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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Approximation methods for nonlinear operator equations
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by C. E. Chidume and H. Zegeye PDF
Proc. Amer. Math. Soc. 131 (2003), 2467-2478 Request permission

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 $\sigma :=\inf _{ n\in N_0 } \frac {\psi (||x_{n+1}-x^*||)}{||x_{n+1}-x^*||}>0$ and that $||Ax_{n+1}-Ax_n||\rightarrow 0$, 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
  • MR Author ID: 232629
  • Email: chidume@ictp.trieste.it
  • H. Zegeye
  • Affiliation: The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
  • Email: habz@ictp.trieste.it
  • 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
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 131 (2003), 2467-2478
  • MSC (2000): Primary 47H04, 47H06, 47H30, 47J05, 47J25
  • DOI: https://doi.org/10.1090/S0002-9939-02-06769-2
  • MathSciNet review: 1974645