Monotone iterative technique for delayed evolution equation periodic problems in Banach spaces

In this paper, we deal with the existence of $\omega$-periodic mild solutions for the abstract evolution equation with delay in an ordered Banach space $E$\[u^\prime (t) + Au(t) = F(t, u(t), u(t - \tau)) , \qquad t\in \mathbb{R},\]where $A : D(A) \subset E \to E$ is a closed linear operator and $-A$ generates a positive $C_0$-semigroup $T(t)(t \geq 0), F : \mathbb{R}\times E \times E \to E$ is a continuous mapping which is $\omega$-periodic in $t$, and $\tau \geq 0$ is a constant. Under some weaker assumptions, we construct monotone iterative method for the delayed evolution equation periodic problems, and obtain the existence of maximal and minimal periodic mild solutions. The results obtained generalize the recent conclusions on this topic. Finally, we present two applications to illustrate the feasibility of our abstract results.

Keywords

evolution equations with delay, upper and lower solutions, existence, monotone iterative technique, positive $C_0$-semigroup

2010 Mathematics Subject Classification

34K30, 47H07, 47H08

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The author’s research was supported by NNSFs of China (11261053, 11361055).

Received 2 January 2018

Accepted 24 December 2018

Published 5 November 2019