Periodic 𝐿²-solutions of an integrodifferential equation in a Hilbert space

Source abstract

Let A A be a closed, densely defined operator in a Hilbert space X X , and let μ , ν \mu ,\;\nu , and η \eta be finite, scalar-valued measures on R {\mathbf {R}} . Consider the abstract integrodifferential equation $$∫ R d d t u ( t − s ) μ ( d s ) + ∫ R u ( t − s ) ν ( d s ) + ∫ R A u ( t − s ) η ( d s ) = f ( t ) , t ∈ R , \int _{\mathbf {R}} {\frac {d} {{dt}}u(t - s)\mu (ds) + \int _{\mathbf {R}} {u(t - s)\nu (ds) + \int _{\mathbf {R}} {Au(t - s)\eta (ds) = f(t),\qquad t \in {\mathbf {R}},} } }$$ where f f is a 2 π 2\pi -periodic L 2 {L^2} function with values in X X . We give necessary and sufficient conditions for this equation to have a mild 2 π 2\pi -periodic L 2 {L^2} -solution with values in X X for all f f , as well as necessary and sufficient conditions for it to have a strong solution for all f f . Furthermore, we give necessary and sufficient conditions for the operator mapping f f into the periodic solution u u to be compact. These results are applied to prove existence of periodic solutions of a nonlinear equation.