Abstract
AbstractThis 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 $$L_{n}u(t)=f\bigl(t,u(t-\tau_{1}),\ldots,u(t- \tau_{m})\bigr),\quad t\in\mathbb{R}, $$Lnu(t)=f(t,u(t−τ1),…,u(t−τm)),t∈R, where $L_{n}u(t)=u^{(n)}(t)+\sum_{i=0}^{n-1}a_{i} u^{(i)}(t)$Lnu(t)=u(n)(t)+∑i=0n−1aiu(i)(t) is the nth-order linear differential operator, $a_{i}\in\mathbb {R}$ai∈R ($i=0,1,\ldots,n-1$i=0,1,…,n−1) are constants, $f: \mathbb{R}\times E^{m}\rightarrow E$f:R×Em→E is a continuous function which is ω-periodic with respect to t, and $\tau_{i}>0$τi>0 ($i=1,2,\ldots,m$i=1,2,…,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.
Funder
the National Natural Science Function of China
the fund of College of Science, Gansu Agricultural University
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Algebra and Number Theory,Analysis
Cited by
1 articles.
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