Ngai-Ching Wong

Equilibrium Problems with Applications to Eigenvalue Problems

In this paper, we consider equilibrium problems and introduce the concept of (S)+ condition for bifunctions. Existence results for equilibrium problems with the (S)+ condition are derived. As special cases, we obtain several existence results for the generalized nonlinear variational inequality studied by Ding and Tarafdar (Ref. 1) and the generalized variational inequality studied by Cubiotti and Yao (Ref. 2). Finally, applications to a class of eigenvalue problems are given.

Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems

The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points F(S) of a nonexpansive mapping S and the set of solutions ΩA of the variational inequality for a monotone, Lipschitz continuous mapping A. We introduce a hybrid extragradient-like approximation method which is based on the well-known extragradient method and a hybrid (or outer approximation) method. The method produces three sequences which are shown to converge strongly to the same common element of

Maps preserving the spectrum of generalized Jordan product of operators

{F(S)\cap\Omega_{A}}

A Unified Hybrid Iterative Method for Solving Variational Inequalities Involving Generalized Pseudocontractive Mappings

. As applications, the method provides an algorithm for finding the common fixed point of a nonexpansive mapping and a pseudocontractive mapping, or a common zero of a monotone Lipschitz continuous mapping and a maximal monotone mapping.

A Generalized Hybrid Steepest-Descent Method for Variational Inequalities in Banach Spaces

The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points of an asymptotically strict pseudocontractive mapping in the intermediate sense and the set of solutions of a variational inequality problem for a monotone and Lipschitz continuous mapping. We introduce an extragradient-like iterative algorithm that is based on the extragradient-like approximation method and the modified Mann iteration process. We establish a strong convergence theorem for two sequences generated by this extragradient-like iterative algorithm. Utilizing this theorem, we also design an iterative process for finding a common fixed point of two mappings, one of which is an asymptotically strict pseudocontractive mapping in the intermediate sense and the other taken from the more general class of Lipschitz pseudocontractive mappings.1991 MSC: 47H09; 47J20.

Subgradients of value functions in parametric dynamic programming

Abstract Let A 1 , A 2 be standard operator algebras on complex Banach spaces X 1 , X 2 , respectively. For k ⩾ 2 , let ( i 1 , … , i m ) be a sequence with terms chosen from { 1 , … , k } , and define the generalized Jordan product T 1 ∘ ⋯ ∘ T k = T i 1 ⋯ T i m + T i m ⋯ T i 1 on elements in A i . This includes the usual Jordan product A 1 ∘ A 2 = A 1 A 2 + A 2 A 1 , and the triple { A 1 , A 2 , A 3 } = A 1 A 2 A 3 + A 3 A 2 A 1 . Assume that at least one of the terms in ( i 1 , … , i m ) appears exactly once. Let a map Φ : A 1 → A 2 satisfy that σ ( Φ ( A 1 ) ∘ ⋯ ∘ Φ ( A k ) ) = σ ( A 1 ∘ ⋯ ∘ A k ) whenever any one of A 1 , … , A k has rank at most one. It is shown in this paper that if the range of Φ contains all operators of rank at most three, then Φ must be a Jordan isomorphism multiplied by an m th root of unity. Similar results for maps between self-adjoint operators acting on Hilbert spaces are also obtained.

Degree theory for generalized variational inequalities and applications

Abstract In this paper, we introduce a new class of generalized co-complementarity problems in Banach spaces. An iterative algorithm for finding approximate solutions of these problems is considered. Some convergence results for this iterative algorithm are derived and several existence results are obtained.

Isometric shifts on C0(X)

We study in this paper the existence and the approximation of solutions of variational inequalities involving generalized pseudocontractive mappings in Banach spaces. The convergence analysis of a proposed hybrid iterative method for approximating common zeros or fixed points of a possibly infinitely countable or uncountable family of such operators will be conducted within the conceptual framework of the “viscosity approximation technique” in reflexive Banach spaces with uniform Gâteaux differentiable norms. This technique should make existing or new results in solving variational inequalities more applicable.