Periodic $L^ 2$-solutions of an integrodifferential equation in a Hilbert space

Abstract:

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