On the oscillation of odd order advanced differential equations

• 767 Accesses

Abstract

The aim of this paper is to study the asymptotic properties and oscillation of the n th order advanced differential equations

$( r ( t ) [ x ( n − 1 ) ( t ) ] γ ) ′ +q(t) x γ [ τ ( t ) ] =0.$

The results obtained are based on the Riccati transformation.

MSC: 34K11, 34C10.

1 Introduction

In this paper, we shall study the asymptotic and oscillation behavior of the solutions of the higher order advanced differential equations

$( r ( t ) [ x ( n − 1 ) ( t ) ] γ ) ′ +q(t) x γ [ τ ( t ) ] =0.$
(1.1)

Throughout the paper, we assume $q,τ∈C([ t 0 ,∞))$, $r∈ C 1 ([ t 0 ,∞))$ and

(H1): n is odd, γ is the ratio of two positive odd integers,

(H2): $r(t)>0$, $r ′ (t)≥0$, $q(t)>0$, $τ(t)≥t$.

Whenever, it is assumed

(1.2)

By a solution of Eq. (1.1), we mean a function $x(t)∈ C n − 1 ([ T x ,∞))$, $T x ≥ t 0$, which has the property $r(t) ( x ( n − 1 ) ( t ) ) γ ∈ C 1 ([ T x ,∞))$ and satisfies Eq. (1.1) on $[ T x ,∞)$. We consider only those solutions $x(t)$ of (1.1) which satisfy $sup{|x(t)|:t≥T}>0$ for all $T≥ T x$. We assume that (1.1) possesses such a solution. A solution of (1.1) is called oscillatory if it has arbitrarily large zeros on $[ T x ,∞)$, and otherwise it is called to be nonoscillatory.

The problem of the oscillation of differential equations has been widely studied by many authors who have provided many techniques especially for lower order delay differential equations. Dong in  improved and extended the Riccati transformation to obtain new oscillatory criteria for the second order delay differential equations

$[ r ( t ) [ x ′ ( t ) ] γ ] ′ +q(t) x γ [ τ ( t ) ] =0.$

Grace et al. in  and the present authors in – used the comparison technique for the third order delay differential equation

$[ r ( t ) [ x ″ ( t ) ] γ ] ′ +q(t) x γ [ τ ( t ) ] =0$

that was compared with the oscillation of certain first order differential equation.

On the other hand, there are comparatively less methods established for the advanced differential equations. The aim of the paper is to fill this gap in the oscillation theory.

Remark 1

All functional inequalities considered in this paper are assumed to hold eventually, that is, they are satisfied for all t large enough.

2 Main results

Our results essentially use the following estimate which is due to Philos and Staikos (see  and ).

Lemma A

Let$z∈ C j ([ t 0 ,∞))$. Assume that$z ( j )$is of fixed sign and not identically zero on a subray of$[ t 0 ,∞)$. If, moreover, $z(t)>0$, $z ( j − 1 ) (t) z ( j ) (t)≤0$, and$lim t → ∞ z(t)≠0$, then for every$k∈(0,1)$there exists$t k ≥ t 0$such that

$z(t)≥ k ( j − 1 ) ! t j − 1 | z ( j − 1 ) (t)|$
(2.1)

holds on$[ t k ,∞)$.

The following useful result will be used later in the proofs of our main results.

Lemma 1

Assume$x(t)>0$, $x ′ (t)>0$, $x ″ (t)>0$, eventually. Then, for arbitrary$k 0 ∈(0,1)$,

$x [ τ ( t ) ] ≥ k 0 τ ( t ) t x(t),$
(2.2)

eventually.

Proof

It follows from the monotonicity of $x ′ (t)$ that

$x [ τ ( t ) ] −x(t)= ∫ t τ ( t ) x ′ (s)ds≥ x ′ (t) ( τ ( t ) − t ) .$

That is,

$x [ τ ( t ) ] x ( t ) ≥1+ x ′ ( t ) x ( t ) ( τ ( t ) − t ) .$
(2.3)

On the other hand, since $x(t)→∞$ as $t→∞$, then for any $k 0 ∈(0,1)$ there exists $t 1$ large enough such that

$k 0 x(t)≤x(t)−x( t 1 )= ∫ t 1 t x ′ (s)ds≤ x ′ (t)(t− t 1 )≤ x ′ (t)t,$

or equivalently,

$x ′ ( t ) x ( t ) ≥ k 0 t .$
(2.4)

Using (2.4) in (2.3), we obtain

$x [ τ ( t ) ] x ( t ) ≥1+ k 0 t ( τ ( t ) − t ) ≥ k 0 τ ( t ) t .$

The proof is complete. □

The positive solutions of (1.1) have the following structure.

