A new oscillation criterion for first-order delay differential equations by iteration

A new oscillation criterion is established for the first-order delay differential equationx′(t)+p(t)x(τ(t))=0,t≥t0,where p(t),τ(t)∈C([t0,+∞),[0,+∞)),τ(t) is non-decreasing, τ(t) < t and limt→∞τ(t)=∞. By discretizing not only x(t) but also ∫τ(t)tp(s)ds, we show that all solutions of the equation are oscillatory iflim supt→∞∫τ(t)tp(s)ds>1+lnλ1λ1−α221−α−α22+α3λ13!1−α−α22+α3λ13!+α4λ124!1−α−α22+α3λ13!+α4λ124!+α5λ135!1−α−⋱,(*)where α=lim inft→∞∫τ(t)tp(s)ds∈(0,1e] and λ1 is the smaller root of the equation λ=eαλ. The right-hand side of (*) is proved to be strictly smaller than previous ones with an illustrative example. In particular, when α=1e, our result leads to lim supt→∞∫τ(t)tp(s)ds>0.5621.