On the existence of positive periodic solutions for second-order functional differential equations with multiple delays (Q1938336)

Summary: Existence results for positive \(\omega\)-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 \(a \in C(\mathbb R)\) is a positive \(\omega\)-periodic function, \(f : \mathbb R \times [0, +\infty)^{n+1} \rightarrow [0, +\infty)\) is a continuous function which is \(\omega\)-periodic in \(t\), and \(\tau_1(t), \dots, \tau_n(t) \in C(\mathbb R, [0, +\infty))\) are \(\omega\)-periodic functions. The existence conditions concern the first eigenvalue of the associated linear periodic boundary value problem. Our discussion is based on the fixed-point index theory in cones.