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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.

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.

Reviewer: Vasile Berinde (Baia Mare)

### 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 |