Skip to content

Advertisement

Open Access

Properties of higher-order half-linear functional differential equations with noncanonical operators

  • Chenghui Zhang1Email author,
  • Ravi P Agarwal2,
  • Martin Bohner3 and
  • Tongxing Li1
Advances in Difference Equations20132013:54

https://doi.org/10.1186/1687-1847-2013-54

Received: 4 December 2012

Accepted: 7 February 2013

Published: 12 March 2013

Abstract

Some new results are presented for the oscillatory and asymptotic behavior of higher-order half-linear differential equations with a noncanonical operator. We study the delayed and advanced equations subject to various conditions.

MSC:34C10, 34K11.

Keywords

asymptotic behavioroscillationfunctional differential equationhigher-order

1 Introduction

Over the past few years, there has been much research activity concerning the oscillation and asymptotic behavior of various classes of differential equations; we refer the reader to [136] and the references cited therein. Half-linear differential equations occur in a variety of real world problems such as in the study of p-Laplace equations, non-Newtonian fluid theory, and the turbulent flow of a polytrophic gas in a porous medium; see the related background details reported in [5]. Many authors have studied the properties of solutions of the higher-order differential equation
L x + q ( t ) x β ( τ ( t ) ) = 0 , L x : = ( r ( x ( n 1 ) ) α ) ( t ) .
(1.1)

The operator Lx is said to be in canonical form if t 0 r 1 / α ( t ) d t = ; otherwise, it is called noncanonical. Throughout the paper, we assume that α and β are ratios of odd positive integers, r C 1 [ t 0 , ) , r ( t ) > 0 , r ( t ) 0 , q , τ C [ t 0 , ) , q ( t ) > 0 , and lim t τ ( t ) = .

Agarwal et al. [6] established a criterion for the existence of bounded solutions of (1.1) under the assumptions that n is even, t 0 q ( t ) d t = , and
t 0 r 1 / α ( t ) d t < .
(1.2)
Zhang et al. [34, 36] obtained some results on asymptotic behavior of (1.1) in the case where (1.2) holds, τ ( t ) < t , and β α . In [34, 36], an unsolved problem can be formulated as follows.
  1. (P)

    Is it possible to establish asymptotic criteria for equation (1.1) in the case where β α ?

     
As a special case when α = 1 and n = 2 , equation (1.1) becomes
( r x ) ( t ) + q ( t ) x β ( τ ( t ) ) = 0 .
(1.3)

Li et al. [24] established the following criterion for (1.3).

Theorem 1.1 (See [[24], Theorem 2.1])

Let (1.2) hold with α = 1 , β 1 , τ ( t ) t , and τ ( t ) > 0 for all t t 0 . Assume that there exists a function ρ C 1 ( [ t 0 , ) , R ) with ρ ( t ) t and ρ ( t ) > 0 such that, for all sufficiently large t 1 and for all positive constants M and L,
[ q ( t ) R β ( τ ( t ) ) β M 1 β τ ( t ) R β 1 ( τ ( t ) ) r ( τ ( t ) ) t 1 t τ ( s ) r ( τ ( s ) ) d s ] d t =
and
[ q ( t ) ξ β ( t ) β ρ ( t ) L β 1 ξ ( t ) r ( ρ ( t ) ) ] d t = ,
(1.4)

where R ( t ) : = t 0 t r 1 ( s ) d s and ξ ( t ) : = ρ ( t ) r 1 ( s ) d s . Then (1.3) is oscillatory.

The purpose of this paper is to solve question (P) and to improve Theorem 1.1. By a solution of equation (1.1) we mean a function x C n 1 [ T x , ) , T x t 0 , which has the property r ( x ( n 1 ) ) α C 1 [ T x , ) and satisfies (1.1) on [ T x , ) . We consider only the solutions satisfying sup { | x ( t ) | : t T } > 0 for all T T x and tacitly assume that (1.1) possesses such solutions. A solution of (1.1) is called oscillatory if it has arbitrarily large zeros on [ T x , ) ; otherwise, it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.

