Mediterr. J. Math. (2020) 17:116 https://doi.org/10.1007/s00009-020-01538-y c  Springer Nature Switzerland AG 2020 On Oscillation of Second Order Delay Differential Equations with a Sublinear Neutral Term Said R. Grace, Irena Jadlovsk´ a and Agacik Zafer Abstract. We derive sufficient conditions for the oscillation of solutions second-order delay differential equation containing a sublinear neutral term. Our conditions differ from the earlier ones even in the special cases, linear or nonlinear, and as illustrated with an example, we not only extend but also improve several results in the literature. Mathematics Subject Classification. 45D05, 34K11, 34K12. Keywords. Second-order differential equation, Nonlinear, Sublinear neu- tral term, Oscillation. 1. Introduction In the present work, we aim to make a contribution to oscillation theory of neutral type delay differential equations by considering a special class of equations of the form (a(t)(x(t)+ p(t)x α (σ(t)) ′ ) γ ) ′ + q(t)x β (τ (t)) = 0, t ≥ t 0 (1.1) where t 0 ≥ 0 is fixed, α,β,γ are ratios of positive odd integers, 0 <α< 1, and β ≥ γ . The functions a,p,q,τ,σ :[t 0 , ∞) → R + are assumed to be smooth enough to guarantee the existence of solutions defined in the neighborhood of the infinity. In addition, we impose that τ (t) <t, σ(t) <t, τ is nondecreasing, τ (t),σ(t) →∞ as t →∞, and that A(t 0 ) :=  ∞ t0 1 a 1/γ (s) ds< ∞. (1.2) By a solution of equation (1.1) we mean a function x ∈C ([t a , ∞), R) with t a = min{τ (t b ),σ(t b )} for some t b ≥ t 0 , which has the property that r(y ′ ) γ ∈ C 1 ([t a , ∞), R), where y(t)= x(t)+ p(t)x α (σ(t)), and that it satisfies (1.1) on [t b , ∞). We only consider those nontrivial solutions of (1.1) which are defined on some half-line [t b , ∞). 0123456789().: V,-vol