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Approximation methods for nonlinear operator equations

Authors:	C E Chidume H Zegeye

Institution:	The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy ; The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

Abstract:	Let be a real normed linear space and $A: E \rightarrow E$ be a uniformly quasi-accretive map. For arbitrary $x_1\in E$ define the sequence $x_n \in E$ by $x_{n+1}:=x_n-\alpha_nAx_n,~n\geq 1,$ where $\{\alpha_n\}$ is a positve real sequence satisfying the following conditions: (i) $\sum\alpha_n=\infty$ ; (ii) $\lim \alpha_n=0$ . For $x^\in N(A):=\{x\in E:Ax=0\}$ , assume that $\sigma :=\inf_{ n\in N_0 } \frac{\psi(\vert\vert x_{n+1}-x^\vert\vert)}{\vert\vert x_{n+1}-x^\vert\vert}>0$ and that $\vert\vert Ax_{n+1}-Ax_n\vert\vert\rightarrow 0$ , where $N_0:=\{n\in N$ (the set of all positive integers): $x_{n+1}\neq x^\}$ and $\psi:0,\infty)\rightarrow 0,\infty)$ is a strictly increasing function with $\psi(0)=0$ . It is proved that a Mann-type iteration process converges strongly to . Furthermore if, in addition, is a uniformly continuous map, it is proved, without the condition on $\sigma$ , that the Mann-type iteration process converges strongly to . As a consequence, corresponding convergence theorems for fixed points of hemi-contractive maps are proved.

Keywords:	Bounded operators nonexpansive retraction uniformly accretive maps uniformly pseudocontractive maps uniformly smooth Banach spaces

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