2 Main results

In the sequel, all functional inequalities are assumed to hold eventually, that is, they are satisfied for all t large enough. We use the notation δ ( t ) : = t r 1 / α ( s ) d s and ( ρ ( t ) ) + : = max { 0 , ρ ( t ) } .

Theorem 2.1 Assume (1.2) and let n 2 , β α , and τ ( t ) < t for all t t 0 . Further, assume that the differential equation
y ( t ) + q ( t ) ( λ 0 τ n 1 ( t ) ( n 1 ) ! r 1 / α ( τ ( t ) ) ) β y β / α ( τ ( t ) ) = 0
(2.1)
is oscillatory for some constant λ 0 ( 0 , 1 ) . If
(2.2)

holds for some constant λ 1 ( 0 , 1 ) and for all constants M > 0 , then every solution of (1.1) is oscillatory or tends to zero as t .

Proof Assume that (1.1) has a nonoscillatory solution x. Without loss of generality, we may assume that x is eventually positive. Moreover, suppose that lim t x ( t ) 0 . It follows from (1.1) that there exist two possible cases:
  1. (1)

    x ( t ) > 0 , x ( n 1 ) ( t ) > 0 , x ( n ) ( t ) < 0 , ( r ( x ( n 1 ) ) α ) ( t ) < 0 ;

     
  2. (2)

    x ( t ) > 0 , x ( n 2 ) ( t ) > 0 , x ( n 1 ) ( t ) < 0 , ( r ( x ( n 1 ) ) α ) ( t ) < 0

     

for t t 1 , where t 1 t 0 is large enough.

Assume that case (1) holds. From [[36], Lemma 2.1], we have
x ( t ) λ t n 1 ( n 1 ) ! r 1 / α ( t ) ( r 1 / α x ( n 1 ) ) ( t )
(2.3)
for every λ ( 0 , 1 ) and for all sufficiently large t. Hence by (1.1), we see that y : = r ( x ( n 1 ) ) α is a positive solution of the differential inequality
y ( t ) + q ( t ) ( λ τ n 1 ( t ) ( n 1 ) ! r 1 / α ( τ ( t ) ) ) β y β / α ( τ ( t ) ) 0 .

Using [[28], Theorem 1], we see that equation (2.1) also has a positive solution, which is a contradiction.

