Abstract:
In this paper, we introduce a new approximation scheme based on the extragradient method and viscosity method for finding a common element of the set of solutions of the set of fixed points of a nonexpansive mapping and the set of the variational inequality for a monotone, Lipschitz continuous mapping. We obtain a strong convergence theorem for the sequences generated by these processes in Hilbert spaces as follows: Let C be a nonempty closed convex subset of a real Hilbert space H. Let A be a monotone and k-Lipschitz continuous mapping of C into H. Let S be a nonexpansive mapping of C into H such that , where and , respectively, denote the set of fixed point of S and the solution set of a variational inequality. Let f be a contraction of H into itself and and be sequences generated by for every n=1,2,…, where and are sequences of numbers satisfying and and . Then, and converge strongly to The results in this paper improve some well-known results in the literature.

Abstract:
A new and interesting model of system of generalized set-valued equilibrium problems which generalizes and unifies the system of set-valued equilibrium problems, the system of generalized implicit vector variational inequalities, the system of generalized vector and vector-like variational inequalities in [1], the system of generalized vector variational inequalities in [2], the system of vector equilibrium problems and the system of vector variational inequalities in [3], the system of scalar variational inequalities in [4,5,9,15,23,28], the system of Ky-Fan variational inequalities in [16] as well as variety of the equilibrium problems in literatures will be introduced, and several existence results of a solution for the system of generalized set-valued equilibrium problems will be shown.

Abstract:
We introduce new and interesting model of system of generalized set-valued equilibrium problems which generalizes and unifies the system of set-valued equilibrium problems, the system of generalized implicit vector variational inequalities, the system of generalized vector and vector-like variational inequalities introduced by Ansari et al. (2002), the system of generalized vector variational inequalities presented by Allevi et al. (2001), the system of vector equilibrium problems and the system of vector variational inequalities given by Ansari et al. (2000), the system of scalar variational inequalities presented by Ansari Yao (1999, 2000), Bianchi (1993), Cohen and Caplis (1988), Konnov (2001), and Pang (1985), the system of Ky-Fan variational inequalities proposed bt Deguire et al. (1999) as well as a variety of equilibrium problems in the literature. Several existence results of a solution for the system of generalized set-valued equilibrium problems will be shown.

Abstract:
In this article, we introduce some new iterative schemes based on the extragradient method (and the hybrid method) for finding a common element of the set of solutions of a generalized equilibrium problem, and the set of fixed points of a family of infinitely nonexpansive mappings and the set of solutions of the variational inequality for a monotone, Lipschitz-continuous mapping in Hilbert spaces. We obtain some strong convergence theorems and weak convergence theorems. The results in this article generalize, improve, and unify some well-known convergence theorems in the literature.

Abstract:
We introduce a new system of generalized vector quasiequilibrium problems which includes system of vector quasiequilibrium problems, system of vector equilibrium problems, and vector equilibrium problems, and so forth in literature as special cases. We prove the existence of solutions for this system of generalized vector quasi-equilibrium problems. Consequently, we derive some existence results of a solution for the system of generalized quasi-equilibrium problems and the generalized Debreu-type equilibrium problem for both vector-valued functions and scalar-valued functions.

Abstract:
The concepts of -well-posedness, -well-posedness in the generalized sense, L--well-posedness and L--well-posedness in the generalized sense for mixed quasi variational-like inequality problems are investigated. We present some metric characterizations for these well-posednesses.

Abstract:
We introduce an Ishikawa iterative scheme by the viscosity approximate method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in Hilbert space. Then, we prove some strong convergence theorems which extend and generalize S. Takahashi and W. Takahashi's results (2007). 1. Introduction Let be a real Hilbert space and let be a nonempty closed convex subset of . Let be a bifunction from to , where is the set of real numbers. The equilibrium problem for is to find such that The set of solutions of (1.1) is denoted by . Given a mapping , let for all . Then, if and only if for all . Numerous problems in physics, optimization, and economics reduce to find a solution of (1.1); for more details, see [1, 2]. Recall that a self-mapping of a closed convex subset of is nonexpansive [3] if there holds that We denote the set of fixed points of by . There are some methods for approximation of fixed points of a nonexpansive mapping. In 2000, Moudafi [4] introduced the viscosity approximation method for nonexpansive mappings (see [5] for further developments in both Hilbert and Banach spaces). Some methods have been proposed to solve the equilibrium problem; see, for instance, [1, 2, 6, 7]. Recently, Combettes and Hirstoaga [6] introduced an iterative scheme of finding the best approximation to the initial data when is nonempty and proved a strong convergence theorem. S. Takahashi and W. Takahashi [7] introduced a Mann iterative scheme by the viscosity approximation method for finding a common element of the set of solution (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert space and proved a strong convergence theorem. On the other hand, Ishikawa [8] introduced the following iterative process defined recursively by where the initial guess is taking in arbitrarily, and are sequences in the interval . In this paper, motivated by the ideas in [4–8], we introduce an Ishikawa iterative scheme by the viscosity approximation method for finding a common element of the set of solution (1.1) and the set of fixed points of a nonexpansive mapping in a Hilbert space. Starting with an arbitrary , define sequences , and by where and . We will prove in Section 3 that if the sequences , and of parameters satisfy appropriate conditions, then the sequences , and generated by (1.4) converge strongly to . The results in this paper extend and generalize S. Takahashi and W. Takahashi's results [7]. 2. Preliminaries Let be a real Hilbert space with inner product , and norm and let be a

Abstract:
We introduce several types of the Levitin-Polyak well-posedness for a generalized vector quasivariational inequality problem with both abstract set constraints and functional constraints. Criteria and characterizations of these types of the Levitin-Polyak well-posednesses with or without gap functions of generalized vector quasivariational inequality problem are given. The results in this paper unify, generalize, and extend some known results in the literature.

Abstract:
The article generalizes the strict efficient solution of multiobjective programming problem (MOP) to Φ strict local efficient solution concept，thus pertains to characterize Φ strict local efficient solution for multiobjective programming problems (MOP) with inequality constraints.To create the necessary framework，we partition the index set of objective of MOP to give rise to subproblem (RMOPJB((P<α,δJB((xJB)),xTX-0.5mmJB))).The Φstrict local efficient solution (Φsles) for MOP is related to the local efficient solution of a subproblem (RMOPJB((P<α,δJB((xJB)),xTX-0.5mmJB)))，having lesser number of objective functions，with MOP.This paper will discuss their relationship.We will through the theorem discuss their relationship.We also generalize the strong convex function；put forward a new concept of convex function —Φstrongly convex function，and by the strong convex function and KKT conditions to characterize Φstrict local efficient solution for MOP.

Abstract:
We first introduce a new notion of the partial and generalized cone subconvexlike set-valued map and give an equivalent characterization of the partial and generalized cone subconvexlike set-valued map in linear spaces. Secondly, a generalized alternative theorem of the partial and generalized cone subconvexlike set-valued map was presented. Finally, Kuhn-Tucker conditions of set-valued optimization problems were established in the sense of globally proper efficiency.