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Convergence theorem for zeros of generalized Lipschitz generalized ${\Phi }$-quasi-accretive operators. (English) Zbl 1072.47062

Summary: Let $E$ be a uniformly smooth real Banach space and let $A:E\to E$ be a mapping with $N\left(A\right)\ne \varnothing$. Suppose that $A$ is a generalized Lipschitz generalized ${\Phi }$-quasi-accretive mapping. Let $\left\{{a}_{n}\right\},\left\{{b}_{n}\right\},$ and $\left\{{c}_{n}\right\}$ be real sequences in [0,1] satisfying the following conditions: (i) ${a}_{n}+{b}_{n}+{c}_{n}=1$; (ii) $\sum \left({b}_{n}+{c}_{n}\right)=\infty$; (iii) $\sum {c}_{n}<\infty$; (iv) $lim{b}_{n}=0·$ Let $\left\{{x}_{n}\right\}$ be generated iteratively from arbitrary ${x}_{0}\in E$ by

${x}_{n+1}={a}_{n}{x}_{n}+{b}_{n}S{x}_{n}+{c}_{n}{u}_{n},n\ge 0,$

where $S:E\to E$ is defined by $Sx:=x-Ax$ forall $x\in E$ and $\left\{{u}_{n}\right\}$ is an arbitrary bounded sequence in $E$. Then there exists ${\gamma }_{0}\in ℝ$ such that if ${b}_{n}+{c}_{n}\le {\gamma }_{0}\phantom{\rule{3.33333pt}{0ex}}\forall \phantom{\rule{3.33333pt}{0ex}}n\ge 0,$ the sequence $\left\{{x}_{n}\right\}$ 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 $\varphi$-hemi-contractive mapping.

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties 47H10 Fixed point theorems for nonlinear operators on topological linear spaces