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A unified hybrid iterative method for solving variational inequalities involving generalized pseudocontractive mappings. (English) Zbl 1262.47091
Let $C$ be a closed subset of a smooth Banach space, $D$ a nonempty closed convex subset of $C$ and let $J: X\rightarrow X^*$ be the normalized duality mapping. Let $\mathcal{F}:C\rightarrow X$ be such that $A=I-\mathcal{F}$ (with $I$ denoting the identity operator) is a generalized $\Phi$-pseudocontractive nonlinear operator with respect to a strictly increasing function $\Phi: [0,\infty)\rightarrow [0,\infty)$ satisfying $\Phi(0)=0$. The main aim of the paper is to approximate solutions of the following generalized variational inequality: $$ \text{find } z\in D \text{ such that } \left\langle\mathcal{F},J(z-v)\right\rangle\leq 0, \text{ for all } v\in D, $$ in the case where $D$ is the set of fixed points, $F(\mathcal{T} )$, of a certain family of mappings. Let $G$ denote an unbounded subset of $\mathbb{R}^{+}$. The most important cases of $D$ are: (1) $D=F(\mathcal{T} )$, where $\mathcal{T} =\{T_s:s\in G\}$ is a family of continuous pseudocontractive mappings (Theorem 3.2). In this case, a Krasnoselskij type net $\{y_s\}_{s\in G}$ is used to approximate the solution. (2) $D=F(\mathcal{T} )$, where $\mathcal{T} =\{T_s:s\in G\}$ is a family of pseudocontractive and nearly uniformly $L$-Lipschitzian mappings (Theorem 4.2), in which case a hybrid iterative algorithm $\{x_n\}$ of the form $$ x_{n+1}=\left(1-\lambda_n (1+\theta_n)\right) x_n+\lambda_n T_{s_n}x_n+\lambda \theta_n A x_n,\, n\geq 0, $$ is considered ($s_n\in G$ is such that $\lim\limits_{n \rightarrow \infty}s_n=+\infty$). A parallel algorithm is also considered in the last section of the paper in order to avoid the assumption of commutativity of the mappings in $\mathcal{T} $: $$ F(T_1T_2\dots T_N)=F(T_NT_1\dots T_{N-1})=\dots =F(T_2T_3\dots T_N T_1), $$ an assumption that is used in proving the preceding convergence results.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
46B20Geometry and structure of normed linear spaces
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