Asymptotically almost periodic solutions for time-varying integro-differential inclusions with nonlocal conditions

This paper studies a class of first-order abstract integro-differential inclusions defined on a Banach space. Our system is characterized by time-varying evolution operators and, notably, nonlocal initial conditions formulated as set inclusions. The motivation for this research stems from the need for more accurate modeling of complex physical and biological systems exhibiting memory effects, where traditional local or single-valued nonlocal conditions prove insufficient.

Employing techniques from resolvent operator theory, fixed point arguments involving measures of non-compact-ness, and the properties of multivalued mappings, we prove the existence of mild solutions for the proposed integro-differential inclusion problem, avoiding the restrictive assumptions on the compactness of the associated resolvent operators. In addition, we establish conditions for the existence of asymptotically almost periodic solutions, necessary for understanding the long-term behavior of systems subjected to time-varying influences and possessing memory. The interplay between time-dependence, hereditary effects, nonlinear multivalued dynamics, and multivalued nonlocal conditions makes the study of such asymptotic behavior particularly relevant.

To illustrate the applicability of our abstract theoretical framework, we conclude the paper with an example.