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Existence theorems of the hemivariational inequality governed by a multi-valued map perturbed with a nonlinear term in Banach spaces

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Abstract

In this paper, we establish some existence results for the hemivariational inequality governed by a multi-valued map perturbed with a nonlinear term in reflexive Banach spaces. Using the concept of the stable \(f\)-quasimonotonicity, the properties of Clarke’s generalized directional derivative, Clarke’s generalized gradient and KKM technique, some existence theorems of solutions are proved when the constrained set is nonempty, bounded (or unbounded), closed and convex. Our main results extend various results existing in the current literatures.

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Acknowledgments

The first author is supported by the “Centre of Excellence in Mathematics” under the Commission on Higher Education, Ministry of Education, Thailand.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand

    Rabian Wangkeeree & Pakkapon Preechasilp

  2. Centre of Excellence in Mathematics, CHE, Si Ayutthaya Road, Bangkok, 10400, Thailand

    Rabian Wangkeeree

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  1. Rabian Wangkeeree
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  2. Pakkapon Preechasilp
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Correspondence to Rabian Wangkeeree.

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Wangkeeree, R., Preechasilp, P. Existence theorems of the hemivariational inequality governed by a multi-valued map perturbed with a nonlinear term in Banach spaces. J Glob Optim 57, 1447–1464 (2013). https://doi.org/10.1007/s10898-012-0018-x

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  • DOI: https://doi.org/10.1007/s10898-012-0018-x

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