Oscillatory Properties of Noncanonical Neutral DDEs of Second-Order

Oscillatory Properties of Noncanonical Neutral DDEs of Second-Order: Comparison

Please note this is a comparison between Version 2 by Lindsay Dong and Version 1 by Clemente Cesarano.

A DDE is a single-variable differential equation, usually called time, in which the derivative of the solution at a certain time is given in terms of the values of the solution at earlier times. Moreover, if the highest-order derivative of the solution appears both with and without delay, then the DDE is called of the neutral type. The neutral DDEs have many interesting applications in various branches of applied science, as these equations appear in the modeling of many technological phenomena. The problem of studying the oscillatory and nonoscillatory properties of DDEs has been a very active area of research in the past few decades.

delay differential equation
neutral
oscillation
noncanonical case

1. Introduction

Consider the 2nd-order delay differential equation (DDE) of the neutral type:

a 0 t v ′ t β ′ + a 2 t u β g 1 t = 0 , (1)

where

t ∈ t 0 , ∞ and

v t : = u t + a 1 t u g 0 t . In this entry, we obtain new sufficient criteria for the oscillation of solutions of (1) under the following hypotheses:

(A1)

β ≥ 1 is a ratio of odd integers;

(A2)

a i ∈ C t 0 , ∞ , 0 , ∞ for

i = 0 , 1 , 2 ,

a 0 t > 0

a 1 ≤ c 0 a constant (this constant plays an important role in the results), and

a 2 does not vanish identically on any half-line

t * , ∞ with

t * ∈ t 0 , ∞ ;

(A3)

g j ∈ C t 0 , ∞ , R

g j t ≤ t

g 0 ′ t ≥ g 0 * > 0 ,

g 0 ∘ g 1 = g 1 ∘ g 0 and

lim t → ∞ g j t = ∞ for

j = 0 , 1

By a proper solution of (1), we mean a

u ∈ C 1 t 0 , ∞ with

a 0 · v ′ β ∈ C 1 t 0 , ∞ and

sup { u t : t ≥ t * } > 0 , for

t * ∈ t 0 , ∞ , and u satisfies (1) on . A solution u of (1) is called nonoscillatory if it is eventually positive or eventually negative; otherwise, it is called oscillatory.

The oscillatory properties of solutions of second-order neutral DDE (

1) in the

) in the

noncanonical case, that is:

, that is:

(2)

where

η t : = ∫ t ∞ a 0 − 1 / β μ d μ .

2. Oscillatory Properties of Noncanonical Neutral DDEs of Second-Order

2.1. Auxiliary Lemmas

2.2. Oscillation Theorems

inWe thbe next theorem, by using the principle of comparison with an equationgin with the following notations: of U + is the first-order, we obtain a new criterion for the oscillationet of all eventually positive solutions of (1).

2.3. Applications

, V t : = a 0 1 / β t v ′ t ,

Lemma 1.

Assume that v ∈ U + and there exists a δ 0 ∈ 0 , 1 such that:

Then, v eventually satisfies:

and: