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...

We consider an analytic center algorithm for solving generalized monotone variational inequalities in R", which adapts a recent result due to Goffin et al. (1993) to the numerical resolution of continuous pseudomonotone variational inequalities.

We introduce and study a new system of variational inclusions with (H ,η)-accretive operators, which contains variational inequalities, variational inclusions, systems of variational inequalities, and systems of variational inclusions in the literature as special cases. By using the resolvent technique for the (H ,η)-accretive operators, we prove the existence and uniqueness of solution and the...

Variational inequalities are an important mathematical tool for modelling free boundary problems that arise in different application areas. Due to the intricate nonsmooth structure of the resulting models, their analysis and optimization is a difficult task that has drawn the attention of researchers for several decades. In this paper we focus on a class of variational inequalities, called of t...

In this paper, we enlighten the role of variational inequalities for obtaining convergence rates in Tikhonov regularization of nonlinear ill-posed problems with convex penalty functionals under convexity constraints in Banach spaces. Variational inequalities are able to cover solution smoothness and the structure of nonlinearity in a uniform manner, not only for unconstrained but, as we indicat...

This paper aims at studying the convergence of some iteration processes for mixed variational inequalities with convex nondifferentiable functionals and J -potential, J -coercive, J -pseudomonotone or J -strongly inverse monotone operators in uniformly smooth Banach spaces. Our results generalize corresponding theorems of [I.B. Badriev, O.A. Zadvornov, A decomposition method for variational ine...

Background Equilibrium is a central concept in numerous disciplines including economics, management science/operations research, and engineering. Methodologies that have been applied to the formulation , qualitative analysis, and computation of equilibria have included • systems of equations, • optimization theory, • complementarity theory, and • fixed point theory. Variational inequality theor...

In this paper, we introduce two classes of generalized variational-like inequalities with compositely monotone multifunctions in Banach spaces. Using the KKM-Fan lemma and the Nadler’s result, we prove the existence of solutions for generalized variational-like inequalities with compositely relaxed η − α monotone multifunctions in reflexive Banach spaces. On the other hand we also derive the so...

It is well known that the variational inequalities are equivalent to the fixed point problems. Using this equivalence, we suggest and consider a new three-step iterative method for solving variational inequalities. The new iterative method is obtained by using three steps under suitable conditions. We prove that the new method is globally convergent. Our results can be viewed as significant ext...