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Oscillation criteria for third-order neutral dynamic equations with continuously distributed delay

Advances in Difference Equations20142014:220

https://doi.org/10.1186/1687-1847-2014-220

  • Received: 17 January 2014
  • Accepted: 19 July 2014
  • Published:

Abstract

It is the purpose of this paper to give oscillation criteria for the third-order neutral dynamic equations with continuously distributed delay,

[ r ( t ) ( [ x ( t ) + a b p ( t , η ) x [ τ ( t , η ) ] Δ η ] Δ Δ ) γ ] Δ + c d q ( t , ξ ) f ( x [ ϕ ( t , ξ ) ] ) Δ ξ = 0 ,

on a time scale T , where γ is the quotient of odd positive integers. By using a generalized Riccati transformation and an integral averaging technique, we establish some new sufficient conditions which ensure that every solution of this equation oscillates or converges to zero.

Keywords

  • oscillation
  • time scales
  • third-order neutral dynamic equation
  • asymptotic behavior

1 Introduction

We are concerned with the oscillatory behavior of third-order neutral dynamic equations with continuously distributed delay,
[ r ( t ) ( [ x ( t ) + a b p ( t , η ) x [ τ ( t , η ) ] Δ η ] Δ Δ ) γ ] Δ + c d q ( t , ξ ) f ( x [ ϕ ( t , ξ ) ] ) Δ ξ = 0 ,
(1)

on an arbitrary time scale T , where γ is a quotient of odd positive integers. Throughout this paper, we will assume the following hypotheses:

(H1) r and q are positive rd-continuous functions on T and
t 0 ( 1 r ( t ) ) 1 γ Δ t = ;
(2)

(H2) p ( t , η ) C r d ( [ t 0 , ) × [ a , b ] , R ) , 0 p ( t ) a b p ( t , η ) Δ η P < 1 ;

(H3) τ ( t , η ) C r d ( [ t 0 , ) × [ a , b ] , T ) is not a decreasing function for η and such that
τ ( t , η ) t and lim t min η [ a , b ] τ ( t , η ) = ;
(H4) ϕ ( t , ξ ) C r d ( [ t 0 , ) × [ c , d ] , T ) is not decreasing function for ξ and such that
ϕ ( t , ξ ) t and lim t min ξ [ c , d ] ϕ ( t , ξ ) = ;

(H5) the function f C r d ( T , R ) is assumed to satisfy u f ( u ) > 0 and there exists a positive rd-continuous function δ ( t ) on T such that f ( u ) u γ δ , for u 0 .

Define the function by
z ( t ) = x ( t ) + a b p ( t , η ) x [ τ ( t , η ) ] Δ η .
(3)
Furthermore, (1) is like the following:
[ r ( t ) ( [ z ( t ) ] Δ Δ ) γ ] Δ + c d q ( t , ξ ) f ( x [ ϕ ( t , ξ ) ] ) Δ ξ = 0 .
(4)

A solution x ( t ) of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is non-oscillatory.

Much recent attention has been given to dynamic equations on time scales, or measure chains, and we refer the reader to the landmark paper of Hilger [1] for a comprehensive treatment of the subject. Since then, several authors have expounded various aspects of this new theory; see the survey paper by Agarwal et al. [2]. A book on the subject of time scales by Bohner and Peterson [3] also summarizes and organizes much of the time scale calculus. In the recent years, there has been increasing interest in obtaining sufficient conditions for the oscillation and non-oscillation of solutions of various equations on time scales; we refer the reader to the papers [419]. Candan [20] considered oscillation of second-order neutral dynamic equations with distributed deviating arguments of the form
( r ( t ) ( ( y ( t ) + p ( t ) y ( τ ( t ) ) ) Δ ) γ ) Δ + c d f ( t , y ( θ ( t , ξ ) ) ) Δ ξ = 0 ,

where γ > 0 is a ratio of odd positive integers with r ( t ) and p ( t ) real-valued rd-continuous positive functions defined on T . He established some new oscillation criteria and gave sufficient conditions to ensure that all solutions of nonlinear neutral dynamic equation are oscillatory on a time scale T .

To the best of our knowledge, there is very little known about the oscillatory behavior of third-order dynamic equations. Erbe et al. [21] are concerned with the oscillatory behavior of solutions of the third-order linear dynamic equation
x Δ Δ Δ ( t ) + p ( t ) x ( t ) = 0 ,
on an arbitrary time scale T , where p ( t ) is a positive real-valued rd-continuous function defined on T . Li et al. [22] considered third-order nonlinear delay dynamic equation
x Δ 3 + p ( t ) x γ ( τ ( t ) ) = 0 ,

on a time scale T , where γ > 0 is quotient of odd positive integers.

