Differential and Integral Equations

Singular differential equations with delay

Abstract

The differential equation $d(Mu(t))/dt=-Lu(t)+L_1u(t-1),$ $t\geq0,$ $u(t)=\varphi(t),$ $-1\leq t\leq0$, for a given strongly continuous $X$-valued function $\varphi$ on $[-1,0]$ is studied, where $M, L, L_1$ are closed linear operators from the complex Banach space $X$ into itself, and $L$ is invertible. Though already in the finite dimensional case in general existence of continuous solutions on $[-1,\infty)$ may fail or it is possible to have continuous solutions only on a finite interval, we indicate classes of operators for which existence results analogous to the ones for regular equations $M=I$ hold. In particular, solutions are given explicitly by a recovery formula.

Article information

Source
Differential Integral Equations, Volume 12, Number 3 (1999), 351-371.

Dates
First available in Project Euclid: 29 April 2013