PERIODIC L2-SOLUTIONS OF AN INTEGRODIFFERENTIAL EQUATION IN A HILBERT-SPACE

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 R . Consider the abstract integrodifferential equation integral(R) d/dt u(t - s)mu(ds) + integral(R) u(t - s)nu(ds) + integral(R) Au(t - s)eta(ds) = f(t), t is-an-element-of R, where f is a 2pi-periodic L2 function with values in X. We give necessary and sufficient conditions for this equation to have a mild 2pi-periodic L2-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.