## 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.

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems 46B20 Geometry and structure of normed linear spaces
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