Assume that case (2) holds. Define the function w by
w ( t ) : = r ( t ) ( x ( n 1 ) ) α ( t ) ( x ( n 2 ) ) α ( t ) , t t 1 .
(2.4)
Then w ( t ) < 0 for t t 1 . Noting that r ( x ( n 1 ) ) α is decreasing, we have
r 1 / α ( s ) x ( n 1 ) ( s ) r 1 / α ( t ) x ( n 1 ) ( t ) , s t t 1 .
Dividing the above inequality by r 1 / α ( s ) and integrating the resulting inequality from t to l, we obtain
x ( n 2 ) ( l ) x ( n 2 ) ( t ) + r 1 / α ( t ) x ( n 1 ) ( t ) t l d s r 1 / α ( s ) .
Letting l , we get
x ( n 2 ) ( t ) r 1 / α ( t ) x ( n 1 ) ( t ) δ ( t ) ,
(2.5)
which yields
r 1 / α ( t ) x ( n 1 ) ( t ) x ( n 2 ) ( t ) δ ( t ) 1 .
Thus, by (2.4), we see that
w ( t ) δ α ( t ) 1 .
(2.6)
Differentiating (2.4), we have
w ( t ) = ( r ( x ( n 1 ) ) α ) ( t ) ( x ( n 2 ) ) α ( t ) α r ( t ) ( x ( n 1 ) ) α + 1 ( t ) ( x ( n 2 ) ) α + 1 ( t ) .
It follows from (1.1) and (2.4) that
w ( t ) = q ( t ) x β ( τ ( t ) ) ( x ( n 2 ) ) α ( t ) α w ( α + 1 ) / α ( t ) r 1 / α ( t ) .
(2.7)
By virtue of (2.5), we have
( x ( n 2 ) δ ) ( t ) 0 .
(2.8)
On the other hand, by [[36], Lemma 2.1], we get
x ( t ) λ ( n 2 ) ! t n 2 x ( n 2 ) ( t )
(2.9)
for every λ ( 0 , 1 ) and for all sufficiently large t. Then from (2.7), (2.8), and (2.9), there exists a constant M > 0 such that
w ( t ) = q ( t ) x β ( τ ( t ) ) ( x ( n 2 ) ( τ ( t ) ) ) β ( x ( n 2 ) ( τ ( t ) ) ) β α ( x ( n 2 ) ( τ ( t ) ) ) α ( x ( n 2 ) ( t ) ) α α w ( α + 1 ) / α ( t ) r 1 / α ( t ) ( M δ ( τ ( t ) ) ) β α q ( t ) ( λ ( n 2 ) ! τ n 2 ( t ) ) β α w ( α + 1 ) / α ( t ) r 1 / α ( t ) .
(2.10)
Multiplying (2.10) by δ α ( t ) and integrating the resulting inequality from t 1 to t, we have
δ α ( t ) w ( t ) δ α ( t 1 ) w ( t 1 ) + α t 1 t r 1 / α ( s ) δ α 1 ( s ) w ( s ) d s + t 1 t ( M δ ( τ ( s ) ) ) β α q ( s ) ( λ ( n 2 ) ! τ n 2 ( s ) ) β δ α ( s ) d s + α t 1 t w ( α + 1 ) / α ( s ) r 1 / α ( s ) δ α ( s ) d s 0 .
Set B : = r 1 / α ( s ) δ α 1 ( s ) , A : = δ α ( s ) / r 1 / α ( s ) , and v : = w ( s ) . Using (2.6) and the inequality
A v ( α + 1 ) / α B v α α ( α + 1 ) α + 1 B α + 1 A α , A > 0 ,
(2.11)
we have
t 1 t [ ( M δ ( τ ( s ) ) ) β α q ( s ) ( λ ( n 2 ) ! τ n 2 ( s ) ) β δ α ( s ) α α + 1 ( α + 1 ) α + 1 1 δ ( s ) r 1 / α ( s ) ] d s δ α ( t 1 ) w ( t 1 ) + 1 ,

which contradicts (2.2). This completes the proof. □

Applying the result of [30] to equation (2.1), we have the following result due to Theorem 2.1.

Corollary 2.2 Assume (1.2) and let n 2 , β > α , τ ( t ) < t , and τ ( t ) > 0 for all t t 0 . Moreover, assume that there exists a continuously differentiable function φ such that
φ ( t ) > 0 and lim t φ ( t ) = ,
(2.12)
lim sup t φ ( τ ( t ) ) τ ( t ) φ ( t ) < α β ,
(2.13)
and
lim inf t q ( t ) ( τ n 1 ( t ) r 1 / α ( τ ( t ) ) ) β e φ ( t ) φ ( t ) > 0 .
(2.14)

If (2.2) holds for some constant λ 1 ( 0 , 1 ) and for all constants M > 0 , then every solution of (1.1) is oscillatory or tends to zero as t .

In the following, we establish some results for (1.1) when n 2 is even.

Theorem 2.3 Assume (1.2) and let n 2 be even, β α , τ ( t ) > 0 , and τ ( t ) t for all t t 0 . Further, assume that there exists a function ρ C 1 ( [ t 0 , ) , ( 0 , ) ) such that
lim sup t t 0 t [ K β α q ( s ) ρ ( s ) 1 ( α + 1 ) α + 1 ( 2 ( n 2 ) ! ) α r ( s ) ( ( ρ ( s ) ) + ) α + 1 ( θ 1 τ ( s ) τ n 2 ( s ) ρ ( s ) ) α ] d s =
(2.15)

holds for all constants θ 1 ( 0 , 1 ) and K > 0 . If (2.2) holds for some constant λ 1 ( 0 , 1 ) and for all constants M > 0 , then every solution of (1.1) is oscillatory or tends to zero as t .

