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Positive periodic solutions for abstract evolution equations with delay

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Abstract

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,

$$\begin{aligned} u'(t)+Au(t)=F(t,u(t),u(t-\tau )),\ \ \ \ t\in \mathbb {R}, \end{aligned}$$

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.

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Authors and Affiliations

  1. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, People’s Republic of China

    Qiang Li

  2. Department of Mathematics, Shanxi Normal University, Linfen, 041000, People’s Republic of China

    Qiang Li

  3. Department of Mathematics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China

    Yongxiang Li

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  1. Qiang Li
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  2. Yongxiang Li
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Correspondence to Qiang Li.

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Research supported by NNSF of China (No. 11261053) and NSF of Shanxi, China (No. 201901D211399).

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Li, Q., Li, Y. Positive periodic solutions for abstract evolution equations with delay. Positivity 25, 379–397 (2021). https://doi.org/10.1007/s11117-020-00768-4

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