Existence and uniqueness of solutions for integrodifferential equations with time delay in Hilbert space (Q1804203)

Let \(H\) be a Hilbert space and \(A : D(A) \subset H \to H\) be a positive self-adjoint operator. Let \(A_1\), \(A_2\) be closed linear operators with domains containing \(D (A)\) and \(a\), a real function defined in \([- h,0]\). The authors study the delay-problem \[ u'(t) + Au (t) + A_1 u(t - h) + \int^0_{-h} a (-s) A_2 u(t + s) ds = f(t), \quad t \in [0,T] \] \(u(0) = x\), \(u(s) = y(s)\), \(- h \leq s < 0\), and under the assumptions that \(f \in L^2 (\varepsilon, T; H)\), \(Ay \in L^2 (- h + \varepsilon, 0; H)\) for each \(\varepsilon>0\), \(f\) and \(Ay\) improperly integrable near 0. They prove the existence and uniqueness of a weak solution such that \(A^{- \alpha} u\) is continuous for each \(\alpha > 0\). Under additional conditions it is proved that \(u \in C([0,T]; X)\). The initial datum \(x\) is supposed only to belong to \(H\). This problem has been studied under more restrictive assumptions by \textit{D. G. Park}, \textit{S. Y. Kim} [Integrodifferential equations with time delay in Hilbert space. Comm. Korean Math. Soc. 7, 189-207 (1992)].