Open access peer-reviewed chapter

# Oscillation Criteria for Second‐Order Neutral Damped Differential Equations with Delay Argument

By Said R. Grace and Irena Jadlovská

Submitted: May 4th 2016Reviewed: September 21st 2016Published: March 15th 2017

DOI: 10.5772/65909

## Abstract

The chapter is devoted to study the oscillation of all solutions to second‐order nonlinear neutral damped differential equations with delay argument. New oscillation criteria are obtained by employing a refinement of the generalized Riccati transformations and integral averaging techniques.

### Keywords

• neutral differential equation
• damping
• delay
• second‐order
• generalized Riccati technique
• oscillation

## 1. Introduction

In the chapter, we are mainly concerned with the oscillatory behavior of solutions to second‐order nonlinear neutral damped differential equations with delay argument of the form

(r(t)(z(t))α)+p(t)(z(t))α+q(t)f(x(σ(t)))=0,tt0,E1

where α1is a quotient of positive odd integers and

z(t)=x(t)+a(t)x(τ(t)).E2

Throughout, we suppose that the following hypotheses hold:

1. r,p,qC(,+), where =[t0,)and +=(0,);

2. aC(,), 0a(t)1;

3. τC(,), τ(t)t, τ(t)as t;

4. σC1(,), σ(t)t, σ(t)0, σ(t)as t;

5. fC(,), such that xf(x)>0and f(x)/xβk>0for x=0, where kis a constant and βis the ratio of odd positive integers.

By a solution of Eq. (1), we mean a nontrivial real‐valued function x(t), which has the property z(t)C1([Tx,)), r(t)(z(t))αC1([Tx,)), Txt0, and satisfies Eq. (1) on [Tx,). In the sequel, we will restrict our attention to those solutions x(t)of Eq. (1) that satisfy the condition

sup{|x(t)|:Tt<}>0forTTx.E3

We make the standing hypothesis that Eq. (1) admits such a solution. As is customary, a solution of Eq. (1) is said to be oscillatory if it is neither eventually positive nor eventually negative on [Tx,)and otherwise, it is termed nonoscillatory. The equation itself is called oscillatory if all its solutions are oscillatory.

Remark 1.All the functional inequalities considered in the sequel are assumed to hold eventually, that is, they are satisfied for all tlarge enough.

Oscillation theory was created in 1836 with a paper of Jacques Charles François Sturm published in Journal des Mathematiqués Pures et Appliqueés. His long and detailed memoir [1] was one of the first contributions in Liouville's newly founded journal and initiated a whole new research into the qualitative analysis of differential equations. Heretofore, the theory of differential equations was primarily about finding solutions of a given equation and so was very limited. Contrarily, the main idea of Sturm was to obtain geometric properties of solutions (such as sign changes, zeros, boundaries, and oscillation) directly from the differential equation, without benefit of solutions themselves.

Henceforth, the oscillation theory for ordinary differential equations has undergone a significant development. Nowadays, it is considered as coherent, self‐contained domain in the qualitative theory of differential equations that is turning mainly toward the study of solution properties of functional differential equations (FDEs).

The problem of obtaining sufficient conditions for asymptotic and oscillatory properties of different classes of FDEs has experienced long‐term interest of many researchers. This is caused by the fact that differential equations, especially those with deviating argument, are deemed to be adequate in modeling of the countless processes in all areas of science. For a summary of the most significant efforts and recent findings in the oscillation theory of FDEs and vast bibliography therein, we refer the reader to the excellent monographs [26].

In a neutral delay differential equation the highest‐order derivative of the unknown function appears both with and without delay. The study of qualitative properties of solutions of such equations has, besides its theoretical interest, significant practical importance. This is due to the fact that neutral differential equations arise in various phenomena including problems concerning electric networks containing lossless transmission lines (as in high‐speed computers where such lines are used to interconnect switching circuits), in the study of vibrating masses attached to an elastic bar or in the solution of variational problems with time delays. We refer the reader to the monograph [7] for further applications in science and technology.

