On solvability of degenerate nonstationary differential-difference equations in Banach spaces

Abstract

The differential problem $$ A(t)u'(t)+B(t)u(t)+M(t)u'(t-\omega)+ N(t)u(t-\omega)=f(t) , \ \ \ t \geq 0, $$ $u(t)=g(t)$, $-\omega \leq t \leq 0,$ where $g$ is a given strongly continuous, $X$-valued function on $[-\omega,0]$, $f$ is strongly continuous from $[0,\infty)$ into $Y$, and $A(t)$, $B(t)$, $M(t)$, $N(t)$ are closed linear operators from the complex Banach space $X$ into the complex Banach space $Y$, is studied. Solvability on $[0,T]$ with $T < \infty$ is considered, too. Moreover, the case where $A(t)$, $B(t)$ are independent of $t$ and $\lambda = 0$ is a multiple pole for $(\lambda B+A)^{-1}$ is investigated.