On the Existence of Positive Periodic Solutions for Second-Order Functional Differential Equations with Multiple Delays

Abstract

The existence results of positive ω-periodic solutions are obtained for the second-order functional differential equation with multiple delays $u″(t)+a(t)u(t)=f(t,u(t),u(t-{\tau }_1(t)),\dots ,u(t-{\tau }_n(t)))$, where $a(t)\in C(ℝ)$ is a positive ω-periodic function, $f:ℝ\times {[0,+\infty )}^{n+1}\to [0,+\infty )$ is a continuous function which is ω-periodic in t, and ${\tau }_1(t),\dots ,{\tau }_n(t)\in C(ℝ,[0,+\infty ))$ are ω-periodic functions. The existence conditions concern the first eigenvalue of the associated linear periodic boundary problem. Our discussion is based on the fixed-point index theory in cones.