An affine Birkhoff--Kellogg type result in cones with applications to functional differential equations

TL;DR

This paper establishes an affine Birkhoff--Kellogg type theorem in cones using fixed point index theory and applies it to analyze the solvability of boundary value problems for functional differential equations.

Contribution

It introduces an affine version of a classical fixed point theorem in cones and demonstrates its application to boundary value problems in functional differential equations.

Findings

Proved an affine Birkhoff--Kellogg type theorem using fixed point index.

Applied the theorem to boundary value problems for functional differential equations.

Provided an illustrative example demonstrating the theoretical results.

Abstract

In this short note we prove, by means of classical fixed point index, an affine version of a Birkhoff--Kellogg type theorem in cones. We apply our result to discuss the solvability of a class of boundary value problems for functional differential equations subject to functional boundary conditions. We illustrate our theoretical results in an example.