Monotone iterative technique for delayed evolution equation periodic problems in Banach spaces

TL;DR

This paper establishes the existence of periodic solutions for delayed evolution equations in Banach spaces using monotone iterative techniques, generalizing previous results and providing practical applications.

Contribution

It introduces a monotone iterative method for delayed evolution equations in Banach spaces, proving the existence of maximal and minimal periodic solutions under weaker assumptions.

Findings

Existence of periodic mild solutions in Banach spaces.

Development of a monotone iterative approach for delayed equations.

Application examples demonstrating the method's feasibility.

Abstract

In this paper, we deal with the existence of $\omega$-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 $t$, and $\tau\geq0$ is a constant. Under some weaker assumptions, we construct monotone iterative method for the delayed evolution equation periodic problems, and obtain the existence of maximal and minimal periodic mild solutions. The results obtained generalize the recent conclusions on this topic. Finally, we present two applications to illustrate the feasibility of our abstract results.