On solvability of the impulsive Cauchy problem for integro-differential inclusions with non-densely defined operators

Philos Trans A Math Phys Eng Sci. 2021 Feb 22;379(2191):20190384. doi: 10.1098/rsta.2019.0384. Epub 2021 Jan 4.


We prove the existence of at least one integrated solution to an impulsive Cauchy problem for an integro-differential inclusion in a Banach space with a non-densely defined operator. Since we look for integrated solution we do not need to assume that A is a Hille Yosida operator. We exploit a technique based on the measure of weak non-compactness which allows us to avoid any hypotheses of compactness both on the semigroup generated by the linear part and on the nonlinear term. As the main tool in the proof of our existence result, we are using the Glicksberg-Ky Fan theorem on a fixed point for a multivalued map on a compact convex subset of a locally convex topological vector space. This article is part of the theme issue 'Topological degree and fixed point theories in differential and difference equations'.

Keywords: fixed point; impulses; integrated semigroup; integrated solution; integro-differential inclusion; measure of weak non-compactness.