Lemma 2

If$x(t)$is a positive solution of (1.1), then$r(t) [ x ( n − 1 ) ( t ) ] γ$is decreasing, all derivatives$x ( i ) (t)$, $1≤i≤n−1$, are of constant signs, and$x(t)$satisfies either

$x ′ (t)>0, x ″ (t)>0, x ( n − 1 ) (t)>0, x ( n ) (t)<0$
(2.5)

or

$( − 1 ) i x ( i ) (t)>0,i=1,2,…,n.$
(2.6)

Proof

Since $x(t)$ is a positive solution of (1.1), then it follows from (1.1) that

$( r ( t ) [ x ( n − 1 ) ( t ) ] γ ) ′ =−q(t) x γ [ τ ( t ) ] <0.$

Thus, $r(t) [ x ( n − 1 ) ( t ) ] γ$ is decreasing, which implies that either $x ( n − 1 ) (t)>0$ or $x ( n − 1 ) (t)<0$. But the case $x ( n − 1 ) (t)<0$ implies $r(t) [ x ( n − 1 ) ( t ) ] γ <−M<0$. An integration from $t 1$ to t yields

$x ( n − 2 ) (t)< x ( n − 2 ) ( t 1 )− M 1 / γ ∫ t 1 t r − 1 / γ (s)ds,$

but in view of (1.2) $x ( n − 2 ) (t)→−∞$ for $t→∞$. Repeating this procedure, we obtain that $x(t)→−∞$ and this is a contradiction, and we conclude that $x ( n − 1 ) (t)>0$. Moreover, $x ( n − 1 ) (t)>0$ implies that either $x ( n − 2 ) (t)>0$ or $x ( n − 2 ) (t)<0$, but the first case leads to $x ( i ) (t)>0$ for $0≤i≤n−2$. Repeating these considerations, we verify that $x(t)$ satisfies either (2.5) or (2.6).

On the other hand, since $x ( n − 1 ) (t)>0$, then using $r ′ (t)>0$ in

$0> ( r ( t ) [ x ( n − 1 ) ( t ) ] γ ) ′ = r ′ (t) [ x ( n − 1 ) ( t ) ] γ +r(t)γ [ x ( n − 1 ) ( t ) ] γ − 1 x ( n ) (t),$

we conclude that $x ( n ) (t)<0$. The proof is complete. □

Now, we offer some criteria for certain asymptotic behavior of all nonoscillatory solutions. For our further references, we set

$Q(t)= ∫ t ∞ q(s) ( τ ( s ) s ) γ ds$

and

$P(t)= 1 r 1 / γ ( t ) [ ∫ t ∞ q ( s ) d s ] 1 / γ .$

Theorem 1

Assume that

$lim inf t → ∞ 1 Q ( t ) ∫ t ∞ s n − 2 Q 1 + 1 / γ ( s ) r 1 / γ ( s ) ds> ( n − 2 ) ! ( γ + 1 ) 1 + 1 / γ$
(2.7)

and

$∫ t 0 ∞ s n − 2 P(s)ds=∞,$
(2.8)

then every nonoscillatory solution$x(t)$of (1.1) satisfies$lim t → ∞ x(t)=0$.

Proof

Assume that $x(t)$ is an eventually positive solution of (1.1). First assume that $x(t)$ satisfies (2.5). By (2.7), it is easy to see that there exists some $k∈(0,1)$ such that

$lim inf t → ∞ k 1 + 1 / γ Q ( t ) ∫ t ∞ s n − 2 Q 1 + 1 / γ ( s ) r 1 / γ ( s ) ds> ( n − 2 ) ! ( γ + 1 ) 1 + 1 / γ .$
(2.9)

We put $k 0 = k 1 / γ$, then setting (2.2) into (1.1), we get

$( r ( t ) [ x ( n − 1 ) ( t ) ] γ ) ′ +kq(t) τ γ ( t ) t γ x γ (t)≤0.$

We define

$w(t)= r ( t ) [ x ( n − 1 ) ( t ) ] γ x γ ( t ) >0.$
(2.10)