So far, most of the results obtained in the literature has centered around the special undampedform of Eq. (1), i.e., when p(t)=0(for example, see Refs. [818]). For instance, in one of the pioneering works on the subject, Grammatikopoulos et al. [8] studied the second‐order neutral differential equation with constant delay of the form

(x(t)+a(t)x(tτ)+q(t)x(tτ)=0E4

and proved that Eq. (4) is oscillatory if

t0q(s)(1a(sτ))ds=.E5

Later on, Grace and Lalli [9] extended the results from [8] to the more general equation

(r(t)(x(t)+a(t)x(tτ))+q(t)f(x(tτ))=0,E6

with

f(x)xk,k>0andt0dsr(s)=E7

and showed that Eq. (6) is oscillatory if there exists a continuously differentiable function ρ(t)such that

t0(ρ(s)q(s)(1a(sτ))(ρ(s))2r(sτ)4kρ(s))ds=.E8

In Ref. [10], Dong has involved to study the oscillation problem for a half‐linear case of Eq. (1) and by defining a sequence of continuous functions has obtained various kinds of better results. Afterward, his approach has been further developed by several authors, see, e.g., [1114]. However, it appears that very little is known regarding the oscillation of Eq. (1) with p(t)0and αβ. Motivated by the results of Ref. [10], this chapter presents some new oscillation criteria, which are applicable on Eq. (1).

On the other hand, Eq. (1) can be considered as a natural generalization of the second‐order delay differential equation of the form

(r(t)(x(t))α)+p(t)(x(t))α+q(t)f(x(σ(t)))=0.E9

Very recently, the authors of [19] studied the oscillation problem of Eq. (9) with p(t)=0and α=β. Their ideas, which are based on careful investigation of classical techniques covering Riccati transformations and integral averages, will be extended to the more general equation (1).

## 2. Main results

For the simplicity and without further mention, we use the following notations:

A(t)=exp(t0tp(s)r(s)ds),Q(t)=kq(t)(1a(σ(t)))β,E10
R(t)=t(A(s)r(s))1αds,Q˜(t)=q(t)(1a(σ(t))R(τ(σ(t)))R(σ(t)))β,E11
P(t)=ϕ(t)ϕ(t)p(t)r(t),q˜(t)=Q(t)+p(t)A(t)r(t)tQ(s)A(s)ds,E12

where ϕ(t)C1(,)is a given function and will be specified later.

The organization of this chapter is as follows. Before stating our main results, we present two lemmas that ensure that any solution x(t)of Eq. (1) satisfies the condition

z(t)>0,z(t)>0,(r(t)(z(t))α)<0,E13

for tsufficiently large. Next, we get our main oscillation results for Eq. (1) by employing the generalized Riccati transformations and integral averaging techniques. We base our arguments on the assumption that the function P(t)is positive or negative.

Lemma 1.Assume that

t0(A(s)r(s))1αds=E14

holds and Eq. (1) has a positive solution x(t)on . Then there exists a T, sufficiently large, such that

z(t)>0,z(t)>0,(r(t)(z(t))α)<0,E15

on [T,).

Proof.Since, x(t)is a positive solution of Eq. (1) on , then, by the assumptions (iii)and (iv), there exists a t1such that x(τ(t))>0and x(σ(t))>0on [t1,). Define the function z(t)as in Eq. (2). Then it is easy to see that z(t)x(t)>0, for tt1, and at the same time, from Eq. (1), we get

(r(t)(z(t))α)+p(t)(z(t))α=q(t)f(x(σ(t)))<0.E16

We assert that r(t)A(t)(z(t))αis decreasing. Clearly, by writing the left‐hand side of Eq. (16) in the form

(r(t)(z(t))α)+p(t)r(t)r(t)(z(t))α<0,E17

we get

(r(t)A(t)(z(t))α)=q(t)A(t)f(x(σ(t)))<0E18

and so the assertion is proved.

Now, we claim that z(t)>0on [t1,). If not, then there exists t2[t1,)such that z(t2)<0. Using the fact that r(t)A(t)(z(t))αis decreasing, we obtain, for tt2,

r(t)A(t)(z(t))α<c:=r(t2)A(t2)(z(t2))α<0.E19

Integrating the above inequality from t2to t, we find that

z(t)<z(t2)+c1αt2t(A(s)r(s))1αdsE20

for tt2.By condition (14), z(t)approaches to as t, which contradicts the fact that z(t)is eventually positive. Therefore, z(t)>0and from Eq. (1), we have that (r(t)(z(t))α)<0. The proof is complete.

Lemma 2.Assume that

t0(A(u)r(u)t0uQ˜(s)Rβ(σ(s))A(s)ds)1αdu=,E21

holds and Eq. (1) has a positive solution x(t)on . Then there exists T, sufficiently large, such that

z(t)>0,z(t)>0,(r(t)(z(t))α)<0,E22

on [T,).

Proof.Similarly to the proof of Lemma 1, we assume that there exists t2such that z(t)<0on [t2,). Taking Eq. (18) into account, we have

z(s)(r(t)A(t)A(s)r(s))1αz(t),E23

for stt2. Integrating the above inequality from tto t, ttt2, we get

z(t)z(t)+(r(t)A(t))1αz(t)tt(r(s)A(s))1αds.E24

Letting t, we have

z(t)R(t)(r(t)A(t))1αz(t),E25

which yields

(z(t)R(t))0E26

and hence we see that z(t)R(t)is nondecreasing. By Eq. (2) and (iii), we have

x(t)=z(t)a(t)x(τ(t))z(t)a(t)z(τ(t))(1a(t)R(τ(t))R(t))z(t),E27

which together with Eq. (1) and the assumption (v)yields

(r(t)(z(t))α)+p(t)(z(t))αkq(t)(1a(σ(t))R(τ(σ(t)))R(σ(t)))βzβ(σ(t))=kQ˜(t)zβ(σ(t)).E28

On the other hand, from Eq. (23), we have

r(t)(z(t))αA(t)r(t2)(z(t2))αA(t2),E29

that is,

r(t)A(t)(z(t))αr(t2)A(t2)(z(t2))α:=γαE30

for some positive constant γ. Setting Eq. (30) into Eq. (25), we obtain

z(t)γR(t)E31

and so, Eq. (28) becomes

(r(t)(z(t))α)+p(t)(z(t))αγ˜Q˜(t)Rβ(σ(t)),E32

where γ˜:=kγβ. Now, if we define the function

U(t)=r(t)(z(t))α>0,E33

then

U(t)+p(t)r(t)U(t)γ˜Q˜(t)Rβ(σ(t)),E34

or, equally

(U(t)A(t))γ˜Q˜(t)Rβ(σ(t))A(t).E35

Integrating the above inequality from t2to t, we get

U(t)γ˜A(t)t2tQ˜(s)Rβ(σ(s))A(s)dsE36

or

r(t)(z(t))αγ˜A(t)t2tQ˜(s)Rβ(σ(s))A(s)ds.E37

It follows from this last inequality that

0<z(t)z(t2)γ˜t2t(A(u)r(u)t2uQ˜(s)Rβ(σ(s))A(s)ds)1αduE38

for tt2.As t, then by condition Eq. (21), z(t)approaches to , which contradicts the fact that z(t)is eventually positive. Therefore, z(t)>0and from Eq. (1), we have (r(t)(z(t))α)<0. The proof is complete.

Lemma 3.Assume that

t0(A(u)r(u)t0uQ˜(s)A(s)ds)1αdu=,E39

holds and Eq. (1) has a positive solution x(t)on . Then there exists T, sufficiently large, such that either

z(t)>0,z(t)>0,(r(t)(z(t))α)<0,E40

on [T,)or limtx(t)=0.

Proof.As in the proof of Lemma 1, we assume that there exists t2such that z(t)<0on [t2,). So, z(t)is decreasing and

limtz(t)=:b0E41

exists. Therefore, there exists t3[t2,)such that

z(σ(t))>z(t)b>0.E42

As in the proof of Lemma 2, we obtain Eq. (27), i.e.,

x(σ(t))(1a(σ(t))R(τ(σ(t)))R(σ(t)))z(σ(t))b(1a(σ(t))R(τ(σ(t)))R(σ(t))),fortt3.E43

Thus,

(r(t)(z(t))α)+p(t)(z(t))αb˜q(t)(1a(σ(t))R(τ(σ(t)))R(σ(t)))β=b˜Q˜(t),E44

where b˜:=kbβ.

Define the function U(t)as in Eq. (103). Then Eq. (44) becomes

(U(t)A(t))b˜Q˜(t)A(t).E45

Integrating the above inequality twice from t3to t, one gets

0<z(t)z(t3)b˜t3t(A(u)r(u)t3uQ˜(s)A(s)ds)1αdu,E46

for tt3.As t, then by condition (39), z(t)approaches to , which contradicts the fact that z(t)is eventually positive. Thus, b=0and from 0x(t)z(t),we see that limtx(t)=0. The proof is complete.

Using results of Lemmas 1 and 2, we can obtain the following oscillation criteria for Eq. (1).

Theorem 1.Let conditions (i)(v)and one of the conditions (14) or (21) hold. Furthermore, assume that there exists a positive continuously differentiable function ϕ(t)such that, for all sufficiently large, T, T1T,

P(t)0E47

on [T,)and

limsupt{ϕ(t)A(t)tQ(s)A(s)ds+T1t[ϕ(s)Q(s)αα(α+1)α+1ϕ(s)r(σ(s))(P(s))α+1(βσ(s)ψ(s))α]ds}=,E48

where

ψ(t)={c1,c1issomepositiveconstantifβ>α1,ifβ=αc2(Ttr1α(s)ds)βαα,c2issomepositiveconstantifβ<α.E49

Then, Eq. (1) is oscillatory.

