Strong convergence of viscosity approximation methods for the fixed-point of pseudo-contractive and monotone mappings

Abstract

In this paper, we introduce a viscosity iterative process, which converges strongly to a common element of the set of fixed points of a pseudo-contractive mapping and the set of solutions of a monotone mapping. We also prove that the common element is the unique solution of certain variational inequality. The strong convergence theorems are obtained under some mild conditions. The results presented in this paper extend and unify most of the results that have been proposed for this class of nonlinear mappings.

The Hybrid Projection Methods for Pseudocontractive, Nonexpansive Semigroup, and Monotone Mapping

“…In this paper, we modify the three-step iterative schemes to prove the strong convergence theorems by using the hybrid projection methods for finding a common element of the set of solutions of fixed points for a pseudocontractive mapping and a nonexpansive semigroup mapping and the set of solutions of a variational inequality problem for a monotone mapping in a Hilbert space under some appropriate control conditions. The results that are presented in this paper extend and improve the corresponding ones announced by Nakajo and Takahashi [25], Takahashi et al [26], Zegeye and Shahzad [27], Tang [28], and many authors.…”

“…for all ∈ and ∈ (0, ∞), where : → is a continuous pseudocontractive mapping and : → is a continuous monotone mapping. In the following year, Tang [28] introduced a viscosity iterative process, which converges strongly to a common element of the set of fixed points of a pseudocontractive mapping and the set of solutions of a monotone mapping as the following:…”

Strong convergence of some iterative algorithms for a general system of variational inequalities

“…The variational inequality problem has been extensively studied in the literature; see [3,5,7,9,10,13,14,16,18] and the references therein. In 2008, Ceng et al [2] considered the following general system of variational inequalities:…”

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

“…Recently viscosity approximation methods for finding fixed points of pseudocontractive mappings have received vast investigations because of their extensive applications in a variety of applied areas of partial differential equations, image recovery, and signal processing. In Hilbert spaces, many authors have studied the fixed-point problems of the nonexpansive mappings and monotone mappings by the viscosity approximation methods and obtained a series of good results; see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] and the reference therein.…”

“…For other related results, see [11][12][13][23][24][25]. On the other side, there are perturbations always occurring in the iterative processes because the manipulations are inaccurate.…”