Erbe et al. [23, 24] established some sufficient conditions which guarantee that every solution of the third-order nonlinear dynamic equation
( c ( t ) ( a ( t ) x Δ ( t ) ) Δ ) Δ + q ( t ) f ( x ( t ) ) = 0 ,
and the third-order dynamic equation
( c ( t ) ( ( a ( t ) x Δ ( t ) ) Δ ) γ ) Δ + f ( t , x ( t ) ) = 0
oscillate or converge to zero. Li et al. [25] considered the third-order delay dynamic equations
( a ( t ) ( [ r ( t ) x Δ ( t ) ] Δ ) γ ) Δ + f ( t , x ( τ ( t ) ) ) = 0 ,

on a time scale T , where γ > 0 is quotient of odd positive integers, a and r are positive rd-continuous functions on T , and the so-called delay function τ : T T satisfies τ ( t ) t , and τ ( t ) as t , f ( x ) C r d ( T × R , R ) is assumed to satisfy u f ( t , u ) > 0 , for u 0 , and there exists a function p on T such that f ( t , u ) u γ p ( t ) > 0 , for u 0 .

Saker [26] considered the third-order nonlinear functional dynamic equations
( p ( t ) ( [ r ( t ) x Δ ( t ) ] Δ ) γ ) Δ + q ( t ) f ( x ( τ ( t ) ) ) = 0 ,
on a time scale T , where γ > 0 is quotient of odd positive integers. Recently Han et al. [27] and Grace et al. [28] considered the third-order neutral delay dynamic equation
( r ( t ) ( x ( t ) a ( t ) x ( τ ( t ) ) ) Δ Δ ) Δ + p ( t ) x γ ( δ ( t ) ) = 0 ,

on a time scale T .

In this paper, we consider third-order neutral dynamic equation with continuously distributed delay on time scales which is not in literature. We obtain some conclusions which contribute to oscillation theory of third-order neutral dynamic equations.

2 Several lemmas

Before stating our main results, we begin with the following lemmas which play an important role in the proof of the main results. Throughout this paper, we let
d + ( t ) : = max { 0 , d ( t ) } , d ( t ) : = max { 0 , d ( t ) } ,
and
β ( t ) : = b ( t ) , 0 < γ 1 , β ( t ) : = b γ ( t ) , γ > 1 , b ( t ) = t σ ( t ) , R ( t , t ) : = t t ( 1 r ( s ) ) 1 γ Δ s ,

where we have sufficiently large t [ t 0 , ) T .

In order to prove our main results, we will use the formula
( z γ ( t ) ) Δ = γ 0 1 [ h z σ + ( 1 h ) z ] γ 1 z Δ ( t ) d h ,

where z ( t ) is delta differentiable and eventually positive or eventually negative, which is a simple consequence of Keller’s chain rule (see Bohner and Peterson [3]).

Lemma 2.1 Let x ( t ) be a positive solution of (1), z ( t ) is defined as in (3). Then z ( t ) has only one of the following two properties:
  1. (I)

    z ( t ) > 0 , z Δ ( t ) > 0 , z Δ Δ ( t ) > 0 ,

     
  2. (II)

    z ( t ) > 0 , z Δ ( t ) < 0 , z Δ Δ ( t ) > 0 ,

     

with t t 1 , t 1 sufficiently large.

Proof Let x ( t ) be a positive solution of (1) on [ t 0 , ) , so that z ( t ) > x ( t ) > 0 , and
[ r ( t ) ( z Δ Δ ( t ) ) γ ] Δ = c d q ( t , ξ ) f ( x [ ϕ ( t , ξ ) ] ) Δ ξ < 0 .
Then r ( t ) ( [ z ( t ) ] Δ Δ ) γ is a decreasing function and therefore eventually of one sign, so z Δ Δ ( t ) is either eventually positive or eventually negative on t t 1 t 0 . We assert that z Δ Δ ( t ) > 0 on t t 1 t 0 . Otherwise, assume that z Δ Δ ( t ) < 0 , then there exists a constant M > 0 , such that
r ( t ) ( z Δ Δ ( t ) ) γ M < 0 .
By integrating the last inequality from t 1 to t, we obtain
z Δ ( t ) z Δ ( t 1 ) M 1 γ t 1 t ( 1 r ( s ) ) 1 γ Δ s .

Let t . Then from (H1), we have ( z ( t ) ) Δ , and therefore eventually z Δ ( t ) < 0 .

