摘要：In this paper, we discuss the existence of the positive time periodic mild solutions for the evolution equation in an ordered Banach space E, u ′ ( t ) + A u ( t ) = f ( t , u ( t ) ) $u'(t)+Au(t)=f(t,u(t))$ , t ∈ R $t\in \mathbb{R}$ , where A : D ( A ) ⊂ E → E $A:D(A)\subset E\rightarrow E$ is a closed linear operator and −A generates a positive compact semigroup T ( t ) $T(t)$ ( t ≥ 0 $t\geq0$ ) in E, the nonlinear function f : R × E → E $f:\mathbb{R}\times E\rightarrow E$ is continuous and f ( t , x ) $f(t,x)$ is ω-periodic in t. We apply the operator semigroup theory and the Leray-Schauder fixed point theorem to obtain the existence of a positive ω-periodic mild solution under the condition that the nonlinear function satisfies a linear growth condition concerning the growth exponent of the semigroup T ( t ) $T(t)$ ( t ≥ 0 $t\geq0$ ). In the end, an example is given to illustrate the applicability of our abstract results.
关键词：abstract evolution equation ; positive periodic mild solutions ; positive compact semigroup ; the growth exponent of the semigroup ; fixed point theorem
