Positive periodic solutions for abstract evolution equations with delay
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 $E$, $$u'(t)+Au(t)=F(t,u(t),u(t-τ)),\ \ \ \ t\in\R,$$ 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 $ω$-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 $ω$-periodic mild solutions by applying operator semigroup theory. In the end, an example is given to illustrate the applicability of our abstract results.