Convergence Theorems For Approximating Research Articles

On Halpern-type sequences with applications in variational inequality problems

We introduce a Halpern-type sequence and give a necessary and sufficient condition for a strong convergence of this sequence. In particular, we obtain two strong convergence theorems for approximation of a fixed point of nonexpansive mappings or of quasi-nonexpansive ones. We also apply our result for various iterative methods in variational inequality problem. For the Lipschitz continuous mappings, we deal with the extragradient method of Korpelevič, the subgradient extragradient method of Censor et al. and the extragradient of Tseng where the step size rule is priorly or posteriorly chosen. For the non-Lipschitz continuous mappings, we use our results to deduce the convergence results of Shehu and Iyiola and of Thong and Gibali. Our approach allows us to conclude many new results with some new assumptions.

A strong convergence theorem for approximation of a zero of the sum of two maximal monotone mappings in Banach spaces

The purpose of this paper is to study the method of approximation for a zero of the sum of two maximal monotone mappings in Banach spaces and prove strong convergence of the proposed method under suitable conditions. The method of proof is of independent interest. In addition, we give some applications to the minimization problems. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

A new convergence theorem for families of asymptotically nonexpansive maps and solution of variational inequality problem

A new strong convergence theorem for approximation of common fixed points of family of uniformly asymptotically regular asymptotically nonexpansive mappings, which is also a unique solution of some variational inequality problem is proved in the framework of a real Banach space. The Theorem presented here extend, generalize and unify many recently announced results.

Convergence Theorems for Equilibrium and Fixed Point Problems

Our purpose in this paper is to prove strong convergence theorems for approximation of a fixed point of a left Bregman strongly relatively nonexpansive mapping which is also a solution to a finite system of equilibrium problems in the framework of reflexive real Banach spaces. We also discuss the approximation of a common fixed point of a family of left Bregman strongly nonexpansive mappings which is also solution to a finite system of equilibrium problems in reflexive real Banach spaces. Our results complement many known recent results in the literature.

General algorithms for split common fixed point problem of demicontractive mappings

In the first part of this paper, we present a new general algorithm for solving the split common fixed point problem for an infinite family of demicontractive mappings. We establish strong convergence of the algorithm in an infinite dimensional Hilbert space. As applications, we consider algorithms for split variational inequality problem and split common null point problem. In the second part of this paper, we present a new algorithm and strong convergence theorem for approximation of solutions of split equality fixed point problems for an infinite family of demicontractive mappings. Our results improve and generalize some recent results in the literature.

Approximation of minimum-norm fixed point of total asymptotically nonexpansive mapping

In this paper, strong convergence theorem for approximation of minimum norm fixed point of total asymptotically nonexpansive mapping in real Hilbert spaces are obtained. As applications, iterative methods for approximation of minimum norm fixed point of continuous pseudocontractive mapping, approximation of solutions of classical equilibrium problems and approximation of solutions of convex minimization problems are proposed. Furthermore, iterative method for approximation of of common minimum norm fixed point of finite family of total asymptotically nonexpansive mapping is proposed. Our theorems unify and complement many recently annouced results.

Approximation of common solutions for system of equilibrium problems and fixed-point problems

In this paper, we complement the result of Zhu and Chang (J Ineq Appl 2013:146, 2013) by proving strong convergence theorems for approximation of a fixed point of a left Bregman strongly relatively nonexpansive mapping which is also a solution to a finite system of equilibrium problems in the framework of reflexive real Banach spaces using the Halpern–Mann’s iterations used in Zhu and Chang (J Ineq Appl 2013:146, 2013). We also discuss the approximation of a common fixed point of a family of left Bregman strongly nonexpansive mappings which is also solution to a finite system of equilibrium problems in reflexive real Banach spaces. Our results complement many known recent results in the literature.

A new iterative scheme for a countable family of relatively nonexpansive mappings and an equilibrium problem in Banach spaces

In this paper, we construct a new iterative scheme and prove strong convergence theorem for approximation of a common fixed point of a countable family of relatively nonexpansive mappings, which is also a solution to an equilibrium problem in a uniformly convex and uniformly smooth real Banach space. We apply our results to approximate fixed point of a nonexpansive mapping, which is also solution to an equilibrium problem in a real Hilbert space and prove strong convergence of general H-monotone mappings in a uniformly convex and uniformly smooth real Banach space. Our results extend many known recent results in the literature.

A Modified Averaging Composite Implicit Iteration Process for Common Fixed Points of a Finite Family of k-Strictly Asymptotically Pseudocontractive Mappings

The composite implicit iteration process introduced by Su and Li [J. Math. Anal. Appl. 320 (2006) 882-891] is modified. A strong convergence theorem for approximation of common fixed points of finite family of k-strictly asymptotically pseudo-contractive mappings is proved in Banach spaces using the modified iteration process.

Strong Convergence Theorems for Families of Weak Relatively Nonexpansive Mappings

We construct a new Halpern type iterative scheme by hybrid methods and prove strong convergence theorem for approximation of a common fixed point of two countable families of weak relatively nonexpansive mappings in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f‐projection operator. Using this result, we discuss strong convergence theorem concerning general H‐monotone mappings. Our results extend many known recent results in the literature.

Iterative Approximation of a Common Zero of a Countably Infinite Family of -Accretive Operators in Banach Spaces

Let E be a real reflexive and strictly convex Banach space which has a uniformly Gâteaux differentiable norm and let C be a closed convex nonempty subset of E. Strong convergence theorems for approximation of a common zero of a countably infinite family of m-accretive mappings from C to E are proved. Consequently, we obtained strong convergence theorems for a countably infinite family of pseudocontractive mappings.

Strong Convergence of an Implicit Iteration Algorithm for a Finite Family of Pseudocontractive Mappings

Strong convergence theorems for approximation of common fixed points of a finite family of pseudocontractive mappings are proven in Banach spaces using an implicit iteration scheme. The results presented in this paper improve and extend the corresponding results of Osilike, Xu and Ori, Chidume and Shahzad, and others.