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Strong convergence of modified implicit iterative algorithms with perturbed mappings for continuous pseudocontractive mappings. (English) Zbl 1168.65350

Authors’ abstract: Let $X$ be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. The first purpose of this paper is to introduce a modified viscosity iterative process with perturbation for a continuous pseudocontractive self-mapping $T$ and prove that this iterative process converges strongly to ${x}^{*}\in F\left(T\right):=\left\{x\in X|x=T\left(x\right)\right\}$, where ${x}^{*}$ is the unique solution in $F\left(T\right)$ to the following variational inequality:

$〈f\left({x}^{*}\right)-{x}^{*},j\left(\nu -{x}^{*}\right)〉⩽0\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}\nu \in F\left(t\right)·$

The second aim of the paper is to propose two modified implicit iterative schemes with perturbation for a continuous pseudocontractive self-mapping $T$ and prove that these iterative schemes strongly converge to the same point ${x}^{*}\in F\left(T\right)$. Basically, we show that if the perturbation mapping is nonexpansive, then the convergence property of the iterative process holds. In this respect, the results presented here extend, improve and unify some very recent theorems in the literature [L. C. Zeng and J. C. Yao, Nonlinear Anal., Theory Methods Appl. 64, No. 11 (A), 2507–2515 (2006; Zbl 1105.47061); H. K. Xu, J. Math. Anal. Appl. 298, No. 1, 279–291 (2004; Zbl 1061.47060); Y. S. Song and R. D. Chen, Nonlinear Anal., Theory Methods Appl. 67, No. 2 (A), 486–497 (2007; Zbl 1126.47054)].

##### MSC:
 65J15 Equations with nonlinear operators (numerical methods)