Positive periodic solutions for abstract evolution equations with delay

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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∈, where A:D(A)⊂ E→ E is a closed linear operator and -A generates a positive C₀-semigroup T(t)(t≥0), F:× E× E→ 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≥0) 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.