Limiting equations of integrodifferential equations in Banach space (Q1342470)

The authors prove that if \(x\) is a bounded continuous weak solution of the equation \[ x'(t) = A \left[ x(t) + \int_ 0^ t F(t-s) x(s) ds \right] + f(t), \quad t \geq 0, \] in a reflexive Banach space, then translates of \(x\) converge in a weak sense toward a weak solution \(y\) of the equation \[ y'(t) = A \left[ y(t) + \int_{-\infty}^ t F(t-s) y(s) ds \right] + g(t), \quad -\infty < t < \infty. \tag{1} \] Here \(A\) is a closed and densely defined operator and \(F(t)\) is a bounded operator for \(t \geq 0\). These results are applied to the study of periodic solutions of (1).