Topological Properties of Solution Sets for <i>τ</i>-Fractional Non-Instantaneous Impulsive Semi-Linear Differential Inclusions with Infinite Delay

Abstract

The knowledge of fractional calculus can be useful in developing models that allow us to better understand and manage some phenomena. In the present paper, we study the topological structure of the mild solution set for a semi-linear differential inclusion containing the τ-Caputo fractional derivative in the presence of non-instantaneous impulses and an infinite delay. We demonstrate that this set is non-empty and an Rδ-set. We use a recent result regarding the existence of solutions for τ-Caputo fractional semi-linear differential inclusions. Unlike many results, we do not suppose that the generating semigroup is compact. An illustrative example is given as an application of our results.

Keywords

• non-instantaneous impulses
• <i>&#x3C4
• </i>-caputo derivative
• semi-linear differential inclusions
• mild solutions
• <i>R_δ</i>-sets