Periodic Solutions of Autonomous Functional-Differential Equations with State Dependent Deviations

Abstract

In this paper, we consider periodic solutions of the functional-differential equation x″ + x(t−kx) = 0. The structure of the set A_k (k ∈ (0, ∞)) of all its nontrivial periodic solutions x satisfying x′ < 1/k on R is described. It is proved that for each k ∈ (0, ∞) and T_* ∈ (2π, ∞), there exists x ∈ A_k having the period T_* and for each k ∈ (0, ∞) and a ∈ (0,1/k), there exists a unique x ∈ A_k such that x(0) = 0 and x′(0) = a.