Since z Δ Δ ( t ) < 0 and z Δ ( t ) < 0 , we have z ( t ) < 0 , which contradicts our assumption z ( t ) > 0 . Therefore, z ( t ) has only one of the two properties (I) and (II).

This completes the proof. □

Lemma 2.2 Let x ( t ) be an eventually positive solution of (1), correspondingly z ( t ) has the property (II). Assume that (2) and
t 0 v [ 1 r ( u ) u q 1 ( s ) Δ s ] 1 γ Δ u Δ v =
(5)

hold. Then lim t x ( t ) = 0 .

Proof Let x ( t ) be an eventually positive solution of (1). Since z ( t ) has the property (II), then there exists finite lim t z ( t ) = I . We assert that I = 0 . Assume that I > 0 , then we have I + ϵ > z ( t ) > I for all ϵ > 0 . Choosing ϵ < I ( 1 P ) P and using (3) and (H2), we obtain
x ( t ) = z ( t ) a b p ( t , η ) [ x ( τ ( t , η ) ) ] Δ η > I a b p ( t , η ) [ x ( τ ( t , η ) ) ] Δ η I p ( t ) [ z ( τ ( t , a ) ) ] I P ( I + ϵ ) > K z ( t ) ,
(6)
where K = I P ( 1 + ϵ ) I + ϵ > 0 . Using (H5) and (6), we find from (1) that
[ r ( t ) ( z Δ Δ ( t ) ) γ ] Δ = c d q ( t , ξ ) f ( x [ ϕ ( t , ξ ) ] ) Δ ξ c d q ( t , ξ ) ( x [ ϕ ( t , ξ ) ] ) γ δ Δ ξ K γ δ c d q ( t , ξ ) ( z [ ϕ ( t , ξ ) ] ) γ Δ ξ .
Note that z ( t ) has property (II) and (H4), and we have
[ r ( t ) ( z Δ Δ ( t ) ) γ ] Δ K γ δ ( z [ ϕ ( t , d ) ] ) γ c d q ( t , ξ ) Δ ξ = q 1 ( t ) ( z ( ϕ 1 ( t ) ) ) γ ,
(7)
where q 1 ( t ) = K γ δ c d q ( t , ξ ) Δ ξ , ϕ 1 ( t ) = ϕ ( t , d ) . Integrating inequality (7) from t to ∞, we obtain
r ( t ) ( z Δ Δ ( t ) ) γ t q 1 ( s ) ( z ( ϕ 1 ( s ) ) ) γ Δ s .
Using ( z ( ϕ 1 ( s ) ) ) γ I γ , we obtain
z Δ Δ ( t ) I r 1 γ [ t q 1 ( s ) ] 1 γ Δ ( s ) .
(8)
Integrating inequality (8) from t to ∞, we have
z Δ ( t ) I t [ 1 r ( u ) u q 1 ( s ) Δ ( s ) ] 1 γ Δ u .
Integrating the last inequality from t 1 to ∞, we obtain
z ( t 1 ) I t 1 v [ 1 r ( u ) u q 1 ( s ) Δ ( s ) ] 1 γ Δ u Δ v .

Because (7) and the last inequality contradict (5), we have I = 0 . Since 0 x ( t ) z ( t ) , lim t x ( t ) = 0 . This completes the proof. □

Lemma 2.3 Assume that x ( t ) is a positive solution of (1), z ( t ) is defined as in (3) such that z Δ Δ ( t ) > 0 , z Δ ( t ) > 0 , on [ t , ) T , t 0 . Then
z Δ ( t ) R ( t , t ) r 1 γ ( t ) z Δ Δ ( t ) .
(9)
Proof Since r ( t ) ( z Δ Δ ( t ) ) γ is strictly decreasing on [ t , ) T , we get for t [ t , ) T
z Δ ( t ) > z Δ ( t ) z Δ ( t ) = t t ( r ( s ) ( z Δ Δ ( t ) ) γ ) 1 γ r 1 γ ( s ) Δ s ( r ( t ) ( z Δ Δ ( t ) ) γ ) 1 γ t t ( 1 r ( s ) ) 1 γ Δ s .
Using the definition of R ( t , t ) , we obtain
z Δ ( t ) > R ( t , t ) r 1 γ ( t ) z Δ Δ ( t ) on  [ t , ) T .

