This paper is concerned with the study of a class of prox-penalization methods for solving variational inequalities of the form Ax + NC(x) 3 0 where H is a real Hilbert space, A : H ⇒ H is a maximal monotone operator and NC is the outward normal cone to a closed convex set C ⊂ H. Given Ψ : H → R ∪ {+∞} which acts as a penalization function with respect to the constraint x ∈ C, and a penalizatio...

The notion of monotone operator in a Hilbert space is extended, and a considerably broader class of possibly multivalued operators is introduced. It is shown that well-known Minty's theorems on maximal monotone operators in Hilbert spaces can be extended to the cases of operators belonging to the class. Typical examples of partial differential operators are given to illustrate that the results ...

In this paper, we study the existence and uniqueness theorem for solving the generalized operator equation of the form F(x) + G(x) + T(x) ∋ 0, where F is a Fréchet differentiable operator, G is a maximal monotone operator and T is a Lipschitzian operator defined on an open convex subset of a Hilbert space. Our results are improvements upon corresponding results of Uko [Generalized equations and...

The paper deals with the properties of a monotone operator defined on a subset of an ordered Banach space. The structure of the set of fixed points between the minimal and maximal ones is described.

In a recent paper by Monteiro and Svaiter, a hybrid proximal extragradient (HPE) framework was used to study the iteration-complexity of a first-order (or, in the context of optimization, second-order) method for solving monotone nonlinear equations. The purpose of this paper is to extend this analysis to study a prox-type first-order method for monotone smooth variational inequalities and incl...

In this paper we analyze the iteration-complexity of the hybrid proximal extragradient (HPE) method for finding a zero of a maximal monotone operator recently proposed by Solodov and Svaiter. One of the key points of our analysis is the use of a new termination criteria based on the εenlargement of a maximal monotone operator. The advantages of using this termination criterion it that its defin...

The sum of (nonlinear) maximal monotone operators is reconsidered from the Yosida approximation and graph-convergence point of view. This leads to a new concept, called variational sum, which coincides with the classical (pointwise) sum when the classical sum happens to be maximal monotone. In the case of subdiierentials of convex lower semicontinuous proper functions, the variational sum is eq...

In this paper, by using Bregman’s technique, we introduce and study the hybrid proximal-point methods for finding a common element of the set of solutions to a system of generalized equilibrium Problems and zeros of a finite family of maximal monotone operators in reflexive Banach spaces. Strong convergence results of the proposed hybrid proximal-point algorithms are also established under some...

In this paper, we introduce and study a new class of generalized nonlinear mixed quasi-variational inequalities involving maximal η-monotone mapping. Using the resolvent operator technique for maximal η-monotone mapping, we prove the existence of solution for this kind of generalized nonlinear multi-valued mixed quasi-variational inequalities without compactness and the convergence of iterative...

We propose two very simple methods, the first one with constant step sizes and second self-adaptive sizes, for finding a zero of sum monotone operators in real reflexive Banach spaces. Our methods require only evaluation single-valued operator at each iteration. Weak convergence results are obtained when set-valued is maximal Lipschitz continuous, strong either these required, addition, to be s...