#### L.C. Ceng, X. Qin, J.C. Yao, Y. Yao, On existence results for a new class of variational-hemivariational-like inequalities in Banach spaces, Vol. 2018 (2018), Article ID 7, pp. 1-14

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DOI: 10.23952/cot.2018.7

Received November 29, 2017; Accepted March 18, 2018; Published March 27, 2018

Abstract. This paper is devoted to studying the existence of solutions for a new class of variational-hemivariational-like inequalities in reflexive Banach spaces. Using the notions of $\Phi$-monotonicity and ${\mathcal H}$-hemicontinuity of a nonempty weakly compact-valued mapping, and the properties of Clarke’s generalized directional derivative and Clarke’s generalized gradient, we prove some existence results of solutions when the constrained set is nonempty, bounded (or unbounded), closed and convex.