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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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Convergence theorem for zeros of generalized Lipschitz generalized phi-quasi-accretive operators
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by C. E. Chidume and C. O. Chidume PDF
Proc. Amer. Math. Soc. 134 (2006), 243-251 Request permission

Abstract:

Let $E$ be a uniformly smooth real Banach space and let $A: E \rightarrow E$ be a mapping with $N(A)\neq \emptyset$. Suppose $A$ is a generalized Lipschitz generalized $\Phi$-quasi-accretive mapping. Let $\{a_{n}\}, \{b_{n}\},$ and $\{c_{n}\}$ be real sequences in [0,1] satisfying the following conditions: (i) $a_{n} + b_{n} + c_{n} = 1$; (ii) $\sum (b_{n} + c_{n} ) = \infty$; (iii) $\sum c_{n} < \infty$; (iv) $\lim b_{n} = 0.$ Let $\{x_{n}\}$ be generated iteratively from arbitrary $x_{0}\in E$ by \[ x_{n+1} = a_{n}x_{n} + b_{n}Sx_{n} + c_{n}u_{n}, n\geq 0,\] where $S: E\rightarrow E$ is defined by $Sx:=x-Ax ~\forall x\in E$ and $\{u_{n}\}$ is an arbitrary bounded sequence in $E$. Then, there exists $\gamma _{0}\in \Re$ such that if $b_{n} + c_{n} \leq \gamma _{0} ~\forall ~ n\geq 0,$ the sequence $\{x_{n}\}$ converges strongly to the unique solution of the equation $Au = 0$. A related result deals with approximation of the unique fixed point of a generalized Lipschitz and generalized $\phi$-hemi-contractive mapping.
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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
  • C. O. Chidume
  • Affiliation: Department of Mathematics and Statistics, Auburn University, Auburn, Alabama
  • Email: chidumeg@hotmail.com
  • Received by editor(s): August 2, 2004
  • Received by editor(s) in revised form: August 30, 2004
  • Published electronically: June 13, 2005
  • Communicated by: Joseph A. Ball
  • © Copyright 2005 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 134 (2006), 243-251
  • MSC (2000): Primary 47H09, 47J25
  • DOI: https://doi.org/10.1090/S0002-9939-05-07954-2
  • MathSciNet review: 2170564