Optimal conditions for unique solvability of the Cauchy problem for first order linear functional differential equations

Abstract

Nonimprovable, in a sense sufficient conditions guaranteeing the unique solvability of the problem

$$u'(t) = \ell (u)(t) + q(t),{\text{ }}u(a) = c,$$

where $\ell :\;\;C(I,\mathbb{R})\;\; \to \;\;L(I,\mathbb{R})$ is a linear bounded operator, $q \in L(I,\mathbb{R})$ and $c \in \mathbb{R}$ are established.