Proof Assume that (1.1) has a nonoscillatory solution x. Without loss of generality, we may assume that x is eventually positive. Moreover, suppose that lim t x ( t ) 0 . It follows from (1.1) that there exist two possible cases (1) and (2) (as those of the proof of Theorem 2.1).

Assume that case (1) holds. From [[4], Lemma 2.1], we see that x ( t ) > 0 for t t 1 . Define the function u by
u ( t ) : = ρ ( t ) r ( t ) ( x ( n 1 ) ) α ( t ) x α ( τ ( t ) / 2 ) , t t 1 .
(2.16)
Then u ( t ) > 0 for t t 1 and
u ( t ) = ρ ( t ) ρ ( t ) u ( t ) + ρ ( t ) ( r ( x ( n 1 ) ) α ) ( t ) x α ( τ ( t ) / 2 ) α ρ ( t ) τ ( t ) 2 r ( t ) ( x ( n 1 ) ) α ( t ) x ( τ ( t ) / 2 ) x α + 1 ( τ ( t ) / 2 ) .
(2.17)
From [[4], Lemma 2.2], there exist a t 2 t 1 and a constant θ 1 with 0 < θ 1 < 1 such that
x ( τ ( t ) / 2 ) θ 1 ( n 2 ) ! τ n 2 ( t ) x ( n 1 ) ( t )
(2.18)
for all t t 2 . It follows from (1.1), (2.16), (2.17), and (2.18) that
u ( t ) ρ ( t ) ρ ( t ) u ( t ) q ( t ) ρ ( t ) x β ( τ ( t ) ) x α ( τ ( t ) / 2 ) α τ ( t ) 2 θ 1 ( n 2 ) ! τ n 2 ( t ) u ( α + 1 ) / α ( t ) ( ρ ( t ) r ( t ) ) 1 / α .
(2.19)
Using x > 0 and (2.19), we get
u ( t ) ( ρ ( t ) ) + ρ ( t ) u ( t ) K β α q ( t ) ρ ( t ) α τ ( t ) 2 θ 1 ( n 2 ) ! τ n 2 ( t ) u ( α + 1 ) / α ( t ) ( ρ ( t ) r ( t ) ) 1 / α
(2.20)
for some constant K > 0 . Set
A : = α τ ( t ) 2 θ 1 ( n 2 ) ! τ n 2 ( t ) ( ρ ( t ) r ( t ) ) 1 / α , B : = ( ρ ( t ) ) + ρ ( t ) , and v : = u ( t ) .
Using inequality (2.11), we obtain
( ρ ( t ) ) + ρ ( t ) u ( t ) α τ ( t ) 2 θ 1 ( n 2 ) ! τ n 2 ( t ) u ( α + 1 ) / α ( t ) ( ρ ( t ) r ( t ) ) 1 / α 1 ( α + 1 ) α + 1 ( 2 ( n 2 ) ! ) α r ( t ) ( ( ρ ( t ) ) + ) α + 1 ( θ 1 τ ( t ) τ n 2 ( t ) ρ ( t ) ) α .
Substituting the last inequality into (2.20), we get
u ( t ) K β α q ( t ) ρ ( t ) + 1 ( α + 1 ) α + 1 ( 2 ( n 2 ) ! ) α r ( t ) ( ( ρ ( t ) ) + ) α + 1 ( θ 1 τ ( t ) τ n 2 ( t ) ρ ( t ) ) α .
(2.21)
Integrating (2.21) from t 2 to t, we have
t 2 t [ K β α q ( s ) ρ ( s ) 1 ( α + 1 ) α + 1 ( 2 ( n 2 ) ! ) α r ( s ) ( ( ρ ( s ) ) + ) α + 1 ( θ 1 τ ( s ) τ n 2 ( s ) ρ ( s ) ) α ] d s u ( t 2 ) ,

which contradicts (2.15). Assume that case (2) holds. Proceeding as in the proof of Theorem 2.1, we can obtain a contradiction to (2.2). This completes the proof. □

Next we establish a result for (1.1) when n = 2 .