Proof.Suppose to the contrary that x(t)is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists Tlarge enough, so that x(t)satisfies the conclusions of Lemma 1 or 2 on [T,)with

x(t)>0,x(τ(t))>0,x(σ(t))>0E50

on [T,). In particular, we have

z(t)>0,z(t)>0,(r(t)(z(t))α)<0,fortT.E51

By Eq. (2) and the assumption (iii), we get

x(t)=z(t)a(t)x(τ(t))z(t)a(t)z(τ(t))(1a(t))z(t),E52

which together with Eq. (1) implies

(r(t)(z(t))α)+p(t)r(t)(z(t))αkq(t)(1a(σ(t)))βzβ(σ(t))=Q(t)zβ(σ(t)).E53

We consider the generalized Riccati substitution

w(t)=ϕ(t)r(t)(z(t))αzβ(σ(t))>0,fortT.E54

As in the proof of Lemma 1, we get Eq. (18), which in view of the assumption (v)yields

(r(t)A(t)(z(t))α)Q(t)A(t)zβ(σ(t)).E55

Integrating Eq. (55) from tto and using the fact that z(t)is increasing, we have

r(t)A(t)(z(t))αtQ(s)A(s)zβ(σ(s))dszβ(σ(t))tQ(s)A(s)ds.E56

So it follows from Eq. (56) and the definition (54) of w(t)that

w(t)=ϕ(t)r(t)(z(t))αzβ(σ(t))ϕ(t)A(t)tQ(s)A(s)ds.E57

By Eq. (53) we can easily prove that

w(t)=(r(t)(z(t))α)ϕ(t)zβ(σ(t))+(ϕ(t)zβ(σ(t)))r(t)(z(t))αϕ(t)zβ(σ(t))(p(t)(z(t))β+Q(t)zβ(σ(t)))+r(t)(z(t))α(ϕ(t)zβ(σ(t))ϕ(t)(zβ(σ(t)))zβ+1(σ(t)))ϕ(t)Q(t)+w(t)(ϕ(t)ϕ(t)p(t)r(t))βϕ(t)r(t)(z(t)βz(σ(t)σ(t)zβ+1(σ(t)).E58

On the other hand, since r(t)(z(t))αis decreasing, we have

z(σ(t))z(t)(r(t)r(σ(t)))1αE59

and thus Eq. (58) becomes

w(t)ϕ(t)Q(t)+P(t)w(t)βϕ(t)σ(t)r1α(σ(t))(w(t)ϕ(t))α+1αzβαα(σ(t)).E60

Now, we consider the following three cases:

Case I: β>α.

In this case, since z(t)>0for tT, then there exists T1Tsuch that z(σ(t))cfor tT1. This implies that

zβαα(σ(t))cβαα:=c1E61

Case II: β=α.

In this case, we see that zβαα(σ(t))=1.

Case III: β<α.

Since r(t)(z(t))αis decreasing, there exists a constant dsuch that

r(t)(z(t))αdE62

for tT. Integrating the above inequality from Tto t, we have

z(t)z(T)+Tt(dr(s))1αds.E63

Hence, there exists T1Tand a constant d1depending on dsuch that

z(t)d1Ttr1α(s)ds,fortT1E64

and thus

zβαα(σ(t))d1βαα(Ttr1α(s)ds)βαα=d2(Ttr1α(s)ds)βααE65

for some positive constant d2.

Using these three cases and the definition of ψ(t), we get

w(t)ϕ(t)Q(t)+P(t)w(t)βσ(t)ψ(t)(ϕ(t)r(σ(t)))1αw1+αα(t)E66

for tT1T. Setting

A:=P(t),E67
B:=βσ(t)ψ(t)(ϕ(t)r(σ(t)))1α,E68

and using the inequality

AuBu1+αααα(α+1)α+1Aα+1Bα,E69

we obtain

w(t)ϕ(t)Q(t)+αα(α+1)α+1ϕ(t)r(σ(t))(P(t))α+1(βσ(t)ψ(t))α.E70

Integrating the above inequality from T1to t, we have

w(t)w(T1)T1t(ϕ(s)Q(s)αα(α+1)α+1ϕ(s)r(σ(s))(P(s))α+1(βσ(s)ψ(s))α)ds.E71

Taking Eq. (57) into account, we get

w(T1)ϕ(t)A(t)tQ(s)A(s)ds+T1t(ϕ(s)Q(s)αα(α+1)α+1ϕ(s)r(σ(s))(P(s))α+1(βσ(s)ψ(s))α)ds.E72

Taking the lim sup on both sides of the above inequality as t, we obtain a contradiction to the condition (48). This completes the proof.

