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,

where \(A:D(A)\subset E\rightarrow E\) is a closed linear operator and \(-A\) generates a positive \(C_{0}\)-semigroup \(T(t)(t\ge 0)\), \(F:\mathbb {R}\times E\times E\rightarrow E\) is a continuous mapping which is \(\omega \)-periodic in t. Under the ordered conditions on the nonlinearity F concerning the growth exponent of the semigroup \(T(t) (t\ge 0)\) or the first eigenvalue of the operator A, we obtain the existence and asymptotic stability 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.

在本文中，我们在一个有序的Banach空间与延迟讨论的存在性和抽象演化方程的正周期温和的渐近稳定性ë，

其中\（A：D（A）\ subset E \ rightarrow E \）是一个封闭的线性算子，而\（-A \）生成一个正数\（C_ {0} \）-半群\（T（t）（t \ ge 0）\），\（F：\ mathbb {R} \ times E \ times E \ rightarrow E \）是一个连续映射，在t中是\（\ omega \）-周期。在关于半群\（T（t）（t \ ge 0）\）的增长指数或算子A的第一特征值的非线性F的有序条件下，我们得到正\（（ \ omega \）算子半群理论的周期周期温和解。最后，给出一个例子来说明我们的抽象结果的适用性。