Convergence and stability of modified Ishikawa iteration sequence with errors
© Yang and Peng; licensee Springer. 2014
Received: 15 July 2014
Accepted: 17 October 2014
Published: 31 October 2014
We show some stability and convergence theorems of the modified Ishikawa iterative sequence with errors for a strongly successively pseudocontractive and strictly asymptotically pseudocontractive mapping in a real Banach space. Additionally, we prove that if T is a uniformly Lipschitzian strongly accretive mapping, the modified Ishikawa iteration sequence with errors converges strongly to the unique solution of the equation . The main results of this paper improve and extend the known results in the current literature.
MSC:47H09, 47H10, 47J25.
Developments in fixed point theory reflect that the iterative construction of fixed points is proposed and vigorously analyzed for various classes of maps in different spaces. The class of pseudocontractive mappings in their relation with iteration procedures has been studied by several researchers under suitable conditions; for more details, see [1–3] and the references therein. Also, the class of nonexpansive mappings via iteration methods has extensively been studied in this regard; see Tan and Xu . The class of strongly pseudocontractive mappings has been studied by many researchers (see [5–7]) under certain conditions. Stability results established in metric space, normed linear space, and Banach space settings are available in the literature. There are several authors whose contributions are of colossal value in the study of stability of the fixed point iterative procedures: Imoru and Olatinwo , Olatinwo , Haghi et al. , Olatinwo and Postolache . Reich and Zaslavski  in Chapter 2 established the existence and uniqueness of a fixed point for a generic mapping, convergence of the iterates of a nonexpansive mapping, stability of the fixed point under small perturbations of a mapping, convergence of Krasnosel’skii-Mann iterations of nonexpansive mappings, generic power convergence of order preserving mappings, and existence and uniqueness of positive eigenvalues and eigenvectors of order-preserving linear operators. They also studied the convergence of iterates of nonexpansive mappings in the presence of computational errors in this chapter. Harker and Hicks in  showed how a stability sequence could arise in practice and demonstrated the importance of investigating the stability of various iterative sequences for some kinds of nonlinear mappings.
The purpose in this paper is to study the modified Ishikawa iteration sequence with errors converging strongly to a fixed point of the uniformly Lipschitzian strongly successively pseudocontractive mapping under the lack of some conditions. On the other hand, the authors show that the modified Ishikawa iteration sequence with errors converges strongly to the unique solution of the equation if T is a Lipschitzian strongly accretive mapping. The results of this paper improve and extend some recent results.
- (i)uniformly Lipschizian if there exists a constant such that
- (ii)strongly successively pseudocontractive if for every there exist and such that(2.1)
- (iii)strongly pseudocontractive if for every there exist and such that(2.2)
for all . That is, T is a strongly successively pseudocontractive mapping.
Lemma 2.2 (see [, Lemma 1.1])
Let E be a Banach space and . Then for all if and only if there exists such that .
Therefore, it follows from Lemma 2.2 and (2.3) that the definition of a strongly successively pseudocontractive mapping is equivalent to the following definition.
for all , and .
is said to be a modified Ishikawa iteration sequence with errors.
The following lemmas will be needed in proving our main results.
Lemma 2.5 (see )
If and , then (i) exists, and (ii) in particular, if has a subsequence converging to 0, then .
From Lemma 2.5 we have the following.
where is a sequence in such that , , and . Then as .
By Lemma 2.5, we see that exists. Therefore, there exists such that .
Note that and , and we have , a contradiction with . Then . This completes the proof of Lemma 2.6. □
Let E be a Banach space and T a self-map of E. Suppose and defines an iteration procedure which yields a sequence of points . Let denote the fixed point of T and let converge to a fixed point p of T. Let and let be a sequence in .
If implies that , then the iteration procedure defined by is said to be T-stable. If implies , then the iteration procedure defined by is said to be almost T-stable.
3 Main results
, as ; ;
converges strongly to a unique fixed point of T in E;
is almost T-stable;
if implies , then the iteration procedure defined by is said to be a weakly T-stable. Thus is also weakly T-stable.
and, since , this implies that .
Since , , it follows from Lemma 2.6 that we have . That is, converges strongly to p.
If , setting , , , , in Lemma 2.6, we have as , i.e., is almost T-stable.
If , setting , , , , in Lemma 2.6, we have as , i.e., is weakly T-stable. This completes the proof. □
Similar to the proof of Theorem 3.1, we have the following.
, as ;
there exists such that ;
converges strongly to a unique fixed point of T in E;
is both almost T-stable and weakly T-stable.
, as ;
, where k is the constant of strongly accretive mapping T, and L is the Lipschitzian constant of mapping ;
converges strongly to a solution p of .
The rest of the proof is similar to the proof of Theorem 3.1. This completes the proof. □
Remark 3.4 (1) Theorem 3.3 extends the main result of  from a uniformly smooth Banach space to a real Banach space and without the boundedness assumption of and ; (2) Theorem 3.3 extends and improves the corresponding results of  by removing the assumptions and .
The authors thank the referees and the editor for their careful reading of the manuscript and their many valuable comments and suggestions for the improvement of this article. The first author was supported in part by the Humanity and Social Sciences Research Planning Foundation of Ministry of Education of China (Grant No. 14YJAZH095). The second author was supported by National Natural Science Foundation of China (Grant No. 61374081) and the Natural Science Foundation of Guangdong Province (Grant No. S2013010013034).
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