On (θ,T)-periodic solutions of abstract generalized ODEs and applications to Volterra–Stieltjes–type integral equations

Abstract

It is known that generalized ordinary differential equations (generalized ODEs for short) encompass other types of equations such as impulsive differential equations as well as dynamic equations on time scales. The present paper concerns the theory of $(\theta ,T)$-periodic solutions in the framework of generalized ODEs in Banach spaces. We exhibit necessary and sufficient conditions for a solution of a generalized ODE to be $(\theta ,T)$-periodic. Moreover, we develop the Floquet theory of homogeneous linear generalized ODEs and, as a consequence, we present a characterization of fundamental matrices for the finite dimensional case. As an illustration, we apply the main results to Volterra–Stieltjes–type integral equations.