Theorem 2.4 Assume (1.2) and let n = 2 , β α , τ ( t ) > 0 , and τ ( t ) t for t t 0 . Further, assume that there exists a function ρ C 1 ( [ t 0 , ) , ( 0 , ) ) such that
lim sup t t 0 t [ K β α q ( s ) ρ ( s ) 1 ( α + 1 ) α + 1 r ( s ) ( ( ρ ( s ) ) + ) α + 1 ( τ ( s ) ρ ( s ) ) α ] d s =
(2.22)
for all constants K > 0 . If
lim sup t t 0 t [ ( M δ ( τ ( s ) ) ) β α q ( s ) δ α ( s ) α α + 1 ( α + 1 ) α + 1 1 δ ( s ) r 1 / α ( s ) ] d s =
(2.23)

holds for all constants M > 0 , then (1.1) is oscillatory.

Proof Assume that (1.1) has a nonoscillatory solution x. Without loss of generality, we may assume that x is eventually positive. It follows from (1.1) that there exist two possible cases (1) and (2) with n = 2 (as those of the proof of Theorem 2.1). Assume that case (1) holds. Define
u ( t ) : = ρ ( t ) r ( t ) ( x ( t ) ) α x α ( τ ( t ) ) , t t 1 .

The rest of the proof is similar to that of Theorem 2.3, and so is omitted. Assume that case (2) holds. Similar as in the proof of Theorem 2.1, we can obtain a contradiction to (2.23). This completes the proof. □

Next we establish some oscillation criteria for (1.1) when n 2 is even and τ ( t ) > t for all t t 0 .

Theorem 2.5 Assume (1.2) and let n 2 be even, β > α , and τ ( t ) > t for all t t 0 . Further, assume that there exists a function ρ C 1 ( [ t 0 , ) , ( 0 , ) ) such that
lim sup t t 0 t [ K β α q ( s ) ρ ( s ) 1 ( α + 1 ) α + 1 ( 2 ( n 2 ) ! ) α r ( s ) ( ( ρ ( s ) ) + ) α + 1 ( θ 1 s n 2 ρ ( s ) ) α ] d s =
(2.24)
holds for all constants θ 1 ( 0 , 1 ) and K > 0 . If
q ( t ) δ β ( τ ( t ) ) ( τ n 2 ( t ) ) β d t = ,
(2.25)

then every solution of (1.1) is oscillatory or tends to zero as t .

Proof Assume that (1.1) has a nonoscillatory solution x. Without loss of generality, we may assume that x is eventually positive. Moreover, suppose that lim t x ( t ) 0 . It follows from (1.1) that there exist two possible cases (1) and (2) (as those of the proof of Theorem 2.1).

Assume that case (1) holds. From [[4], Lemma 2.1], we see that x ( t ) > 0 for t t 1 . Define the function u by
u ( t ) : = ρ ( t ) r ( t ) ( x ( n 1 ) ) α ( t ) x α ( t / 2 ) , t t 1 .
Then u ( t ) > 0 for t t 1 and
u ( t ) = ρ ( t ) ρ ( t ) u ( t ) + ρ ( t ) ( r ( x ( n 1 ) ) α ) ( t ) x α ( t / 2 ) α ρ ( t ) 2 r ( t ) ( x ( n 1 ) ) α ( t ) x ( t / 2 ) x α + 1 ( t / 2 ) .
From [[4], Lemma 2.2], there exist a t 2 t 1 and a constant θ 1 with 0 < θ 1 < 1 such that
x ( t / 2 ) θ 1 ( n 2 ) ! t n 2 x ( n 1 ) ( t )
for all t t 2 . Thus
u ( t ) ρ ( t ) ρ ( t ) u ( t ) q ( t ) ρ ( t ) x β ( τ ( t ) ) x α ( t / 2 ) α 2 θ 1 ( n 2 ) ! t n 2 u ( α + 1 ) / α ( t ) ( ρ ( t ) r ( t ) ) 1 / α .
Similar as in the proof of Theorem 2.3, we can get a contradiction to (2.24). Assume that case (2) holds. We have (2.5) and (2.9) for every λ ( 0 , 1 ) and for all sufficiently large t. Thus, we get by (1.1), (2.5), and (2.9) that
( r ( x ( n 1 ) ) α ) ( t ) q ( t ) ( λ ( n 2 ) ! τ n 2 ( t ) δ ( τ ( t ) ) ) β ( r 1 / α x ( n 1 ) ) β ( τ ( t ) ) 0 .
Let u : = r ( x ( n 1 ) ) α . Then y : = u > 0 is a solution of the advanced inequality
y ( t ) q ( t ) ( λ ( n 2 ) ! τ n 2 ( t ) δ ( τ ( t ) ) ) β y β / α ( τ ( t ) ) 0 .
It follows from [[8], Lemma 2.3] that the corresponding advanced differential equation
y ( t ) q ( t ) ( λ ( n 2 ) ! τ n 2 ( t ) δ ( τ ( t ) ) ) β y β / α ( τ ( t ) ) = 0

