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

AbstractSufficient 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