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

Authors:	C E Chidume C O Chidume

Institution:	The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy ; Department of Mathematics and Statistics, Auburn University, Auburn, Alabama

Abstract:	Let be a uniformly smooth real Banach space and let $A: E \rightarrow E$ be a mapping with $N(A)\neq \emptyset$ . Suppose 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 . 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 . A related result deals with approximation of the unique fixed point of a generalized Lipschitz and generalized $\phi$ -hemi-contractive mapping.

Keywords:	Generalized Lipschitz maps generalized $\Phi$-quasi-accretive maps generalized $\Phi$-hemicontractive maps uniformly smooth real Banach spaces

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