 □

Lemma 2.4 Assume that x ( t ) is a positive solution of (1), correspondingly z ( t ) has the property (I). Such that z Δ ( t ) > 0 , z Δ Δ ( t ) > 0 , on [ t , ) T , t t 0 . Furthermore,
t 2 t q 2 ( s ) ϕ 2 γ ( s ) Δ s = .
(10)
Then there exists a T [ t , ) T , sufficiently large, so that
z ( t ) > t z Δ ( t ) ,

z ( t ) / t is strictly decreasing, t [ T , ) T .

Proof Let U ( t ) = z ( t ) t z Δ ( t ) . Hence U Δ ( t ) = σ ( t ) z Δ Δ ( t ) < 0 . We claim there exists a t 1 [ t , ) T such that U ( t ) > 0 , z ( ϕ ( t , ξ ) ) > 0 on [ t 1 , ) T . Assume not. Then U ( t ) < 0 on [ t 1 , ) T . Therefore,
( z ( t ) t ) Δ = t z Δ ( t ) z ( t ) t σ ( t ) = U ( t ) t σ ( t ) > 0 , t [ t 1 , ) T ,
which implies that z ( t ) / t is strictly increasing on [ t 1 , ) T . Pick t 2 [ t 1 , ) T so that ϕ ( t , ξ ) ϕ ( t 1 , ξ ) , for t t 2 . Then
z ( ϕ ( t , ξ ) ) ϕ ( t , ξ ) z ( ϕ ( t 1 , ξ ) ) ϕ ( t 1 , ξ ) = d > 0 ,
so that z ( ϕ ( t , ξ ) ) > d ϕ ( t , ξ ) , for t t 2 . By (1), (3), and (H2), we obtain
x ( t ) = z ( t ) a b p ( t , η ) x [ τ ( t , η ) ] Δ η z ( t ) a b p ( t , η ) z [ τ ( t , η ) ] Δ η z ( t ) z [ τ ( t , b ) ] a b p ( t , η ) Δ η ( 1 a b p ( t , η ) Δ η ) z ( t ) ( 1 P ) z ( t ) .
(11)
Using (11), (H4), and (H5), we have
[ r ( t ) ( [ z ( t ) ] Δ Δ ) γ ] Δ = c d q ( t , ξ ) f ( x [ ϕ ( t , ξ ) ] ) Δ ξ δ ( 1 P ) γ c d q ( t , ξ ) z γ ( ϕ ( t , ξ ) ) Δ ξ δ ( 1 P ) γ z γ ( ϕ ( t , c ) ) c d q ( t , ξ ) Δ ξ q 2 ( t ) z γ ( ϕ 2 ( t ) ) ,
(12)

where q 2 ( t ) = δ ( 1 P ) γ c d q ( t , ξ ) Δ ξ , ϕ 2 ( t ) = ϕ ( t , c ) .

Now by integrating both sides of last equation from t 2 to t, we have
r ( t ) ( z Δ Δ ( t ) ) γ r ( t 2 ) ( z Δ Δ ( t 2 ) ) γ + t 2 t q 2 ( s ) z γ ( ϕ 2 ( s ) ) Δ s 0 .
This implies that
r ( t 2 ) ( z Δ Δ ( t 2 ) ) γ t 2 t q 2 ( s ) ( z ( ϕ 2 ( s ) ) ) γ Δ s d γ t 2 t q 2 ( s ) ϕ 2 γ ( s ) Δ s ,
which contradicts (10). So U ( t ) > 0 on t [ t 1 , ) T and consequently,
( z ( t ) t ) Δ = t z Δ ( t ) z ( t ) t σ ( t ) = U ( t ) t σ ( t ) < 0 , t [ t 1 , ) T ,

and we find that z ( t ) / t is strictly decreasing on t [ t 1 , ) T . The proof is now complete. □

3 Main results

In this section we give some new oscillation criteria for (1).

Theorem 3.1 Assume that (2), (5), and (10) hold. Furthermore, assume that there exists a positive function ρ C r d 1 ( [ t 0 , ) T , R ) , for all sufficiently large T 1 [ t 0 , ) T , there is a T > T 1 such that
lim sup t T t [ ρ σ ( s ) q 2 ( s ) ( ϕ 2 ( s ) σ ( s ) ) γ ( ( ρ Δ ( s ) ) + ) γ + 1 ( γ + 1 ) γ + 1 ( β ( s ) ρ σ ( s ) R ( s , t ) ) γ ] Δ s = .
(13)

Then every solution of (1) is either oscillatory or tends to zero.

