Boundary Value Problems for Generalized ODEs.

The aim of this paper is to investigate the existence and uniqueness of solutions of the following boundary value problem concerning generalized ODEs dx d τ = D [ A (t) x + F (t) ] , ∫ a b d [ K (s) ] x (s) = r , <graphic href="12220_2022_1090_Article_Equ52.gif"></graphic> for operators taking values in general Banach spaces. Up to now, this problem was only tackled in the case of finite dimensional space-valued functions. We establish necessary and sufficient conditions not only for the existence of at least one solution, but also for the uniqueness of a solution. Another important result describes the solution in terms of a Green function and the fundamental operator of the corresponding homogeneous problem. We also explore the problem dx d τ = D [ A (t) x + F (t) ] <graphic href="12220_2022_1090_Article_Equ53.gif"></graphic> and we give necessary and sufficient conditions for the existence of a (θ , T) -periodic solution, where θ ∈ R , with θ ≠ 0 , and T > 0 , recalling that the notion of (θ , T) -periodic solution generalizes the notions of periodic, anti-periodic, almost periodic, and quasi-periodic solution. Examples are given to illustrate the main results. In addition, we apply the main theorems to abstract ODEs, as well as to a Volterra-Stieltjes-type integral equation, and we include examples of these as well. [ABSTRACT FROM AUTHOR]