Abstract
Abstract The main purpose of this paper is to study the oscillatory properties of solutions of the third-order quasi-linear delay differential equation ℒ y ( t ) + f ( t ) y β ( σ ( t ) ) = 0 {\cal L}y(t) + f(t){y^\beta }(\sigma (t)) = 0 where ℒy(t) = (b(t)(a(t)(y 0(t)) )0)0 is a semi-canonical differential operator. The main idea is to transform the semi-canonical operator into canonical form and then obtain new oscillation results for the studied equation. Examples are provided to illustrate the importance of the main results.
Highlights
In this paper, we are concerned with the third-order quasi-linear delay di erential equationLy(t) + f (t)yβ(σ(t)) =, t ≥ t >, (1.1)where L denote the di erential operator Ly(t) = (b(t)(a(t)(y′(t))α)′)′
The main idea is to transform the semi-canonical operator into canonical form and obtain new oscillation results for the studied equation
Throughout, we will assume that: (H ) a ∈ C( )([t, ∞)), b ∈ C ([t, ∞)), a(t) > and b(t) > for all t ≥ t ; (H ) f (t) ∈ C([t, ∞)) is a positive for all t ≥ t ; (H ) σ(t) ∈ C ([t, ∞)), σ(t) < t, limt→∞ σ(t) = ∞ and σ′(t) ≥ for all t ≥ t ; (H ) α and β are ratios of odd positive integers; (H ) The operator L is in semi-canonical form, that is
Summary
The main idea is to transform the semi-canonical operator into canonical form and obtain new oscillation results for the studied equation. Oscillation Results for Third-Order Semi-Canonical Quasi-Linear Delay Di erential Equations Assume that y(t) is an eventually positive solution of (1.1), y(t) satis es one of the following three options: (I) y(t) > , (y′(t))α > , (a(t)(y′(t))α)′ > , (b(t)(a(t)(y′(t))α)′)′ < , (II) y(t) > , (y′(t))α < , (a(t)(y′(t))α)′ > , (b(t)(a(t)(y′(t))α)′)′ < , (III) y(t) > , (y′(t))α > , (a(t)(y′(t))α)′ < , (b(t)(a(t)(y′(t))α)′)′ < , eventually for all su ciently large t.
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