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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 $T:D\rightarrow K(E)$ a nonself multimap such that $F(T)\neq \varnothing$ and $P_T$ is nonexpansive, where $F(T)$ is the fixed point set of $T$, $K(E)$ is the family of nonempty compact subsets of $E$ and $P_T(x)=\{u_x\in T_X: \|x-u_x\| = d(x,Tx)\}$. Suppose that $D$ is a nonexpansive retract of $E$ and that, for each $v\in D$ and $t\in (0,1)$, the contraction $S_t$ defined by $S_tx=tP_Tx+(1 - t)v$ has a fixed point $x_t\in D$. Let $\{\alpha _n\},\{\beta _n\}$ and $\{\gamma _n\}$ be three real sequences in (0,1) satisfying approximate conditions. Then, for fixed $u\in D$ and arbitrary $x_{0}\in D$, the sequence $\{x_n\}$ generated by $$x_n\in \alpha _nu+\beta _nx_{n-1}+\gamma _nP_T(x_n), \quad n\ge 0,$$ converges strongly to a fixed point of $T$.

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H05Monotone operators (with respect to duality) and generalizations
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
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References:
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