Positive periodic solutions for high-order differential equations with multiple delays in Banach spaces

Advances in Difference Equations

article

Liang, Yue¹  Li, Hong¹ 

[1] Center for Quantitative Biology, College of Science, Gansu Agricultural University

关键词: n th-order differential equation;    Delays;    Positive ω -periodic solution;    Existence;    Fixed point index theory;   

DOI  :  10.1186/s13662-020-02595-z

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

This paper deals with the existence of positive ω-periodic solutions for nth-order ordinary differential equation with delays in Banach space E of the form $$L_{n}u(t)=f\bigl(t,u(t-\tau_{1}),\ldots,u(t- \tau_{m})\bigr),\quad t\in\mathbb{R}, $$ where $L_{n}u(t)=u^{(n)}(t)+\sum_{i=0}^{n-1}a_{i} u^{(i)}(t)$ is the nth-order linear differential operator, $a_{i}\in\mathbb {R}$ ($i=0,1,\ldots,n-1$) are constants, $f: \mathbb{R}\times E^{m}\rightarrow E$ is a continuous function which is ω-periodic with respect to t, and $\tau_{i}>0$ ($i=1,2,\ldots,m$) are constants which denote the time delays. We first prove the existence of ω-periodic solutions of the corresponding linear problem. Then the strong positivity estimation is established. Finally, two existence theorems of positive ω-periodic solutions are proved. Our discussion is based on the theory of fixed point index in cones.

【 授权许可】

CC BY