Remark 2.Note that the presence of the term ϕ(t)A(t)tQ(s)A(s)dsin Eq. (57) improves a number of related results in, e.g., [9, 1318, 20].

Setting ϕ(t)=tin Eq. (57), then the following corollary becomes immediate.

Corollary 1.Let conditions (i)(v)and one of the conditions (14) or (21) hold. Assume that, for all sufficiently large, T, T1T,

tp(t)r(t)E73

on [T,)and

limsupt{tA(t)tQ(s)A(s)ds+T1t[sQ(s)αα(α+1)α+1sr(σ(s))(1sp(s)r(s))α+1(βσ(s)ψ(s))α]ds}=,E74

where ψ(t)is as in Theorem 1. Then Eq. (1) is oscillatory.

Corollary 2.Assume that the conditions (39) and (74) hold. Then Eq. (1) is oscillatory or limtx(t)=0.

Next, we present some complementary oscillation results for Eq. (1) by using an integral averaging technique due to Philos. We need the class of functions F. Let

D0={(t,s):t>st0}andD={(t,s):t>st0}E75

The function H(t,s)C(D,)is said to belong to a class Fif

(a)H(t,t)=0for tT, H(t,s)>0for (t,s)D0

(b)H(t,s)has a continuous and nonpositive partial derivative on D0with respect to the second variable such that

s(H(t,s)ϕ(s))H(t,s)ϕ(s)p(s)r(s)=h(t,s)(H(t,s)ϕ(s))αα+1E76

for all (t,s)D0.

Theorem 2.Let conditions (i)(v)and one of the conditions (14) or (21) hold. Furthermore, assume that there exist functions H(t,s), h(t,s)Fsuch that, for all sufficiently large, T, for T1T,

limsupt1H(t,T1)T1t(H(t,s)(ϕ(s)Q(s)+ρ(s)ϕ(s)p(s))αα(α+1)α+1hα+1(t,s)r(σ(s))βα(σ(s)ψ(s))α)ds=E77

where ϕ(t)and ρ(t)are continuously differentiable functions and ψ(t)is as in Theorem 1. Then Eq. (1) is oscillatory.

Proof.Suppose to the contrary that x(t)is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists Tlarge enough, so that x(t)satisfies the conclusions of Lemma 1 or 2 on [T,)with

x(t)>0,x(τ(t))>0,x(σ(t))>0E78

on [T,). In particular, we have

z(t)>0,z(t)>0,(r(t)(z(t))α)<0,fortT.E79

Define the function w(t)as

w(t)=ϕ(t)r(t)((z(t))αzβ(σ(t))+ρ(t))ϕ(t)r(t)ρ(t),E80

where ρ(t)C1(,). Similarly to the proof of Theorem 1, we obtain the inequality

w(t)ϕ(t)Q(t)+ϕ(t)(r(t)ρ(t))+(ϕ(t)ϕ(t)p(t)r(t))w(t)βσ(t)ψ(t)(ϕ(t)r(σ(t)))1α(w(t)ϕ(t)r(t)ρ(t))1+αα.E81

Multiplying Eq. (81) by H(t,s), integrating with respect to sfrom T1to tfor tT1T, and using (a)and (b), we find that

T1tH(t,s)ϕ(s)(Q(s)(r(s)ρ(s)))dsTtH(t,s)w(s)ds+T1tH(t,s)(ϕ(s)ϕ(s)p(s)r(s))w(s)dsT1tβH(t,s)σ(s)ψ(s)(ϕ(s)r(σ(s)))1α(w(s)ϕ(s)r(s)ρ(s))1+ααds=H(t,s)w(s)|T1t+T1t(sH(t,s)+H(t,s)(ϕ(s)ϕ(s)p(s)r(s)))w(s)dsT1tβH(t,s)σ(s)ψ(s)(ϕ(s)r(σ(s)))1α(w(s)ϕ(s)r(s)ρ(s))1+ααds=H(t,T1)w(T1)+T1th(t,s)ϕ(s)(H(t,s)ϕ(s))αα+1w(s)dsT1tβH(t,s)σ(s)ψ(s)(ϕ(s)r(σ(s)))1α(w(s)ϕ(s)r(s)ρ(s))1+ααdsE82

