Yeong-Cheng Liou

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.

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 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 ) .

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.

An Extragradient Method for Fixed Point Problems and Variational Inequality Problems

We present an extragradient method for fixed point problems and variational inequality problems. Using this method, we can find the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for monotone mapping.

Iterative Algorithms for General Multivalued Variational Inequalities

We introduce and study some new classes of variational inequalities and the Wiener-Hopf equations. Using essentially the projection technique, we establish the equivalence between these problems. This equivalence is used to suggest and analyze some iterative methods for solving the general multivalued variational in equalities in conjunction with nonexpansive mappings. We prove a strong convergence result for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general multivalued variational inequalities under some mild conditions. Several special cases are also discussed.

Strong Convergence of a Modified Extragradient Method to the Minimum-Norm Solution of Variational Inequalities

We suggest and analyze a modified extragradient method for solving variational inequalities, which is convergent strongly to the minimum-norm solution of some variational inequality in an infinite-dimensional Hilbert space.

Strong convergence of a self-adaptive method for the split feasibility problem

Self-adaptive methods which permit step-sizes being selected self-adaptively are effective methods for solving some important problems, e.g., variational inequality problems. We devote this paper to developing and improving the self-adaptive methods for solving the split feasibility problem. A new improved self-adaptive method is introduced for solving the split feasibility problem. As a special case, the minimum norm solution of the split feasibility problem can be approached iteratively.MSC:47J25, 47J20, 49N45, 65J15.

Hierarchical convergence of an implicit double-net algorithm for nonexpansive semigroups and variational inequality problems

AbstractIn this paper, we show the hierarchical convergence of the following implicit double-net algorithm: xs,t=s[tf(xs,t)+(1-t)(xs,t-μAxs,t)]+(1-s)1λs∫0λsT(v)xs,tdν,∀s,t∈(0,1), where f is a ρ-contraction on a real Hilbert space H, A : H → H is an α-inverse strongly monotone mapping and S = {T(s)}s ≥ 0: H → H is a nonexpansive semi-group with the common fixed points set Fix(S) ≠ ∅, where Fix(S) denotes the set of fixed points of the mapping S, and, for each fixed t ∈ (0, 1), the net {xs, t} converges in norm as s → 0 to a common fixed point xt ∈ Fix(S) of {T(s)}s ≥ 0and, as t → 0, the net {xt} converges in norm to the solution x* of the following variational inequality: x*∈Fix(S);〈Ax*,x-x*〉≥0,∀x∈Fix(S).MSC(2000): 49J40; 47J20; 47H09; 65J15.