A Class of Pseudo-Monotone Operators and its Applications in PDE

In this paper we give a criterion of pseudo-monotone operators by which a class of operators related to elliptic partial differential equations is recognized as pseudo-monotone operators. Some applications in boundary value problems for elliptic equations and obstacje problems are given.

SYSTEM OF GENERALIZED VECTOR VARIATIONAL INEQUALITIES

In this paper, the authors introduce and study system of generalized vector variational inequalities. Under suitable conditions, the existence of solutions for system of generalized vector variational inequalities is presented by Kakutani-Fan-Glicksberg fixed point theorem.

Hybrid Extragradient-Type Methods for Finding a Common Solution of an Equilibrium Problem and a Family of Strict Pseudo-Contraction Mappings

This paper proposes a new hybrid variant of extragradient methods for finding a common solution of an equilibrium problem and a family of strict pseudo-contraction mappings. We present an algorithmic scheme that combine the idea of an extragradient method and a successive iteration method as a hybrid variant. Then, this algorithm is modified by projecting on a suitable convex set to get a better convergence property. The convergence of two these algorithms are investigated under certain assumptions.

Generalized Quasi Variational-Type Inequalities

In this paper, we define the concepts of (η,h)-quasi pseudo-monotone operators on compact set in locally convex Hausdorff topological vector spaces and prove the existence results of solutions for a class of generalized quasi variational type inequalities in locally convex Hausdorff topological vector spaces.

Sub-Differential Characterizations of Non-Smooth Lower Semi-Continuous Pseudo-Convex Functions on Real Banach Spaces

In this paper, we characterize lower semi-continuous pseudo-convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the pseudo-monotonicity of its Clarke-Rockafellar Sub-differential. We extend the results on the characterizations of non-smooth convex functions f : X → R ∪ {+ ∞} on convex subset of real Banach spaces K ⊂ X with respect to the monotonicity of its sub-differentials to the lower semi-continuous pseudo-convex functions on real Banach spaces.

MODIFIED STOCHASTIC EXTRAGRADIENT METHODS FOR STOCHASTIC VARIATIONAL INEQUALITY

In this paper,we consider two kinds of extragradient methods to solve the pseudo-monotone stochastic variational inequality problem.First,we present the modified stochastic extragradient method with constant step-size(MSEGMC)and prove the convergence of it.With the strong pseudo-monotone operator and the exponentially growing sample sequences,we establish the R-linear convergence rate in terms of the mean natural residual and the oracle complexity O(1/ǫ).Second,we propose a modified stochastic extragradient method with adaptive step-size(MSEGMA).In addition,the step-size of MSEGMA does not depend on the Lipschitz constant and without any line-search procedure.Finally,we use some numerical experiments to verify the effectiveness of the two algorithms.

Existence of Solutions to Nonlinear Neumann Boundary Value Problems with p-Laplacian Operator and Iterative Construction

By using some results of pseudo-monotone operator, we discuss the existence and uniqueness of the solution of one kind nonlinear Neumann boundary value problems involving the p-Laplacian operator. We also construct an iterative scheme converging strongly to this solution.

The Keldys-Fichera Obstacle Problem

In this paper we consider a quasilinear obstacle problem with coefficients of higher growth orders and the Keldys-Fichera boundary condition.