Setting

A:=h(t,s)ϕ(s)[H(t,s)ϕ(s)]αα+1,B:=βH(t,s)σ(s)ψ(s)(ϕ(s)r(σ(s)))1αE83

and

C:=ϕ(s)r(s)ρ(s)E84

and using the inequality

AuB(uC)1+ααAC+αα(α+1)α+1Aα+1Bα,E85

we obtain

T1tH(t,s)ϕ(s)(Q(s)(r(s)ρ(s)))dsH(t,T1)w(T1)+T1th(t,s)r(s)ρ(s)[H(t,s)ϕ(s)]αα+1ds+T1tαα(α+1)α+1hα+1(t,s)r(σ(s))βα(σ(s)ψ(s))αdsE86

Thus,

H(t,T1)w(T1)T1tH(t,s)ϕ(s)(Q(s)(r(s)ρ(s)))ds+T1th(t,s)r(s)ρ(s)[H(t,s)ϕ(s)]αα+1dsT1tαα(α+1)α+1hα+1(t,s)r(σ(s))βα(σ(s)ψ(s))αds.E87

That is,

H(t,T1)w(T1)T1tH(t,s)ϕ(s)(Q(s)(r(s)ρ(s)))ds+T1tr(s)ρ(s)(s(H(t,s)ϕ(s))H(t,s)ϕ(s)p(s)r(s))dsT1tαα(α+1)α+1hα+1(t,s)r(σ(s))βα(σ(s)ψ(s))αds=T1tH(t,s)(ϕ(s)Q(s)+ρ(s)ϕ(s)p(s))dsH(t,s)ϕ(s)r(s)ρ(s)|T1tT1tαα(α+1)α+1hα+1(t,s)r(σ(s))βα(σ(s)ψ(s))αdsE88

It follows that

T1tH(t,s)(ϕ(s)Q(s)+ρ(s)ϕ(s)p(s))dsT1tαα(α+1)α+1hα+1(t,s)r(σ(s))βα(σ(s)ψ(s))αdsH(t,T1)(w(T1)ϕ(T1)r(T1)ρ(T1)),E89

which is a contradiction to Eq. (77). The proof is complete.

Remark 3.Authors in [15, 20] studied a partial case of Eq. (1) by employing the generalized Riccati substitution (80). Note that the function ρ(t)used in the generalized Riccati substitution (80) finally becomes unimportant. Thus, we can put ρ(t)=0and obtain similar results to those from [15, 20].

In the next part, we provide several oscillation results for Eq. (1) under the assumption that the function P(t)is nonpositive. These results generalize those from [10] for Eq. (1) in such sense that αβand p(t)0.

Theorem 3.Let conditions (i)(v)and one of the conditions (14) or (21) hold. Furthermore, assume that there exists a continuously differentiable function ϕ(t)such that, for all sufficiently large, T, T1T,

P(t)0E90

on [T,)and

limsupt[ϕ(t)A(t)tQ(s)A(s)ds+T1tϕ(s)(Q(s)A(s)P(s)sQ(u)A(u)du)ds]=.E91

Then Eq. (1) is oscillatory.

Proof.Suppose to the contrary that x(t)is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists Tlarge enough, so that x(t)satisfies the conclusions of Lemma 1 or 2 on [T,)with

x(t)>0,x(τ(t))>0,x(σ(t))>0E92

on [T,). In particular, we have

z(t)>0,z(t)>0,(r(t)(z(t))α)<0,fortT.E93

Proceeding as in the proof of Theorem 1, we obtain the inequality (66), i.e.,

w(t)ϕ(t)Q(t)+P(t)w(t)βσ(t)ψ(t)(ϕ(t)r(σ(t)))1αw1+αα(t)E94

for tT1T. Using Eq. (90), and setting Eq. (57) in Eq. (94), we get

w(t)ϕ(t)Q(t)+ϕ(t)A(t)P(t)tQ(s)A(s)dsβσ(t)ψ(t)(ϕ(t)r(σ(t)))1αw1+αα(t)ϕ(t)Q(t)+ϕ(t)A(t)P(t)tQ(s)A(s)ds,E95

that is,

w(t)+ϕ(t)Q(t)ϕ(t)A(t)P(t)tQ(s)A(s)ds0.E96

Integrating the above inequality from T1to t, we have

w(T1)w(t)+T1t(ϕ(s)Q(s)ϕ(s)A(s)P(s)sQ(u)A(u)du)dsϕ(t)A(t)tQ(s)A(s)ds+T1t(ϕ(s)Q(s)ϕ(s)A(s)P(s)sQ(u)A(u)du)dsE97

Taking the lim sup on both sides of the above inequality as t, we obtain a contradiction to condition Eq. (91). This completes the proof.

