Oscillation of deviating differential equations

Consider the first-order linear delay (advanced) differential equation x ′(t) + p(t)x(τ(t)) = 0 (x′(t)− q(t)x(σ(t)) = 0), t > t0, where p (q) is a continuous function of nonnegative real numbers and the argument τ(t) (σ(t)) is not necessarily monotone. Based on an iterative technique, a new oscillation criterion is established when the well-known conditions lim sup t→∞ ∫ t τ(t) p(s) ds > 1 ( lim sup t→∞ ∫ σ(t) t q(s) ds > 1 ) and lim inf t→∞ ∫ t τ(t) p(s) ds > 1 e ( lim inf t→∞ ∫ σ(t) t q(s) ds > 1 e ) are not satisfied. An example, numerically solved in MATLAB, is also given to illustrate the applicability and strength of the obtained condition over known ones.