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Zbl 1072.47062
Chidume, C.E.; Chidume, C.O.
Convergence theorem for zeros of generalized Lipschitz generalized $\Phi$-quasi-accretive operators.
(English)
[J] Proc. Am. Math. Soc. 134, No. 1, 243-251 (2006). ISSN 0002-9939; ISSN 1088-6826/e

Summary: Let $E$ be a uniformly smooth real Banach space and let $A: E \rightarrow E$ be a mapping with $N(A)\neq \emptyset$. Suppose that $A$ 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 $\gamma_{0}\in \bbfR$ 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.
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H09 Mappings defined by "shrinking" properties
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: uniformly smooth real Banach space; generalized Lipschitz generalized $\Phi$-quasi-accretive mapping; approximation of zeros; strong convergence; approximation of fixed points; generalized $\phi$-hemi-contractive mapping

Cited in: Zbl 1173.47048

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