Oscillation Criteria for First Order Neutral Differential Equations with Positive and Negative Coefficients

Abstract

Sufficient conditions for the oscillation of the equation ddt[x(t)-R(t)x(t-r)]+P(t)x(t-τ)-Q(t)x(t-δ)=0, where P,Q,R∈C([t0,∞),R+), r∈(0,∞), τ,δ∈R+, τ≥δ, are obtained where R(t)+∫tt-τ+δQ(u)du-1 is allowed to oscillate and the condition ∫∞t0s[P(s)-Q(s-τ+δ)]∫∞s[P(u)-Q(u-τ+δ)]duds=∞ is not necessary. Some examples are given, which show that the results here are almost sharp. © 2001 Academic Press.