Abstract
We study the half-linear delay differential equation , , We establish a new a priori bound for the nonoscillatory solution of this equation and utilize this bound to derive new oscillation criteria for this equation in terms of oscillation criteria for an ordinary half-linear differential equation. The presented results extend and improve previous results of other authors. An extension to neutral equations is also provided.
Highlights
In this paper we study oscillatory properties of the delay second-order half-linear differential equation (r (t) Φ (x (t))) + c (t) Φ (x (τ (t))) = 0, (1)Φ (x) := |x|p−2x, p > 1.We suppose that r, c, τ are continuous functions defined on [t0, ∞) such that r(t) > 0, c(t) > 0 for large t, τ(t) ≤ t for all t, and limt → ∞τ(t) = ∞
The assumptions used in the paper ensure that the positive solutions are eventually increasing and concave down
The nonlinearity of the equation causes, that the method from [13, 14] does not extend to (1) directly and we have to use an indirect approach which originates in the fact that the half-linear extension does not yield (13) as its special case, but includes the term τ(s) instead of s
Summary
The nonlinearity of the equation causes, that the method from [13, 14] does not extend to (1) directly and we have to use an indirect approach which originates in the fact that the half-linear extension does not yield (13) as its special case, but includes the term τ(s) instead of s. This estimate suggests a new tool which can be used to improve some oscillation criteria for (1)
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