has an eventually positive solution. Using condition (2.25) and [[22], Theorem 1], one can obtain a contradiction. This completes the proof. □

Finally, we establish a result for (1.1) when n = 2 .

Theorem 2.6 Assume (1.2) and let n = 2 , β > α , and τ ( t ) > t for t t 0 . Further, assume that there exists a function ρ C 1 ( [ t 0 , ) , ( 0 , ) ) such that
lim sup t t 0 t [ K β α q ( s ) ρ ( s ) 1 ( α + 1 ) α + 1 r ( s ) ( ( ρ ( s ) ) + ) α + 1 ρ α ( s ) ] d s =
(2.26)
for all constants K > 0 . If
q ( t ) δ β ( τ ( t ) ) d t = ,
(2.27)

then (1.1) is oscillatory.

Proof Assume that (1.1) has a nonoscillatory solution x. Without loss of generality, we may assume that x is eventually positive. It follows from (1.1) that there exist two possible cases (1) and (2) with n = 2 (as those of the proof of Theorem 2.1). Assume that case (1) holds. Define
u ( t ) : = ρ ( t ) r ( t ) ( x ( t ) ) α x α ( t ) , t t 1 .

The rest of the proof is similar to that of Theorem 2.3, and so is omitted. Assume that case (2) holds. Similar as in the proof of Theorem 2.5, we can obtain a contradiction to (2.27). This completes the proof. □

3 Examples and discussions

In the following, we illustrate possible applications with two examples.

Example 3.1 For t 1 , consider the second-order delay differential equation
( e t x ( t ) ) + e 5 t / 2 x 3 ( t 2 ) = 0 .
(3.1)

Let α = 1 , β = 3 , and ρ ( t ) = 1 . Note that δ ( t ) = e t . Using Theorem 2.4, equation (3.1) is oscillatory. It is not difficult to see that Theorem 1.1 fails to apply due to condition (1.4).

Example 3.2 For t 1 , consider the second-order advanced differential equation
( e t x ( t ) ) + e 6 t t x 3 ( 2 t ) = 0 .
(3.2)

Let α = 1 , β = 3 , and ρ ( t ) = 1 . Note that δ ( t ) = e t . Using Theorem 2.6, equation (3.2) is oscillatory.

In this paper, we suggested some new results on the oscillation and asymptotic behavior of differential equation (1.1). Theorem 2.1 can be applied in the odd-order and even-order equations.

We stress that the study of equation (1.1) in the case (1.2) brings additional difficulties. Since the sign of x ( n 1 ) is not known, our criteria include a pair of assumptions; see, e.g., (2.2) and (2.15). We utilized two different methods (Riccati substitution and comparison method) to deal with the cases τ ( t ) t and τ ( t ) > t .

Declarations

Acknowledgements

This research is supported by NNSF of P.R. China (Grant Nos. 61034007, 51277116, 50977054).

Authors’ Affiliations

(1)
School of Control Science and Engineering, Shandong University, Jinan, P.R. China
(2)
Department of Mathematics, Texas A&M University-Kingsville, Kingsville, USA
(3)
Department of Mathematics and Statistics, Missouri S&T, Rolla, USA

References

  1. Agarwal RP, Bohner M, Li W Monographs and Textbooks in Pure and Applied Mathematics 267. In Nonoscillation and Oscillation: Theory for Functional Differential Equations. Marcel Dekker, New York; 2004.View ArticleGoogle Scholar
  2. Agarwal RP, Grace SR: Oscillation of certain functional differential equations. Comput. Math. Appl. 1999, 38: 143-153.MathSciNetView ArticleGoogle Scholar
  3. Agarwal RP, Grace SR, O’Regan D: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic, Dordrecht; 2000.View ArticleGoogle Scholar
  4. Agarwal RP, Grace SR, O’Regan D: Oscillation criteria for certain n th order differential equations with deviating arguments. J. Math. Anal. Appl. 2001, 262: 601-622. 10.1006/jmaa.2001.7571MathSciNetView ArticleGoogle Scholar
  5. Agarwal RP, Grace SR, O’Regan D: Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Kluwer Academic, Dordrecht; 2002.View ArticleGoogle Scholar
  6. Agarwal RP, Grace SR, O’Regan D: The oscillation of certain higher-order functional differential equations. Math. Comput. Model. 2003, 37: 705-728. 10.1016/S0895-7177(03)00079-7MathSciNetView ArticleGoogle Scholar
  7. Agarwal RP, Shieh S-L, Yeh C-C: Oscillation criteria for second-order retarded differential equations. Math. Comput. Model. 1997, 26: 1-11.MathSciNetView ArticleGoogle Scholar
  8. Baculíková B: Properties of third-order nonlinear functional differential equations with mixed arguments. Abstr. Appl. Anal. 2011, 2011: 1-15.View ArticleGoogle Scholar
  9. Baculíková B, Džurina J: Oscillation of third-order nonlinear differential equations. Appl. Math. Lett. 2011, 24: 466-470. 10.1016/j.aml.2010.10.043MathSciNetView ArticleGoogle Scholar
  10. Baculíková B, Džurina J: Oscillation of third-order functional differential equations. Electron. J. Qual. Theory Differ. Equ. 2010, 43: 1-10.View ArticleGoogle Scholar
  11. Baculíková B, Džurina J, Graef JR: On the oscillation of higher order delay differential equations. Nonlinear Oscil. 2012, 15: 13-24.Google Scholar
  12. Dahiya RS: Oscillation criteria of even-order nonlinear delay differential equations. J. Math. Anal. Appl. 1976, 54: 653-665. 10.1016/0022-247X(76)90184-0MathSciNetView ArticleGoogle Scholar
  13. Džurina J, Baculíková B: Oscillation and asymptotic behavior of higher-order nonlinear differential equations. Int. J. Math. Math. Sci. 2012, 2012: 1-9.View ArticleGoogle Scholar
  14. Džurina J, Stavroulakis IP: Oscillation criteria for second order delay differential equations. Appl. Math. Comput. 2003, 140: 445-453. 10.1016/S0096-3003(02)00243-6MathSciNetView ArticleGoogle Scholar
  15. Erbe L, Kong Q, Zhang B: Oscillation Theory for Functional Differential Equations. Marcel Dekker, New York; 1995.Google Scholar
  16. Grace SR: Oscillation theorems for n th-order differential equations with deviating arguments. J. Math. Anal. Appl. 1984, 101: 268-296. 10.1016/0022-247X(84)90066-0MathSciNetView ArticleGoogle Scholar
  17. Grace SR: Oscillation theorems for certain functional differential equations. J. Math. Anal. Appl. 1994, 184: 100-111. 10.1006/jmaa.1994.1187MathSciNetView ArticleGoogle Scholar
  18. 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.2008.01.025MathSciNetView ArticleGoogle Scholar