Setting ϕ(t)=1, we have the following consequence.

Corollary 3.Let conditions (i)(v)and one of the conditions (14) or (21) hold. Assume that

limsupt[A(t)tQ(s)A(s)ds+T1tq˜(s)ds]=,E98

for all sufficiently large T, for T1T. Then Eq. (1) is oscillatory.

Define a sequence of functions {yn(t)}n=0as

y0(t)=tq˜(s)ds,tTE99
yn(t)=tβσ(s)ψ(s)r1α(σ(s))(yn1(s))1+ααds+y0(t),tT,n=1,2,3,,E100

for Tt0sufficiently large.

By induction, we can see that ynyn+1, n=1,2,3,.

Lemma 4.Let conditions (i)(v)and one of the conditions (14) or (21) hold. Assume that x(t)is a positive solution of Eq. (1) on . Then there exists T, sufficiently large, such that

w(t)yn(t),E101

where w(t)and yn(t)are defined as Eqs. (54) and (100), respectively. Furthermore, there exists a positive function y(t)on [T1,), T1T, such that

limnyn(t)=y(t)E102

and

y(t)=tβσ(s)ψ(s)r1α(σ(s))(y(s))1+ααds+y0(t).E103

Proof.Similarly to the proof of Theorem 3, we obtain Eq. (95). Setting ϕ(t)=1in Eq. (95), we get

w(t)+Q(t)+p(t)A(t)r(t)tQ(s)A(s)ds+βσ(t)ψ(t)r1α(σ(t))w1+αα(t)0E104

for tT1T. Integrating Eq. (104) from tto t, we get

w(t)w(t)+ttq˜(s)ds+ttβσ(s)ψ(s)r1α(σ(s))w1+αα(s)ds0E105

or

w(t)w(t)+ttβσ(s)ψ(s)r1α(σ(s))w1+αα(s)ds0.E106

We assert that

tβσ(s)ψ(s)r1α(σ(s))w1+αα(s)ds<.E107

If not, then

w(t)w(t)ttβσ(s)ψ(s)r1α(σ(s))w1+αα(s)dsE108

as t, which contradicts to the positivity of w(t)and thus the assertion is proved. By Eq. (104), we see that w(t)is decreasing that means

limtw(t)=k,k0.E109

By virtue of Eq. (107), we have k=0. Thus, letting tin Eq. (105), we get

w(t)tq˜(s)ds+tβσ(s)ψ(s)r1α(σ(s))w1+αα(s)ds=y0(t)+tβσ(s)ψ(s)r1α(σ(s))w1+αα(s)ds,E110

that is,

w(t)tq˜(s)ds=y0(t).E111

Moreover, by induction, we have that

w(t)yn(t),fortT1,n=1,2,3,.E112

Thus, since the sequence {yn(t)}n=0is monotone increasing and bounded above, it converges to y(t). Letting nand using Lebesgue monotone convergence theorem in Eq. (100), we get Eq. (103). The proof is complete.

Theorem 4.Let conditions (i)(v)and one of the conditions (14) or (21) hold. If

liminft(1y0(t)tβσ(s)ψ(s)r1α(σ(s))(y0(s))1+ααds)>α(α+1)1+αα,E113

where ψ(t)is as in Theorem 1, then Eq. (1) is oscillatory.

Proof.Suppose to the contrary that x(t)is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists Tlarge enough, so that x(t)satisfies the conclusions of Lemma 1 or 2 on [T,)with

x(t)>0,x(τ(t))>0,x(σ(t))>0E114

on [T,). In particular, we have

z(t)>0,z(t)>0,(r(t)(z(t))α)<0,fortT.E115

By Eq. (113), there exists a constant γ>α(α+1)1+ααsuch that

liminft1y0(t)tβσ(s)ψ(s)r1α(σ(s))(y0(s))1+ααds>γ.E116

Proceeding as in the proof of Lemma 4, we obtain Eq. (110) and from that, we have

w(t)y0(t)1+1y0(t)tβσ(s)ψ(s)r1α(σ(s))(y0(s))1+αα(w(s)y0(s))1+ααdsE117

Let

λ=inftt1w(t)y0(t).E118

Then it is easy to see that λ1and

λ1+λ1+ααγ,E119

which contradicts the admissible value of λand γ, and thus completes the proof.

