On second order nonautonomous semilinear differential inclusions with the nonlinear part depending on speed

Abstract

In this paper, we establish the existence of an admissible pair for a Cauchy problem governed by a hyperbolic integro-differential inclusion involving a Balakrishnan–Taylor-type viscous damping term. To this end, we prove the existence of ${\mathcal{C}}^1$-mild solutions for an abstract Cauchy problem driven by a nonautonomous semilinear second-order differential inclusion, where the linear part generates a fundamental system and the nonlinear term also depends on the first derivative of the solution. Our result provides a broad extension of recent existence theorems available in the literature.