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Solvability of Generalized Variational Inequality Problems for Unbounded Sets in Reflexive Banach Spaces

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Abstract

By employing the notion of exceptional family of elements, we establish some existence results for generalized variational inequality problems in reflexive Banach spaces provided that the mapping is upper sign-continuous. We show that the nonexistence of an exceptional family of elements is a necessary condition for the solvability of the dual variational inequality. For quasimonotone variational inequalities, we present some sufficient conditions for the existence of strong solutions. For the pseudomonotone case, the nonexistence of an exceptional family of elements is proved to be an equivalent characterization of the problem having strong solutions. Furthermore, we establish several equivalent conditions for the solvability for the pseudomonotone case. As a byproduct, a quasimonotone generalized variational inequality is proved to have a strong solution if it is strictly feasible. Moreover, for the pseudomonotone case, the strong solution set is nonempty and bounded if it is strictly feasible.

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Authors and Affiliations

  1. Department of Mathematics, Guangxi Normal University, Guilin, Guangxi, 541004, P.R. China

    J. H. Fan

  2. Le’An Middle School, Xiamen, Fujian, 361021, P.R. China

    X. G. Wang

Authors
  1. J. H. Fan
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  2. X. G. Wang
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Correspondence to J. H. Fan.

Additional information

Communicated by J.P. Crouzeix.

This work was supported by the Guangxi Science Foundation (Grant No. 0832101, 0640063). We are grateful to the referees for useful suggestions which have contributed to the final presentation of the paper.

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Fan, J.H., Wang, X.G. Solvability of Generalized Variational Inequality Problems for Unbounded Sets in Reflexive Banach Spaces. J Optim Theory Appl 143, 59–74 (2009). https://doi.org/10.1007/s10957-009-9556-x

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  • DOI: https://doi.org/10.1007/s10957-009-9556-x

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