On Oscillation of Second Order Delay Differential Equations with a Sublinear Neutral Term
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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 neutral 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)xα (σ(t)) )γ ) + q(t)xβ (τ (t)) = 0,
t ≥ t0
(1.1)
where t0 ≥ 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 : [t0 , ∞) → 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 ∞ 1 A(t0 ) := ds < ∞. (1.2) 1/γ (s) a t0 By a solution of equation (1.1) we mean a function x ∈ C([ta , ∞), R) with ta = min{τ (tb ), σ(tb )} for some tb ≥ t0 , which has the property that r(y )γ ∈ C 1 ([ta , ∞), R), where y(t) = x(t) + p(t)xα (σ(t)), and that it satisfies (1.1) on [tb , ∞). We only consider those nontrivial solutions of (1.1) which are defined on some half-line [tb , ∞). 0123456789().: V,-vol
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Neutral type delay differential equations arise in many areas of applied mathematics and so they have received great attention in the last several decades. The literature on the oscillation theory of such equations is also quite extensive. In reviewing the literature, it becomes apparent that most oscillation results for Eq. (1.1) were given in a particular case α = 1 under either (1.2) holds or fails, see, e.g., [2–5,7–9,11,12] respectively. However, there are few results dealing with the oscillation of second order differential equations with a sublinear neutral term. For an important initial contribution to the problem we refer the reader to [1], where (1.1) was studied in the linear case γ = β = 1 under the assumptions (1.2) and 21−α − 1 1−α A(τ (σ(t))) + < 1 for all constants K > 0. p(τ (t)) α2 A(τ (t)) KA(t) Further contributions to studying oscillatory properties of solutions in the special case γ = 1 can be found in Ref. [10] under the conditions (1.2) and p(t)Aα (σ(t)) p(t) , < 1. (1.3) max A1−α (σ(t)) A2−α (t) Another approach was developed in Ref. [6] when the condition (1.3) is replaced by a condition strongly depending on the choice of suitable auxiliary functions. The purpose of the paper is to continue further in studying the oscillation of (1.1). Our conditions differ from the earlier ones even in the special cases,
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