Yonghong Yao

On modified iterative method for nonexpansive mappings and monotone mappings

Abstract In this paper, we introduce a new iterative scheme for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for α-inverse-strongly monotone mappings. We show that the sequence converges strongly to a common element of the above two sets under some parameters controlling conditions.

Convergence Theorem for Equilibrium Problems and Fixed Point Problems of Infinite Family of Nonexpansive Mappings

We introduce an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of common fixed points of infinite nonexpansive mappings in a Hilbert space. We prove a strong-convergence theorem under mild assumptions on parameters.

Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems☆

In this paper, we present an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem and the set of fixed points of an infinite family of nonexpansive mappings and the set of a variational inclusion in a real Hilbert space. Furthermore, we prove that the proposed iterative algorithm has strong convergence under some mild conditions imposed on algorithm parameters.

Regularized Methods for the Split Feasibility Problem

Many applied problems such as image reconstructions and signal processing can be formulated as the split feasibility problem (SFP). Some algorithms have been introduced in the literature for solving the (SFP). In this paper, we will continue to consider the convergence analysis of the regularized methods for the (SFP). Two regularized methods are presented in the present paper. Under some different control conditions, we prove that the suggested algorithms strongly converge to the minimum norm solution of the (SFP).

Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method

In this paper, we introduce an iterative method based on the extragradient method for finding a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping in a real Hilbert space. Furthermore, we prove that the studied iterative method strongly converges to a common element of the set of a general system of variational inequalities and the set of fixed points of a strictly pseudocontractive mapping under some mild conditions imposed on algorithm parameters.

A Hybrid Projection Algorithm for Finding Solutions of Mixed Equilibrium Problem and Variational Inequality Problem

We propose a modified hybrid projection algorithm to approximate a common fixed point of a -strict pseudocontraction and of two sequences of nonexpansive mappings. We prove a strong convergence theorem of the proposed method and we obtain, as a particular case, approximation of solutions of systems of two equilibrium problems.

On a Two-Step Algorithm for Hierarchical Fixed Point Problems and Variational Inequalities

A common method in solving ill-posed problems is to substitute the original problem by a family of well-posed (i.e., with a unique solution) regularized problems. We will use this idea to define and study a two-step algorithm to solve hierarchical fixed point problems under different conditions on involved parameters.

A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem

Abstract Let C be a nonempty closed convex subset of a real Hilbert space H . Let f : C → H be a ρ -contraction. Let S : C → C be a nonexpansive mapping. Let B , B ˜ : H → H be two strongly positive bounded linear operators. Consider the triple-hierarchical constrained optimization problem of finding a point x ∗ such that x ∗ ∈ Ω , 〈 ( B ˜ − γ f ) x ∗ − ( I − B ) S x ∗ , x − x ∗ 〉 ≥ 0 , ∀ x ∈ Ω , where Ω is the set of the solutions of the following variational inequality: x ∗ ∈ E P ( F , A ) , 〈 ( B ˜ − S ) x ∗ , x − x ∗ 〉 ≥ 0 , ∀ x ∈ E P ( F , A ) , where E P ( F , A ) is the set of the solutions of the equilibrium problem of finding z ∈ C such that F ( z , y ) + 〈 A z , y − z 〉 ≥ 0 , ∀ y ∈ C . Assume Ω ≠ 0 . The purpose of this paper is the solving of the above triple-hierarchical constrained optimization problem. For this purpose, we first introduce an implicit double-net algorithm. Consequently, we prove that our algorithm converges hierarchically to some element in E P ( F , A ) which solves the above triple-hierarchical constrained optimization problem. As a special case, we can find the minimum norm x ∗ ∈ E P ( F , A ) which solves the monotone variational inequality 〈 ( I − S ) x ∗ , x − x ∗ 〉 ≥ 0 , ∀ x ∈ E P ( F , A ) .

Strong convergence of a proximal point algorithm with general errors

In this paper we construct a proximal point algorithm for maximal monotone operators with appropriate regularization parameters. We obtain the strong convergence of the proposed algorithm, which affirmatively answer the open question put forth by Boikanyo and Morosanu (Optim Lett 4:635–641, 2010).

A New Hybrid Iterative Algorithm for Fixed-Point Problems, Variational Inequality Problems, and Mixed Equilibrium Problems

We introduce a new hybrid iterative algorithm for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of the variational inequality of a monotone mapping, and the set of solutions of a mixed equilibrium problem. This study, proves a strong convergence theorem by the proposed hybrid iterative algorithm which solves fixed-point problems, variational inequality problems, and mixed equilibrium problems.