Weak and Strong Convergence Theorems for Asymptotically Strict Pseudocontractive Mappings in the Intermediate Sense
© Jing Zhao and Songnian He. 2010
Received: 23 June 2010
Accepted: 19 October 2010
Published: 24 November 2010
We study the convergence of Ishikawa iteration process for the class of asymptotically -strict pseudocontractive mappings in the intermediate sense which is not necessarily Lipschitzian. Weak convergence theorem is established. We also obtain a strong convergence theorem by using hybrid projection for this iteration process. Our results improve and extend the corresponding results announced by many others.
1. Introduction and Preliminaries
The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al. . It is known  that if is a nonempty closed convex bounded subset of a uniformly convex Banach space and is asymptotically nonexpansive in the intermediate sense, then has a fixed point. It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense contains properly the class of asymptotically nonexpansive mappings.
The class of asymptotically -strict pseudocontractions was introduced by Qihou  in 1996 (see also ). Kim and Xu  studied weak and strong convergence theorems for this class of mappings. It is important to note that every asymptotically -strict pseudocontractive mapping with sequence is a uniformly -Lipschitzian mapping with .
They obtained a weak convergence theorem of modified Mann iterative processes for the class of mappings which is not necessarily Lipschitzian. Moreover, a strong convergence theorem was also established in a real Hilbert space by hybrid projection methods; see  for more details.
In order to prove our main results, we also need the following lemmas.
Lemma 1.2 (see ).
Lemma 1.3 (see ).
Lemma 1.4 (see ).
Lemma 1.5 (see ).
Lemma 1.7 (see ).
Let be a nonempty subset of a Hilbert space and a uniformly continuous asymptotically -strict pseudocontractive mapping in the intermediate sense with sequence . Let be a sequence in such that and as , then as .
Lemma 1.8 (see [7, Proposition ]).
Let be a nonempty closed convex subset of a Hilbert space and a continuous asymptotically -strict pseudocontractive mapping in the intermediate sense. Then is demiclosed at zero in the sense that if is a sequence in such that and , then .
Lemma 1.9 (see ).
2. Main Results
Next, we modify Ishikawa iterative process to get a strong convergence theorem.
We break the proof into six steps.
Since is reflexive and is bounded, we get that is nonempty. First, we show that is a singleton. Assume that is subsequence of such that . Observe that is uniformly continuous and as , for any we have as . From Lemma 1.8, we see that .
Hence by the uniqueness of the nearest point projection of onto . Since is an arbitrary weakly convergent subsequence, it follows that and hence . It is easy to see as (2.34) that . Since has the Kadec-Klee property, we obtain that , that is, as . This completes the proof.
This research is supported by Fundamental Research Funds for the Central Universities (ZXH2009D021) and supported by the Science Research Foundation Program in Civil Aviation University of China (no. 09CAUC-S05) as well.
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