On fourth order retarded equations with functional boundary conditions: a unified approach

TL;DR

This paper develops a unified approach to analyze the solvability of a broad class of fourth order retarded differential equations with functional boundary conditions using a Birkhoff-Kellogg type theorem, including an illustrative example.

Contribution

It introduces a new method leveraging a Birkhoff-Kellogg theorem to establish solvability for complex retarded differential equations with functional boundary conditions.

Findings

Established conditions for existence of solutions

Applied the approach to a specific example

Extended the theory to parameter-dependent equations

Abstract

By means of a recent Birkhoff-Kellogg type theorem, we discuss the solvability of a fairly general class of parameter-dependent fourth order retarded differential equations subject to functional boundary conditions. We seek solutions within a translate cone of nonnegative functions. We provide an example to illustrate our theoretical results.