A study on positive solutions for evolution equation with delay in Banach space

Abstract

We study the existence of the positive mild solutions of evolution systems with delay in an ordered Banach space. Firstly, the existence of mild solution is obtained by means of the characteristics of $C_0$-semigroup $T(t)(t\geq 0)$ and monotone iterative technique. Secondly, without assuming the existence of upper and lower solutions, we deal with the existence of positive mild solutions of an evolution equation with delay on infinite intervals by introducing the generalized Arzel\`{a}-Ascoli theorem and Kuratowski measure of noncompactness. The results in this paper are generalization and continuation of the recent results on this issue. Finally, as the application of abstract results, an example is given to illustrate our main results.