Decay solutions to abstract impulsive fractional mobile–immobile equations involving superlinear nonlinearities

Abstract

We study the existence of decay solutions for a class of abstract fractional mobile-immobile equations involving impulses and nonlinear perturbations in Hilbert spaces, where the nonlinear terms are supposed to be of superlinear growth. Concerning the associated linear problem, we make use of the theory of completely positive functions to establish a variation of parameters formula in terms of the relaxation functions and to prove the differentiability of resolvent families. In addition, based on local estimates on Hilbert scales and fixed point arguments, we obtain some results on the global existence and the existence of a compact set of decay solutions to our problem.