Existence of variational quasi-hemivariational inequalities involving a set-valued operatorand a nonlinear term

This paper is devoted to the existence of solutions for a class of variational quasi-hemivariational inequalities involving a lower hemicontinuous set-valued operator and a nonlinear term in reflexive Banach spaces. In the case when the constraint set is bounded, under certain generalized monotonicity conditions, we prove an existence result of solutions for the problem by means of F-KKM theorem. In the case when the constraint set is unbounded, under certain coercivity conditions, we construct an existence theorem of solutions and a boundedness theorem of the solution set for the problem, respectively. Moreover, a necessary and sufficient condition to the existence of solutions is also derived. The results presented in this paper generalize and improve some known results.

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