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Strong convergence of an iterative algorithm for nonself multimaps in Banach spaces. (English) Zbl 1222.47092

Summary: Let $E$ be a uniformly convex Banach space having a uniformly Gâteaux differentiable norm, $D$ a nonempty closed convex subset of $E$, and $T:D\to K\left(E\right)$ a nonself multimap such that $F\left(T\right)\ne ⌀$ and ${P}_{T}$ is nonexpansive, where $F\left(T\right)$ is the fixed point set of $T$, $K\left(E\right)$ is the family of nonempty compact subsets of $E$ and ${P}_{T}\left(x\right)=\left\{{u}_{x}\in {T}_{X}:\parallel x-{u}_{x}\parallel =d\left(x,Tx\right)\right\}$. Suppose that $D$ is a nonexpansive retract of $E$ and that, for each $v\in D$ and $t\in \left(0,1\right)$, the contraction ${S}_{t}$ defined by ${S}_{t}x=t{P}_{T}x+\left(1-t\right)v$ has a fixed point ${x}_{t}\in D$. Let $\left\{{\alpha }_{n}\right\},\left\{{\beta }_{n}\right\}$ and $\left\{{\gamma }_{n}\right\}$ be three real sequences in (0,1) satisfying approximate conditions. Then, for fixed $u\in D$ and arbitrary ${x}_{0}\in D$, the sequence $\left\{{x}_{n}\right\}$ generated by

${x}_{n}\in {\alpha }_{n}u+{\beta }_{n}{x}_{n-1}+{\gamma }_{n}{P}_{T}\left({x}_{n}\right),\phantom{\rule{1.em}{0ex}}n\ge 0,$

converges strongly to a fixed point of $T$.

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H05 Monotone operators (with respect to duality) and generalizations 47H10 Fixed point theorems for nonlinear operators on topological linear spaces