A positive eigenvalue result for semilinear differential equations in Banach spaces with functional initial conditions

TL;DR

This paper establishes the existence of positive eigenvalues and nonnegative eigenfunctions for a broad class of semilinear differential equations in Banach spaces, including models with nonlocal initial conditions.

Contribution

It introduces a novel theoretical framework combining nonlinear analysis and semigroup theory to handle nonlocal initial conditions in Banach space differential equations.

Findings

Existence of positive eigenvalues proven for abstract initial value problems

Applicable to models with periodic, multipoint, and integral average conditions

Illustrated with a reaction-diffusion equation with nonlocal initial conditions

Abstract

We study the existence of positive eigenvalues with associated nonnegative mild eigenfunctions for a class of abstract initial value problems in Banach spaces with functional, possibly nonlocal, initial conditions. The framework includes periodic, multipoint, and integral average conditions. Our approach relies on nonlinear analysis, topological methods, and the theory of strongly continuous semigroups, yielding results applicable to a wide range of models. As an illustration, we apply the abstract theory to a reaction-diffusion equation with a nonlocal initial condition arising from a heat flow problem.