Differentiating $w(t)$, one gets

$w ′ ( t ) = [ r ( t ) [ x ( n − 1 ) ( t ) ] γ ] ′ x γ ( t ) − γ r ( t ) [ x ( n − 1 ) ( t ) ] γ x γ ( t ) x ′ ( t ) x ( t ) ≤ − k q ( t ) τ γ ( t ) t γ − γ w ( t ) x ′ ( t ) x ( t ) .$
(2.11)

On the other hand, Lemma A implies

$x ′ (t)≥ k ( n − 2 ) ! t n − 2 x ( n − 1 ) (t).$

Setting the last inequality into (2.11), we obtain

$w ′ (t)≤−k [ q ( t ) ( τ ( t ) t ) γ + γ w 1 + 1 / γ ( t ) t n − 2 ( n − 2 ) ! r 1 / γ ( t ) ] .$

Integrating the last inequality from t to ∞, we have

$w(t)≥k [ Q ( t ) + γ ( n − 2 ) ! ∫ t ∞ w 1 + 1 / γ ( s ) s n − 2 r 1 / γ ( s ) d s ]$
(2.12)

or

$w ( t ) k Q ( t ) ≥1+ γ k 1 + 1 / γ ( n − 2 ) ! Q ( t ) ∫ t ∞ s n − 2 r 1 / γ ( s ) Q 1 + 1 / γ (s) ( w ( s ) k Q ( s ) ) 1 + 1 / γ ds,$

eventually, let us say $t≥ t 1$. Since $w(t)>kQ(t)$, then

$inf t ≥ t 1 w ( t ) k Q ( t ) =λ≥1.$

Thus,

$w ( t ) k Q ( t ) ≥1+ γ ( k λ ) 1 + 1 / γ ( n − 2 ) ! Q ( t ) ∫ t ∞ s n − 2 r 1 / γ ( s ) Q 1 + 1 / γ (s)ds.$
(2.13)

From (2.9), we see that there exists some positive η such that

$k 1 + 1 / γ ( n − 2 ) ! Q ( t ) ∫ t ∞ s n − 2 r 1 / γ ( s ) Q 1 + 1 / γ (s)ds>η> ( γ + 1 ) − γ + 1 γ .$
(2.14)

Combining (2.13) together with (2.14), we have

$w ( t ) k Q ( t ) ≥1+γ λ 1 + 1 / γ η.$

Therefore,

$λ≥1+γ λ 1 + 1 / γ η>1+γ λ 1 + 1 / γ ( γ + 1 ) − γ + 1 γ$

or equivalently,

$0> 1 γ + 1 + γ γ + 1 ( λ γ + 1 ) 1 + 1 / γ − λ γ + 1 .$

This contradicts the fact that the function

$f(α)= 1 γ + 1 + γ γ + 1 α 1 + 1 / γ −α$

is nonnegative for all $α>0$, and we conclude that $x(t)$ cannot satisfy (2.5).

Now we assume that $x(t)$ satisfies (2.6). Then there exists a finite $lim t → ∞ x(t)=ℓ$. We claim that $ℓ=0$. Assume that $ℓ>0$. Integrating (1.1) from t to ∞, we obtain

$r(t) ( x ( n − 1 ) ( t ) ) γ ≥ ∫ t ∞ q(s) x γ [ τ ( s ) ] ds≥ ℓ γ ∫ t ∞ q(s)ds,$

which implies

$x ( n − 1 ) (t)≥ℓP(t).$

Integrating the last inequality twice from t to ∞, we get

$x ( n − 3 ) (t)≥ℓ ∫ t ∞ ∫ u ∞ P(s)dsdu=ℓ ∫ t ∞ P(s)(s−t)ds.$

Repeating this procedure, we arrive at

$− x ′ (t)≥ ℓ ( n − 3 ) ! ∫ t ∞ ( s − t ) n − 3 P(s)ds.$

Now, integrating from $t 1$ to ∞, we see that

$x( t 1 )≥ ℓ ( n − 2 ) ! ∫ t 1 ∞ ( s − t 1 ) n − 2 P(s)ds≥ ℓ 2 n − 2 ( n − 2 ) ! ∫ 2 t 1 ∞ s n − 2 P(s)ds,$

which contradicts (2.8), and so we have verified that $lim t → ∞ x(t)=0$. □

Example 1

Consider the odd order ($n≥3$) nonlinear differential equation

$( t ( x ( n − 1 ) ( t ) ) 3 ) ′ + β t 3 n − 3 x 3 (λt)=0,β>0,λ>1.$
(2.15)

