Positive periodic solutions for abstract evolution equations with delay

TL;DR

This paper establishes the existence and stability of positive periodic solutions for a class of abstract evolution equations with delay in ordered Banach spaces, using semigroup theory and order conditions.

Contribution

It introduces new conditions on the nonlinearity and operator spectrum to guarantee positive periodic solutions and their stability in delayed evolution equations.

Findings

Proves existence of positive periodic mild solutions under specified conditions.

Shows asymptotic stability of these solutions.

Provides an example demonstrating applicability of the theoretical results.

Abstract

In this paper, we discuss the existence and asymptotic stability of the positive periodic mild solutions for the abstract evolution equation with delay in an ordered Banach space $E$, $$u'(t)+Au(t)=F(t,u(t),u(t-\tau)),\ \ \ \ t\in\R,$$ where $A:D(A)\subset E\rightarrow E$ is a closed linear operator and $-A$ generates a positive $C_{0}$-semigroup $T(t)(t\geq0)$, $F:\R\times E\times E\rightarrow E$ is a continuous mapping which is $\omega$-periodic in $t$. Under order conditions on the nonlinearity $F$ concerning the growth exponent of the semigroup $T(t)(t\geq0)$ or the first eigenvalue of the operator $A$, we obtain the existence and asymptotic stability results of the positive $\omega$-periodic mild solutions by applying operator semigroup theory. In the end, an example is given to illustrate the applicability of our abstract results.