Conditions for the solvability of the Cauchy problem for linear first-order functional differential equations

TL;DR

This paper establishes necessary and sufficient conditions for the unique solvability of the Cauchy problem in a family of scalar linear first-order functional differential equations, focusing on equations with operators into bounded functions.

Contribution

It provides new solvability criteria for a family of functional differential equations with operators into bounded functions, extending previous results.

Findings

Derived necessary and sufficient conditions for solvability.

Applicable to equations with operators into essentially bounded functions.

Unified criteria for entire family of equations.

Abstract

Conditions for the unique solvability of the Cauchy problem for a family of scalar functional differential equations are obtained. These conditions are sufficient for the solvability of the Cauchy problem for every equation from the family and are necessary for the solvability of the Cauchy problem for all equations from the family. In contrast to many known articles, we consider equations with functional operators acting into the space of essentially bounded functions.