Here $q(t)=β/ t 3 n − 3$ and $τ(t)=λt$, so that

$Q ( t ) = ∫ t ∞ q ( s ) ( τ ( s ) s ) 3 d s = λ 3 β ( 3 n − 4 ) t 3 n − 4 , P ( t ) = 1 r 1 / 3 ( t ) [ ∫ t ∞ q ( s ) d s ] 1 / 3 = ( β 3 n − 4 ) 1 / 3 1 t n − 1 .$

Consequently,

$∫ t 0 ∞ s n − 2 P(s)ds= ( β 3 n − 4 ) 1 / 3 ∫ t 0 ∞ 1 s ds=∞,$

i.e., (2.8) holds; moreover, (2.7) reduces to

$λ β 1 / 3 > ( 3 n − 4 4 ) 4 / 3 (n−2)!,$

which, by Theorem 1, guarantees that all nonoscillatory solutions of (2.15) tend to zero at infinity.

Let ${ A m ( t ) } m = 0 ∞$ be a sequence of continuous functions defined as follows,

and

$A m + 1 (t)= A 0 (t)+ k γ ( n − 2 ) ! ∫ t ∞ A m 1 + 1 / γ (s) s n − 2 r 1 / γ ( s ) ds,m=0,1,….$
(2.16)

Then we have the following result.

Theorem 2

Assume that (2.8) holds and there exists some$A m (t)$such that

$∫ t 0 ∞ q(t) ( τ ( t ) t ) γ exp ( k γ ( n − 2 ) ! ∫ t 0 t A m 1 / γ ( s ) s n − 2 r 1 / γ ( s ) d s ) dt=∞$
(2.17)

for some$k∈(0,1)$. Then every nonoscillatory solution$x(t)$of (1.1) satisfies$lim t → ∞ x(t)=0$.

Proof

Assume that $x(t)$ is an eventually positive solution of (1.1). By Lemma 2, $x(t)$ satisfies either (2.5) or (2.6). It follows from the proof of Theorem 1 that if $x(t)$ satisfies (2.6), then (2.8) insures that it tends to zero at infinity.

Assume that $x(t)$ satisfies (2.5). It follows from the proof of Theorem 1 that (2.12) holds for every $k∈(0,1)$.

By induction, using (2.12), it is easy to see that the sequence ${ A m ( t ) } m = 0 ∞$ is nondecreasing and $w(t)≥ A m (t)$. Thus the sequence ${ A m ( t ) } m = 0 ∞$ converges to $A(t)$. By the Lebesgue monotone convergence theorem and letting $m→∞$ in (2.16), we get

$A(t)= A 0 (t)+ k γ ( n − 2 ) ! ∫ t ∞ A 1 + 1 / γ (s) s n − 2 r 1 / γ ( s ) ds,$

which in view of $A(t)≥ A m (t)$ implies

$A ′ ( t ) = − k q ( t ) ( τ ( t ) t ) γ − k γ ( n − 2 ) ! A 1 + 1 / γ ( t ) t n − 2 r 1 / γ ( t ) ≤ − k q ( t ) ( τ ( t ) t ) γ − k γ ( n − 2 ) ! A ( t ) A m 1 / γ ( t ) t n − 2 r 1 / γ ( t ) ,$

eventually, let us say $t≥ t 1$. Therefore,

$[ A ( t ) exp ( k γ ( n − 2 ) ! ∫ t 1 t A m 1 / γ ( s ) s n − 2 r 1 / γ ( s ) d s ) ] ′ ≤ − k q ( t ) ( τ ( t ) t ) γ exp ( k γ ( n − 2 ) ! ∫ t 1 t A m 1 / γ ( s ) s n − 2 r 1 / γ ( s ) d s ) .$

An integration from $t 1$ to t yields

$0 ≤ A ( t ) exp ( k γ ( n − 2 ) ! ∫ t 1 t A m 1 / γ ( s ) s n − 2 r 1 / γ ( s ) d s ) ≤ A ( t 1 ) − k ∫ t 1 t q ( u ) ( τ ( u ) u ) γ exp ( k γ ( n − 2 ) ! ∫ t 1 u A m 1 / γ ( s ) s n − 2 r 1 / γ ( s ) d s ) d u .$

Letting $t→∞$, we obtain a contradiction. The proof is complete. □

Theorem 3

Assume that (2.8) holds and there exist some$k∈(0,1)$and$A m (t)$such that

$k lim sup t → ∞ t ( n − 1 ) γ r ( t ) A m (t)> ( ( n − 1 ) ! ) γ .$
(2.18)

Then every nonoscillatory solution$x(t)$of (1.1) satisfies$lim t → ∞ x(t)=0$.

