In this paper, we consider a generalized nonlinear variational inequality problem involving single valued and multivalued nonlinear operators. We also study criteria of its solvability. Iterative methods for approximate solution are also proposed and a convergence result is established. Further, we study iterative methods for finding common element of fixed point set of nonexpansive mapping and...

This paper deals with the characterization of solutions for vector equilibrium problems by means of conjugate duality. By using the Fenchel duality we establish variational principles, that is, optimization problems with set-valued objective functions, the solution sets of which contain the ones of the vector equilibrium problems. The set-valued objective mappings depend on the data, but not on...

The aim of this paper is to show connections between the sensitivity of the solutions of variational inequalitites and properties of the solution sets of vector variational inequalities in the infinite dimensional setting. We will show that, in some cases, the solution set of a vector variational inequality is a bijective and continuous image of a connected subset of the surface of the unit bal...

Since the fundamental theory of variational inequality was founded in the 1960s, the variational inequality theory with applications has made powerful progress and has become an important part of nonlinear analysis. It has been applied intensively to mechanics, differential equation, cybernetics, quantitative economics, optimization theory, nonlinear programming, and so forth (see [2]). In virt...

In this paper, we introduce the extragradient method for finding a common element of the set of solutions of generalized mixed equilibrium problem, the set of common fixed point of family of nonexpansive mappings the set of variational inequality for monotone, Lipschitz continuous mapping in a Hilbert space. Then we prove the strong convergence of iterative algorithm to a common element of this...

We construct a new iterative scheme by hybrid methods and prove strong convergence theorem for approximation of a common fixed point of two countable families of closed relatively quasinonexpansive mappings which is also a solution to a system of equilibrium problems in a uniformly smooth and strictly convex real Banach space with Kadec-Klee property using the properties of generalized f -proje...

The purpose of this paper is, by using a new hybrid method, to prove a strong convergence theorem for finding a common element of the set of solutions for a generalized mixed equilibrium problem, the set of solutions for a variational inequality problem, and the set of common fixed points for a pair of quasi-φ-asymptotically nonexpansive mappings. Under suitable conditions some strong convergen...