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Existence Results for a Class of Variational Quasi-Mixed Hemivariational-Like Inequalities

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Abstract

The paper aims to explore the existence results for a class of variational quasi-mixed hemivariational-like inequality problems with nonlinear terms in reflexive Banach spaces, which contain variational and hemivariational inequalities. We make use of stable \((\eta ,\psi )\)-quasimonotonicity, KKM theorem, Clarke’s generalized directional derivative and Clarke’s generalized gradient to derive the existence theorems for the condition of the constrained set being bounded. Further, we obtain the solution’s existence results when the constrained set is unbounded by utilizing suitable coercive conditions. Moreover, we present some sufficient conditions to assure the boundedness of the solutions set. Besides, we also demonstrate a necessary and sufficient criteria for a restricted class of variational quasi-mixed hemivariational-like inequality problems. Several applications of the main results are illustrated. The new developments improve and generalize some well-known works.

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Acknowledgements

The first author (K Mahalik) was supported by the “Council of Scientific and Industrial Research”, India (Grant No-09/081(1306)/2017-EMR-I).

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  1. Department of Mathematics, Indian Institute of Technology, Kharagpur, 721302, India

    K. Mahalik & C. Nahak

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  1. K. Mahalik
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  2. C. Nahak
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Correspondence to C. Nahak.

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Communicated by Rosihan M. Ali.

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Mahalik, K., Nahak, C. Existence Results for a Class of Variational Quasi-Mixed Hemivariational-Like Inequalities. Bull. Malays. Math. Sci. Soc. 45, 1877–1901 (2022). https://doi.org/10.1007/s40840-021-01214-8

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  • DOI: https://doi.org/10.1007/s40840-021-01214-8

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