  19. Grace SR, Lalli BS: Oscillation theorems for n th-order delay differential equations. J. Math. Anal. Appl. 1983, 91: 352-366. 10.1016/0022-247X(83)90157-9MathSciNetView ArticleGoogle Scholar
  20. Grace SR, Lalli BS: Oscillation of even order differential equations with deviating arguments. J. Math. Anal. Appl. 1990, 147: 569-579. 10.1016/0022-247X(90)90371-LMathSciNetView ArticleGoogle Scholar
  21. Kartsatos AG: On oscillation of solutions of even order nonlinear differential equations. J. Differ. Equ. 1969, 6: 232-237. 10.1016/0022-0396(69)90014-XMathSciNetView ArticleGoogle Scholar
  22. Kitamura Y, Kusano T: Oscillation of first-order nonlinear differential equations with deviating arguments. Proc. Am. Math. Soc. 1980, 78: 64-68. 10.1090/S0002-9939-1980-0548086-5MathSciNetView ArticleGoogle Scholar
  23. Ladde GS, Lakshmikantham V, Zhang BG: Oscillation Theory of Differential Equations with Deviating Arguments. Marcel Dekker, New York; 1987.Google Scholar
  24. Li T, Han Z, Zhang C, Sun S: On the oscillation of second-order Emden-Fowler neutral differential equations. J. Appl. Math. Comput. 2011, 37: 601-610. 10.1007/s12190-010-0453-0MathSciNetView ArticleGoogle Scholar
  25. Li T, Thandapani E: Oscillation of solutions to odd-order nonlinear neutral functional differential equations. Electron. J. Differ. Equ. 2011, 23: 1-12. 10.1007/s10884-010-9200-3Google Scholar
  26. Mahfoud WE: Oscillation and asymptotic behavior of solutions of n th order nonlinear delay differential equations. J. Differ. Equ. 1977, 24: 75-98. 10.1016/0022-0396(77)90171-1MathSciNetView ArticleGoogle Scholar
  27. Philos CG: A new criterion for the oscillatory and asymptotic behavior of delay differential equations. Bull. Acad. Pol. Sci., Sér. Sci. Math. 1981, 39: 61-64.Google Scholar
  28. Philos CG: On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delays. Arch. Math. 1981, 36: 168-178. 10.1007/BF01223686MathSciNetView ArticleGoogle Scholar
  29. Rogovchenko YV, Tuncay F: Oscillation theorems for a class of second order nonlinear differential equations with damping. Taiwan. J. Math. 2009, 13: 1909-1928.MathSciNetGoogle Scholar
  30. Tang X: Oscillation for first order superlinear delay differential equations. J. Lond. Math. Soc. 2002, 65: 115-122. 10.1112/S0024610701002678View ArticleGoogle Scholar
  31. Xu Z, Xia Y: Integral averaging technique and oscillation of certain even order delay differential equations. J. Math. Anal. Appl. 2004, 292: 238-246. 10.1016/j.jmaa.2003.11.054MathSciNetView ArticleGoogle Scholar
  32. Yildiz MK, Öcalan Ö: Oscillation results of higher order nonlinear neutral delay differential equations. Selçuk J. Appl. Math. 2010, 11: 55-62.Google Scholar
  33. Zhang B: Oscillation of even order delay differential equations. J. Math. Anal. Appl. 1987, 127: 140-150. 10.1016/0022-247X(87)90146-6MathSciNetView ArticleGoogle Scholar
  34. Zhang C, Agarwal RP, Bohner M, Li T: New results for oscillatory behavior of even-order half-linear delay differential equations. Appl. Math. Lett. 2013, 26: 179-183. 10.1016/j.aml.2012.08.004MathSciNetView ArticleGoogle Scholar
  35. Zhang C, Agarwal RP, Bohner M, Li T: Oscillation of third-order nonlinear delay differential equations. Taiwan. J. Math. 2013, 17(2):545-558.MathSciNetGoogle Scholar
  36. Zhang C, Li T, Sun Bo, Thandapani E: On the oscillation of higher-order half-linear delay differential equations. Appl. Math. Lett. 2011, 24: 1618-1621. 10.1016/j.aml.2011.04.015MathSciNetView ArticleGoogle Scholar

Copyright

© Zhang et al.; licensee Springer 2013

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

Advertisement