Inequality problems of quasi‐hemivariational type involving set‐valued operators and a nonlinear term

Inequality problems of quasi‐hemivariational type involving set‐valued operators and a nonlinear term

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Article ID: iaor20123059
Volume: 52
Issue: 4
Start Page Number: 743
End Page Number: 756
Publication Date: Apr 2012
Journal: Journal of Global Optimization
Authors: ,
Keywords: global optimization, Banach space, inequality problems
Abstract:

The aim of this paper is to establish the existence of at least one solution for a general inequality of quasi‐hemivariational type, whose solution is sought in a subset K of a real Banach space E. First, we prove the existence of solutions in the case of compact convex subsets and the case of bounded closed and convex subsets. Finally, the case when K is the whole space is analyzed and necessary and sufficient conditions for the existence of solutions are stated. Our proofs rely essentially on the Schauder’s fixed point theorem and a version of the KKM principle due to Ky Fan (1984).

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