On solvability of degenerate nonstationary differential-difference equations in Banach spaces
The authors study the following differential problem \[ A(t)u^\prime (t) + B(t)u(t) + M(t)u^\prime (t-\omega) + N(t)u(t-\omega)=f(t), \quad t \in J \subset\mathbb{R}, \] \[ u(t) = g(t),\qquad - \omega \leq t \leq 0, \] where the operators \(A(t), B(t), M(t), N(t) (t \in J)\) are closed linear operators from a Banach space \(X\) into a Banach space \(Y\) and the operator \(A(t)\) is not supposed to be invertible. The problem is studied in the case when the point \(\mu = 0\) is a simple pole of the resolvent operator \((\mu B(t) + A(t))^{-1}\) for all \(t \in J\). The authors also consider the case of stationary operators \(A,B\) when the resolvent \((\mu B + A)^{-1}\) has a multiple pole at \(\mu = 0\). Concrete applications to partial differential equations with delay are given.
partial differential equations with delay
scientific article; zbMATH DE number 1832847