In this paper, we propose a split nonconvex VARIATIONAL inequality problem which is a natural extension of split convex VARIATIONAL inequality problem in two different Hilbert spaces. Relying on the prox-regularity notion, we introduce and establish the convergence of an iterative method for the new split nonconvex VARIATIONAL inequality problem. Further, we also establish the convergence of an iterative method for the split convex VARIATIONAL inequality problem. The results presented in this paper are new and different form the previously known results for nonconvex (convex) VARIATIONAL inequality problems. These results also generalize, unify, and improve the previously known results of this area.

In this paper, we introduce a new iterative algorithm for approximating a common solution of certain class of multiple{sets split VARIATIONAL inequality problems. The sequence of the proposed iterative algorithm is proved to converge strongly in Hilbert spaces. As application, we obtain some strong convergence results for some classes of multiple{sets split convex minimization problems.

The problem of generalized equilibrium problem is very general in the different subjects. Optimization problems, VARIATIONAL inequalities, Nash equilibrium problem and minimax problems are as special cases of generalized equilibrium problem. The purpose of this paper is to investigate the problem of approximating a common element of the set of generalized equilibrium problem, VARIATIONAL inequality problem and fixed point problem. In this article, a new iterative algorithm is introduced based on theextragradient method. Under suitable conditions, a weak convergence theorem for finding a common solution of a generalized equilibrium problem, a VARIATIONAL inequality problem and the set of fixed points of a finite family of strictly pseudo contraction mappings is proved. Our results improve and generalize some recent results in the literature. Finally, we give a numerical example to show the validity of the results.

In this paper, we introduce a new type of modified generalized α,-nonexpansive mapping and establish some fixed point properties and demiclosedness principle for this class of mappings in the framework of uniformly convex Banach spaces. We further propose a new iterative method for approximating a common fixed point of two modified generalized α,-nonexpansive mappings and present some weak and strong convergence theorems for these mappings in uniformly convex Banach spaces. In addition, we apply our result to solve a convex-constrained minimization problem, VARIATIONAL inequality and split feasibility problem and present some numerical experiments in infinite dimensional spaces to establish the applicability and efficiency of our proposed algorithm. The obtained results in this paper improve and extend some related results in the literature.

We introduce a new iterative algorithm based on a viscosity approximation method for finding the common solution of VARIATIONAL inequality problems for an inverse strongly accretive operator and the solution of fixed point problems for Lipschitzian semigroup mappings in Banach spaces. In controlling suitable conditions, strong convergence theorems are proven. Our results extend and improve the recent results of some authors in the literature in this field.

In this article, we prove some properties of a demicontractive mapping defined on a nonempty closed convex subset of a Hilbert space. By using these properties, we obtain strong convergence theorems of a hybrid shrinking projection method for finding a common element of the set of common fixed points of a finite family of demicontractive mappings and the set of common solutions of a finite family of VARIATIONAL inequality problems in a Hilbert space. A numerical example is presented to illustrate the proposed method and convergence result. Our results improve and extend the corresponding results existing in the literature.

We introduce a general implicit algorithm for finding a common element of the set of solutions of systems of equilibrium problems and the set of common fixed points of a sequence of nonexpansive mappings and a continuous representation of nonexpansive mappings. Then we prove the strong convergence of the proposed implicit scheme to the unique solution of the minimization problem on the solution of systems of equilibrium problems and the common fixed points of a sequence of nonexpansive mappings and a continuous representation of nonexpansive mappings.

In this paper we prove that if X is a Banach space, then for every lower semi-continuous bounded below function f; there exists a ('1; '2)-convex function g; with arbitrarily small norm, such that f + g attains its strong minimum on X: This result extends some of the well-known varitional principles as that of Ekeland [On the VARIATIONAL principle, J. Math. Anal. Appl. 47 (1974) 323{ 353], that of Borwein-Preiss [A smooth VARIATIONAL principle with applications to subdi erentiability and to di erentiability of convex functions, Trans. Amer. Math. Soc. 303 (1987) 517{527] and that of Deville-Godefroy-Zizler [Un principe variationel utilisant des fonctions bosses, C. R. Acad. Sci. (Paris). Ser. I 312 (1991) 281{286] and [A smooth VARIATIONAL principle with applications to Hamilton-Jacobi equations in in nite dimensions, J. Funct. Anal. 111 (1993) 197{212].

In this paper, we introduce the notion of f-weakly compatible mapping for a pair of mappings. A fixed point theorem for two pairs of f- weakly compatible mappings satisfying a rational type contraction in a normed space is also established. Subsequently we use our result to find existence of solutions of VARIATIONAL inequalities.

In this paper, we consider different types of generalized vector VARIATIONAL-like inequalities and study the relationships between their  olutions . We study the general forms of Stampacchia and Minty type vector VARIATIONAL inequalities for bifunctions and establish the existence of their solutions in the setting of topological vector spaces. We extend these vector VARIATIONAL inequalities for the Clarke’s sub differential of non-differentiable locally Lipschitz functions and prove the existence of their solutions. As applications, we establish some existence results for a solution of the vector optimization problem by using Stampacchia and Minty type vector VARIATIONAL inequalities.