3.7) implies that . Since satisfies condition (A) and the limit exists, we get . The rest of the proof follows now directly from Theorem 3.2. This completes the proof.
Theorems 3.2 and 3.3 extend [12, Theorem 3] essentially since the following hold.
Hilbert spaces are extended to uniformly convex Banach spaces.
The requirement of compactness on domain on [12, Theorem 3] is dropped.
A single mapping is replaced by a family of mappings.
Moreover, if , then is well defined by (II). Hence, Theorems 3.2 and 3.3 are also answers to the question proposed by Qing .
Theorem 3.6 extends the corresponding result [6, Theorem 3.1].
The authors would like to thank the referees very much for helpful comments and suggestions. The work was supported partly by the National Natural Science Foundation of China, the Specialized Research Fund for the Doctoral Program of Higher Education of China, the NCET-04-0572 and Research Fund for the Key Program of the Chinese Academy of Sciences.
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