Viscosity Approximation Methods and Strong Convergence Theorems for the Fixed Point of Pseudocontractive and Monotone Mappings in Banach Spaces

Abstract

Suppose thatCis a nonempty closed convex subset of a real reflexive Banach spaceEwhich has a uniformly Gateaux differentiable norm. A viscosity iterative process is constructed in this paper. A strong convergence theorem is proved for a common element of the set of fixed points of a finite family of pseudocontractive mappings and the set of solutions of a finite family of monotone mappings. And the common element is the unique solution of certain variational inequality. The results presented in this paper extend most of the results that have been proposed for this class of nonlinear mappings.

Viscosity Approximation Methods with Errors and Strong Convergence Theorems for a Common Point of Pseudocontractive and Monotone Mappings: Solutions of Variational Inequality Problems

“…In this paper, motivated and inspired by the above results, we introduce two iterations with perturbations which converge strongly to a common element of the set of fixed points of a finite family of pseudocontractive mappings more general than nonexpansive mappings and the solution set of a finite family of monotone mappings more general thaninverse strongly monotone mappings or maximal monotone mappings. Our theorems presented in this paper improve and extend the corresponding results of Yao and Shahzad [24], Zegeye and Shahzad [21], and Tang [22] and some other results in this direction.…”

“…In particular, Theorem 6 extends Theorem 6 of Yao and Shahzad [24] in the sense that our convergence is for the more general class of continuous pseudocontractive and continuous monotone mappings. Theorem 6 also extends Theorem 3.2 of Tang [22] in the sense that our convergence is for the more general algorithm with perturbations.…”

“…Lemma 5 (see [22]). Let be a nonempty closed convex and bounded subset of a Hilbert space , and let {Γ : → , = 1, 2, .…”

“…Very recently, Tang [22] introduced the following sequence and obtained the strong convergence theorems:…”