Abstract
Our objective in this paper is to study the oscillatory and asymptotic behavior of the solutions of third-order neutral differential equations with damping and distributed deviating arguments. New oscillation criteria are established, which are based on a refinement generalized Riccati transformation. An important tool for this investigation is the integral averaging technique. Moreover, we provide an example to illustrate the main results.
Highlights
We consider a third-order half-linear neutral differential equation with damping and distributed deviating arguments of the form α2(ξ ) α1(ξ ) y (ξ ) γ + α3(ξ ) α1(ξ ) y (ξ ) γ d+ F ξ, s, x g(ξ, s) dη(s) = 0 (1)c for ξ > ξ0, where y(ξ ) = x(ξ ) + b a p(ξ, σ )x(τ (ξ )) dσ, γ is a quotient of odd positive integers.Throughout the manuscript, we assume the following conditions hold
We study the oscillation criteria of third-order neutral delay (TOND) differential equations with distributed deviating arguments and damping
E.g., the papers [24,25,26,27] for more details
Summary
We consider a third-order half-linear neutral differential equation with damping and distributed deviating arguments of the form α2(ξ ) α1(ξ ) y (ξ ) γ + α3(ξ ) α1(ξ ) y (ξ ) γ d. Let x(ξ ) be a positive solution of Eq (1), and y(ξ ) satisfies case 2 in Lemma 2.2. Proof Suppose to the contrary that there exists an eventually positive solution x(ξ ) of. For each l ≥ ξ0, there exists a positive function ρ(ξ ) ∈ C1([ξ0, ∞), R+) satisfying ρ (ξ ) < 0 such that (ξ ) g(ξ,c) 1. Proof Suppose the contrary, without loss of generality, we can assume that x(ξ ) is an eventually positive solution of Eq (1) for ξ > ξ0. Similar to the proof of Theorem 3.1, case 2 always hold and case 1 still satisfies (23), (26), (27), and (28) for ξ > ξ1. Assume that there exists a positive function ρ(ξ ) ∈ C1([ξ0, ∞), R+), satisfying ρ (ξ ) < 0 such that lim sup θn(ξ ) g(ξ,c) 1. By the same proof, we get new oscillation criteria
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