A Halpern-Type Iteration Method for Bregman Nonspreading Mapping and Monotone Operators in Reflexive Banach Spaces

In this paper, we introduce an iterative method for approximating a common solution of monotone inclusion problem and fixed point of Bregman nonspreading mappings in a reflexive Banach space. Using the Bregman distance function, we study the composition of the resolvent of a maximal monotone operator and the antiresolvent of a Bregman inverse strongly monotone operator and introduce a Halpern-type iteration for approximating a common zero of a maximal monotone operator and a Bregman inverse strongly monotone operator which is also a fixed point of a Bregman nonspreading mapping. We further state and prove a strong convergence result using the iterative algorithm introduced. This result extends many works on finding a common solution of the monotone inclusion problem and fixed-point problem for nonlinear mappings in a real Hilbert space to a reflexive Banach space.

Existence and Convergence Theorems by an Iterative Shrinking Projection Method of a Strict Pseudocontraction in Hilbert Spaces

The aim of this paper is to provide some existence theorems of a strict pseudocontraction by the way of a hybrid shrinking projection method, involving some necessary and sufficient conditions. The method allows us to obtain a strong convergence iteration for finding some fixed points of a strict pseudocontraction in the framework of real Hilbert spaces. In addition, we also provide certain applications of the main theorems to confirm the existence of the zeros of an inverse strongly monotone operator along with its convergent results.

A Penalization-Gradient Algorithm for Variational Inequalities

This paper is concerned with the study of a penalization-gradient algorithm for solving variational inequalities, namely, findx̅∈Csuch that〈Ax̅,y-x̅〉≥0for ally∈C, whereA:H→His a single-valued operator,Cis a closed convex set of a real Hilbert spaceH. GivenΨ:H→R ∪ {+∞}which acts as a penalization function with respect to the constraintx̅∈C, and a penalization parameterβk, we consider an algorithm which alternates a proximal step with respect to∂Ψand a gradient step with respect toAand reads asxk=(I+λkβk∂Ψ)-1(xk-1-λkAxk-1). Under mild hypotheses, we obtain weak convergence for an inverse strongly monotone operator and strong convergence for a Lipschitz continuous and strongly monotone operator. Applications to hierarchical minimization and fixed-point problems are also given and the multivalued case is reached by replacing the multivalued operator by its Yosida approximate which is always Lipschitz continuous.

Strong Convergence of a Hybrid Projection Algorithm for Equilibrium Problems, Variational Inequality Problems and Fixed Point Problems in a Banach Space

We introduce and study a new hybrid projection algorithm for finding a common element of the set of solutions of an equilibrium problem, the set of common fixed points of relatively quasi-nonexpansive mappings, and the set of solutions of the variational inequality for an inverse-strongly-monotone operator in a Banach space. Under suitable assumptions, we show a strong convergence theorem. Using this result, we obtain some applications in a Banach space. The results obtained in this paper extend and improve the several recent results in this area.

The Shrinking Projection Method for Solving Variational Inequality Problems and Fixed Point Problems in Banach Spaces

We consider a hybrid projection algorithm based on the shrinking projection method for two families of quasi-ϕ-nonexpansive mappings. We establish strong convergence theorems for approximating the common element of the set of the common fixed points of such two families and the set of solutions of the variational inequality for an inverse-strongly monotone operator in the framework of Banach spaces. As applications, at the end of the paper we first apply our results to consider the problem of finding a zero point of an inverse-strongly monotone operator and we finally utilize our results to study the problem of finding a solution of the complementarity problem. Our results improve and extend the corresponding results announced by recent results.