Journal of Global Optimization

, Volume 52, Issue 4, pp 743–756 | Cite as

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

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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 (Math Ann 266:519–537, 1984).

Keywords

Quasi-hemivariational inequality Set-valued operator Lower semicontinuous set-valued operator Clarke’s generalized gradient Generalized monotonicity KKM mapping 

Mathematics Subject Classification (2000)

47J20 47H04 49J53 54C60 47H05 
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Copyright information

© Springer Science+Business Media, LLC. 2011

Authors and Affiliations

  1. 1.Institute of Mathematics “Simion Stoilow” of the Romanian AcademyBucharestRomania
  2. 2.Department of MathematicsUniversity of CraiovaCraiovaRomania

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