On the nature of the nonoscillatory solutions of a class of neutral delay differential equations

AbstractConsider the neutral delay differential equation ddt[x(t) + px(t − τ)] + q[x(t − σ1) − x(t − σ2)] = 0, (E) where p, q, τ, σ1, and σ2 are nonnegative constants. For τ ≦ σ2, we prove that all nonoscillatory solutions of (E) are bounded if and only if: (C) The characteristic equation λ + pλe−λτ + q(e−λσ1 − e−λσ2) = 0 (∗) of (E) has no positive roots and zero is a simple root if (∗). We also prove that condition (C) is equivalent to 1 + p > q(σ1 − σ2) if τ ≦ σ2. Moreover, for the case where τ > σ2, we show that (C) is a necessary and sufficient condition for every nonoscillatory solution x of (E) to be bounded or such that 0<lim inft→∞1t∫01|x(s)|ds≦lim supt→∞1t∫01|x(s)|ds<∞