On Variational Inequalities with Multivalued Operators with Semi-bounded Variation

  • O. V. Solonoukha
  • Published 1997


In this paper we explore some problems for the steady-state variational inequalities with multivalued operators (VIMO). As far as we know, in this variant the term VIMO had been first introduced in [1]. The results of this paper with respect to VIMO extend and/or improve analogous ones from [1-7]. We refused the regularity conditions of the monotonic disturbance of the multivalued mapping ([1]) and some other properties of the objects ([2]). Besides we considered a wider class of operators with respect to [3,4]. We are studying the connections between the class of radially semi-continuous operators with semi-bounded variation, the class of pseudo-monotone mappings, which is used earlier (for example, in [2]) on selector’s language, and the class of monotone mappings. Moreover, for new class of operators the property of local boundedness is substituted for a weaker one with respect to [1,2,5,7,8]. Also we refuse the condition that A(y) is a convex closet set owing to the forms of support functions. Let X be a reflexive Banach space, X be its topological dual space, by 〈·, ·〉 we denote the dual pairing on X ×X, 2 ∗ be the totality of all nonempty subsets of the space X, A : X → 2 ∗ be a multivalued mapping with DomA = {y ∈ X : A(y) 6= ∅}. A : X → 2 ∗ is called strong iff DomA = X . Further for simplicity we will consider only strong mappings A. Let us consider the upper and lower support functions which are associated to A: [A(y), ξ]+ = sup d∈A(y) 〈d, ξ〉, [A(y), ξ]− = inf d∈A(y) 〈d, ξ〉,

Cite this paper

@inproceedings{Solonoukha1997OnVI, title={On Variational Inequalities with Multivalued Operators with Semi-bounded Variation}, author={O. V. Solonoukha}, year={1997} }