Positive periodic solutions for abstract evolution equations with delay

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.