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Convergence theorem for zeros of generalized Lipschitz generalized phi-quasi-accretive operators

Authors: C. E. Chidume and C. O. Chidume
Journal: Proc. Amer. Math. Soc. 134 (2006), 243-251
MSC (2000): Primary 47H09, 47J25
Published electronically: June 13, 2005
MathSciNet review: 2170564
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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

\begin{displaymath}x_{n+1} = a_{n}x_{n} + b_{n}Sx_{n} + c_{n}u_{n}, n\geq 0,\end{displaymath}

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

C. O. Chidume
Affiliation: Department of Mathematics and Statistics, Auburn University, Auburn, Alabama

Keywords: Generalized Lipschitz maps, generalized $\Phi$-quasi-accretive maps, generalized $\Phi$-hemicontractive maps, uniformly smooth real Banach spaces
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
Article copyright: © Copyright 2005 American Mathematical Society

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