Positive periodic solutions for third-order ordinary differential equations with delay

• Main Authors: He Yang, Yujia Chen
• Format: Article
• Language: English
• Published: SpringerOpen 2018-08-01
• Series: Advances in Difference Equations
• Subjects: Third-order differential equation; Positive ω-periodic solution; Positive cone; Delay; Fixed point index theory in cones
• Online Access: http://link.springer.com/article/10.1186/s13662-018-1727-3

Abstract This paper deals with the existence of positive ω-periodic solutions for third-order ordinary differential equation with delay u‴(t)+Mu(t)=f(t,u(t),u(t−τ)),t∈R, $$u'''(t)+Mu(t)=f\bigl(t,u(t),u(t-\tau) \bigr),\quad t\in {\mathbb{R}}, $$ where ω>0 $\omega>0$ and M>0 $M>0$ are constants, f:R3→R $f: {\mathbb{R}}^{3}\rightarrow {\mathbb{R}}$ is continuous, f(t,x,y) $f(t,x,y)$ is ω-periodic in t, and τ>0 $\tau>0$ is a constant denoting the time delay. We show the existence of positive ω-periodic solutions when 0<M<(2π3ω)3 $0< M<(\frac{2\pi}{\sqrt {3}\omega})^{3}$ and f satisfies some order conditions. The discussion is based on the theory of fixed point index.