Theorem 5.Let conditions (i)(v), one of the conditions (14) or (21) hold, and yn(t)be defined as in Eq. (100). If there exists some yn(t)such that, for Tsufficiently large,

limsuptyn(t)(Tσ(t)r1α(s)ds)α>1ψ(t),E120

where ψ(t)is as in Theorem 1, then Eq. (1) is oscillatory.

Proof.Suppose to the contrary that x(t)is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists Tlarge enough, so that x(t)satisfies the conclusions of Lemma 1 or 2 on [T,)with

x(t)>0,x(τ(t))>0,x(σ(t))>0E121

on [T,). In particular, we have

z(t)>0,z(t)>0,(r(t)(z(t))α)<0,fortT.E122

Proceeding as in the proof of Theorem 3 and using defining w(t)as in Eq. (54), for T1T, we get

1w(t)=zβ(σ(t))r(t)(z(t))αψ(t)r(t)(z(σ(t))z(t))α=ψ(t)r(t)(z(T1)+T1σ(t)r1α(s)r1α(s)z(s)dsz(t))αψ(t)(T1σ(t)r1α(s)ds)αE123

Thus,

w(t)(Tσ(t)r1α(s)ds)α1ψ(t)(Tσ(t)r1α(s)dsT1σ(t)r1α(s)ds)αE124

And therefore,

limsuptw(t)(Tσ(t)r1α(s)ds)α1ψ(t),E125

which contradicts Eq. (120). The proof is complete.

Theorem 6.Let conditions (i)(v), one of the conditions (14) or (21) hold, and yn(t)be defined as in Eq. (100). If there exists some yn(t)such that

T1q˜(t)exp(T1tβσ(s)ψ(s)r1α(σ(s))yn1α(s)ds)dt=E126

or

T1βσ(t)ψ(t)yn1α(t)y0(t)r1α(σ(t))exp(T1tβσ(s)ψ(s)r1α(σ(s))yn1α(s)ds)dt=,E127

for Tsufficiently large and T1T, where ψ(t)is as in Theorem 1, then Eq. (1) is oscillatory.

Proof.Suppose to the contrary that x(t)is a nonoscillatory solution of Eq. (1). Then, without loss of generality, we may assume that there exists Tlarge enough, so that x(t)satisfies the conclusions of Lemma 1 or 2 on [T,)with

x(t)>0,x(τ(t))>0,x(σ(t))>0E128

on [T,). In particular, we have

z(t)>0,z(t)>0,(r(t)(z(t))α)<0,fortT.E129

From Eq. (103), we have

y(t)=βσ(t)ψ(t)r1α(σ(t))(y(t))1+ααq˜(t),E130

for all tT1T. Since y(t)yn(t), Eq. (130) yields

y(t)βσ(t)ψ(t)r1α(σ(t))yn1α(t)y(t)q˜(t).E131

Multiplying the above inequality by the integration factor

exp(T1tβσ(s)ψ(s)r1α(σ(s))yn1α(s)ds),E132

one gets

y(t)exp(T1tβσ(s)ψ(s)r1α(σ(s))yn1α(s)ds)×(y(t1)T1tq˜(s)exp(T1sβσ(u)ψ(u)r1α(σ(u))yn1α(u)du)ds),E133

from which we have that

T1tq˜(s)exp(T1sβσ(u)ψ(u)r1α(σ(u))yn1α(u)du)dsy(T1)<.E134

This is a contradiction with Eq. (126).

Now denote

u(t)=tβσ(s)ψ(s)r1α(σ(s))(y(s))1+ααdsE135

Taking the derivative of u(t), one gets

u(t)=βσ(t)ψ(t)r1α(σ(t))(y(t))1+ααβσ(t)ψ(t)r1α(σ(t))yn1α(t)y(t)=βσ(t)ψ(t)r1α(σ(t))yn1α(t)(u(t)+y0(t))E136

Proceeding in a similar manner to that above, we conclude that

T1βσ(t)ψ(t)r1α(σ(t))yn1α(t)y0(t)exp(T1tβσ(s)ψ(s)r1α(σ(s))yn1α(s)ds)dt<,E137

which contradicts to Eq. (127). The proof is complete.

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Said R. Grace and Irena Jadlovská (March 15th 2017). Oscillation Criteria for Second‐Order Neutral Damped Differential Equations with Delay Argument, Dynamical Systems - Analytical and Computational Techniques, Mahmut Reyhanoglu, IntechOpen, DOI: 10.5772/65909. Available from:

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