On fractional differential inclusion with damping driven by variational-hemivariational inequality

In this paper we study an evolution problem (FDIVHVI) which constitutes of the nonlinear fractional differential inclusion with damping driven by a variational-hemivariational inequality (VHVI) in Banach spaces. More precisely, first, it is shown that the solution set for VHVI is nonempty, bounded, convex and closed under the surjectivity theorem and the Minty formula. Then, we introduce an associated multivalued map with the solution set of the VHVI, and prove that it is upper semicontinuous and measurable. Finally, by utilizing the fixed point theorem of condensing multivalued operators, properties of \((\alpha , \mu )\)-regularized families of operators and properties of measure of noncompactness, we show the existence of mild solutions for FDIVHVI.