Positive Periodic Solutions of Third-Order Ordinary Differential Equations with Delays

Abstract

The existence results of positive $\omega$-periodic solutions are obtained for the third-order ordinary differential equation with delays $u^{\mathrm{\prime \prime \prime }}(t)+a(t)u(t)=f(t,u(t-{\tau }_{\mathrm{0}}),u^{\prime }(t-{\tau }_{\mathrm{1}}),u^{\mathrm{\prime \prime }}(t-{\tau }_{\mathrm{2}})),t\in \mathrm{ℝ,}$ where $a\in C(ℝ,(\mathrm{0},\infty ))$ is $\omega$-periodic function and $f:ℝ\times [0,\infty )\times {ℝ}^2\to [0,\infty )$ is a continuous function which is $\omega$-periodic in $t,\text{and\hspace{0.17em}}{\tau }_{\mathrm{0}},{\tau }_{\mathrm{1}},{\tau }_{\mathrm{2}}$ are positive constants. The discussion is based on the fixed-point index theory in cones.