Proof

Assume that $x(t)$ is an eventually positive solution of (1.1) satisfying (2.5). It follows from Lemma A that

$x(t)≥ k 1 / γ ( n − 1 ) ! t n − 1 x ( n − 1 ) (t),$

eventually, where $k∈(0,1)$ is the same as in $A m (t)$. Then

$1 w ( t ) = 1 r ( t ) ( x ( t ) x ( n − 1 ) ( t ) ) γ ≥ 1 r ( t ) k ( ( n − 1 ) ! ) γ t ( n − 1 ) γ ,$

or equivalently,

$( ( n − 1 ) ! ) γ ≥k t ( n − 1 ) γ r ( t ) w(t)≥k t ( n − 1 ) γ r ( t ) A m (t),$

which contradicts (2.18). □

Letting $m=0$ in Theorem 3, we have the following result.

Corollary 1

Assume that (2.8) holds and

$lim sup t → ∞ t ( n − 1 ) γ r ( t ) ∫ t ∞ q(s) ( τ ( s ) s ) γ ds> ( ( n − 1 ) ! ) γ .$
(2.19)

Then every nonoscillatory solution$x(t)$of (1.1) satisfies$lim t → ∞ x(t)=0$.

Proof

It follows from (2.19) that there exists some $k∈(0,1)$ such that

$k 2 lim sup t → ∞ t ( n − 1 ) γ r ( t ) ∫ t ∞ q(s) ( τ ( s ) s ) γ ds> ( ( n − 1 ) ! ) γ ,$

which is equivalent to

$k lim sup t → ∞ t ( n − 1 ) γ r ( t ) A 0 (t)> ( ( n − 1 ) ! ) 3 .$

The assertion now follows from Theorem 3. □

Example 2

Consider the third order nonlinear differential equation

$( t 2 ( x ( n − 1 ) ( t ) ) 3 ) ′ + β t 3 n − 4 x 3 (λt)=0,β>0,λ≥1,t≥1.$
(2.20)

A simple calculation leads to

$Q(t)= λ 3 β ( 3 n − 5 ) t 3 n − 5 ,P(t)= ( β 3 n − 5 ) 1 / 3 1 t n − 1 .$

Then (2.8) holds and (2.19) reduces to

$β λ 3 >(3n−5) ( ( n − 1 ) ! ) 3 ,$

and thus, by Corollary 1, every nonoscillatory solution $x(t)$ of (2.20) tends to zero as $t→∞$.

Our results are based on Lemma 1, i.e., we essentially utilize the estimate (2.2). It is easy to see that for $x(t)= t 1 / 2$ and $τ(t)=2t$, estimate (2.2) does not hold, that is, for

$x(t)>0, x ′ (t)>0, x ″ (t)<0,$
(2.21)

relationship (2.2) fails, and so our result here cannot be applied for n even. Hence, it remains an open problem how to obtain the corresponding results also for n even.

References

1. 1.

Dong JG: Oscillation behaviour of second order nonlinear neutral differential equations with deviating argument. Comput. Math. Appl. 2010, 59: 3710-3717. 10.1016/j.camwa.2010.04.004

2. 2.

Grace SR, Agarwal RP, Pavani R, Thandapani E: On the oscillation of certain third order nonlinear functional differential equations. Appl. Math. Comput. 2008, 202: 102-112. 10.1016/j.amc.2008.01.025

3. 3.

Baculíková B, Džurina J: Oscillation of third-order neutral differential equations. Math. Comput. Model. 2010, 52: 215-226. 10.1016/j.mcm.2010.02.011

4. 4.

Baculíková B, Džurina J: Oscillation of third-order nonlinear differential equations. Appl. Math. Lett. 2011, 24: 466-470. 10.1016/j.aml.2010.10.043

5. 5.

Baculíková B: Properties of third order nonlinear functional differential equations with mixed arguments. Abstr. Appl. Anal. 2011., 2011: 10.1155/2011/857860

6. 6.

Baculíková B, Džurina J: Oscillation of third-order functional differential equations. Electron. J. Qual. Theory Differ. Equ. 2010., 2010:

7. 7.

Philos, CG: Oscillation and asymptotic behavior of linear retarded differential equations of arbitrary order. Tech. Report No. 57, Univ. Ioannina (1981)

8. 8.

Philos CG: On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delay. J. Aust. Math. Soc. 1984, 36: 176-186. 10.1017/S1446788700024630

Author information

Correspondence to Jozef Džurina.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions 