Applied Mathematics & Information Sciences

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Consider the differential equation with a retarded argument of the form x′(t) + p(t)x(τ(t)) = 0, t ≥ t0, (1) where the functions p, τ ∈ C([t0,∞), R+), (here R+ = [0, ∞)), τ(t) ≤ t for t ≥ t0 and limt→∞ τ(t) = ∞ and the equation with a constant positive delay τ of the form x′(t) + p(t)x(t − τ) = 0, t ≥ t0, (2) Optimal conditions for the oscillation of all solutions to these equations are presented when the well-known oscillation conditions limsup t→∞ Zτt(t) p(s)ds > 1 and liminft→∞ Zτt(t) p(s)ds > 1e are not satisfied and also in the critical case where liminft→∞ p(t) = e1τ in Eq. (2). In the case that the function Rtt−τ p(s)ds is slowly varying at infinity, then under mild additional assumptions limsup t→∞ Zt−t τ p(s)ds > 1e is a sharp condition for the oscillation of all solutions to Eq. (2). Examples illustrating the results are given.

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