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Approximation methods for nonlinear operator equations


Authors: C. E. Chidume and H. Zegeye
Journal: Proc. Amer. Math. Soc. 131 (2003), 2467-2478
MSC (2000): Primary 47H04, 47H06, 47H30, 47J05, 47J25
Published electronically: November 13, 2002
MathSciNet review: 1974645
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Abstract: Let $E$ 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 $x^*$. Furthermore if, in addition, $A$ is a uniformly continuous map, it is proved, without the condition on $\sigma$, that the Mann-type iteration process converges strongly to $x^*$. As a consequence, corresponding convergence theorems for fixed points of hemi-contractive maps are proved.


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Additional Information

C. E. Chidume
Affiliation: The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
Email: chidume@ictp.trieste.it

H. Zegeye
Affiliation: The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
Email: habz@ictp.trieste.it

DOI: https://doi.org/10.1090/S0002-9939-02-06769-2
Keywords: Bounded operators, nonexpansive retraction, uniformly accretive maps, uniformly pseudocontractive maps, uniformly smooth Banach spaces
Received by editor(s): December 8, 2001
Received by editor(s) in revised form: March 18, 2002
Published electronically: November 13, 2002
Additional Notes: The second author undertook this work with the support of the “ICTP Programme for Training and Research in Italian Laboratories, Trieste, Italy".
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2002 American Mathematical Society