Abstract
The aim of this study was to examine the asymptotic properties and oscillation of the even-order neutral differential equations. The results obtained are based on the Riccati transformation and the theory of comparison with first- and second-order delay equations. Our results improve and complement some well-known results. We obtain Hille and Nehari type oscillation criteria to ensure the oscillation of the solutions of the equation. One example is provided to illustrate these results.
Highlights
Research activity has focused on the oscillatory behavior of solutions to different classes of neutral differential equations
The asymptotic behavior of the solutions to delay and neutral delay differential equations was discussed in many works, awith fruitful achievements [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]
One of the main reasons for this lies in delay differential equations arising in many applied problems in natural sciences, technology, and automatic control [25]
Summary
Research activity has focused on the oscillatory behavior of solutions to different classes of neutral differential equations. Throughout this paper, we assume that the following conditions are satisfied:. A function y ∈ C n−1 [ς y , ∞), ς y ≥ ς 0 , is called a solution of Equation (1), if r y(n−1). Π (s) ds > (n − 1)!, δ (ς) ≥ 0 where π (ς) := δn−1 (ς) (1 − p (δ (ς))) q (ς), every solution of Equation (4) is oscillatory. In this paper, using the theory of comparison with first- and second-order delay equations, new oscillatory criteria are established of Equation (1). We establish the oscillatory behavior of Equation (1) under the conditions δ (ς) < g (ς) , δ0 (ς) ≥ 0 and and ς0 r (s) exp −
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