Proof Assume (1) has a non-oscillatory solution x ( t ) on [ t 0 , ) T . We may assume without loss of generality that x ( t ) > 0 , t t 1 ; x ( τ ( t , η ) ) > 0 , ( t , η ) [ t 1 , ) × [ a , b ] and x ( ϕ ( t , ξ ) ) > 0 , ( t , ξ ) [ t 1 , ) × [ c , d ] for all t 1 [ t 0 , ) T . z ( t ) is defined as in (3). We suppose that z ( t ) > 0 . We shall consider only this case, since the proof when z ( t ) is eventually negative is similar. Therefore Lemma 2.1 and Lemma 2.2, we get
[ r ( t ) ( [ z ( t ) ] Δ Δ ) γ ] Δ < 0 , z Δ Δ ( t ) > 0 , t [ t 1 , ) T ,

and either z Δ ( t ) > 0 for t t 2 t 1 or lim t x ( t ) = 0 . Let z Δ ( t ) > 0 on [ t 2 , ) T .

By (11) and (12), we have
[ r ( t ) ( [ z ( t ) ] Δ Δ ) γ ] Δ q 2 ( t ) z γ ( ϕ 2 ( t ) ) ,

where q 2 ( t ) = δ ( 1 P ) γ c d q ( t , ξ ) Δ ξ , ϕ 2 ( t ) = ϕ ( t , c ) .

Define the function w ( t ) by the Riccati substitution
w ( t ) = ρ ( t ) r ( t ) ( [ z ( t ) ] Δ Δ ) γ z γ ( t ) .
(14)
Then
w Δ ( t ) = ρ Δ ( t ) r ( t ) ( [ z ( t ) ] Δ Δ ) γ z γ ( t ) + ρ σ ( t ) [ r ( t ) ( [ z ( t ) ] Δ Δ ) γ z γ ( t ) ] Δ = ρ Δ ( t ) r ( t ) ( [ z ( t ) ] Δ Δ ) γ z γ ( t ) + ρ σ ( t ) [ r ( t ) ( [ z ( t ) ] Δ Δ ) γ ] Δ z γ σ ( t ) ρ σ ( t ) r ( t ) ( [ z ( t ) ] Δ Δ ) γ ( z γ ( t ) ) Δ z γ ( t ) z γ σ ( t ) .
From (1), the definition of w ( t ) and using the fact z ( t ) / t is strictly decreasing for t [ t 3 , ) T , t 3 t 2 , it follows that
w Δ ( t ) ρ Δ ( t ) ρ ( t ) w ( t ) ρ σ ( t ) q 2 ( t ) z γ ( ϕ 2 ( t ) ) z γ σ ( t ) ρ σ ( t ) r ( t ) ( [ z ( t ) ] Δ Δ ) γ ( z γ ( t ) ) Δ z γ ( t ) z γ σ ( t ) , w Δ ( t ) ρ Δ ( t ) ρ ( t ) w ( t ) ρ σ ( t ) q 2 ( t ) ( ϕ 2 ( t ) σ ( t ) ) γ ρ σ ( t ) r ( t ) ( [ z ( t ) ] Δ Δ ) γ ( z γ ( t ) ) Δ z γ ( t ) z γ σ ( t ) .
(15)
Now we consider the following two cases: 0 < γ 1 and γ > 1 . In the first case 0 < γ 1 . Using the Keller chain rule (see [3]), we have
( z γ ( t ) ) Δ = γ 0 1 [ h z σ + ( 1 h ) z ] γ 1 z Δ ( t ) d h γ ( z σ ( t ) ) γ 1 z Δ ( t ) ,
(16)
in view of (16), Lemma 2.2, Lemma 2.3, and (9), we have
w Δ ( t ) ρ σ ( t ) q 2 ( t ) ( ϕ 2 ( t ) σ ( t ) ) γ + ( ρ Δ ( t ) ) + ρ ( t ) w ( t ) γ ρ σ ( t ) r ( t ) ( z Δ Δ ( t ) ) γ z Δ ( t ) z ( t ) z γ + 1 ( t ) z σ ( t ) ρ σ ( t ) q 2 ( t ) ( ϕ 2 ( t ) σ ( t ) ) γ + ( ρ Δ ( t ) ) + ρ ( t ) w ( t ) γ ρ σ ( t ) R ( t , t ) r γ + 1 γ ( t ) ( z Δ Δ ( t ) ) γ + 1 z ( t ) z γ + 1 ( t ) z ( σ ( t ) ) ρ σ ( t ) q 2 ( t ) ( ϕ 2 ( t ) σ ( t ) ) γ + ( ρ Δ ( t ) ) + ρ ( t ) w ( t ) γ ρ σ ( t ) R ( t , t ) t σ ( t ) w γ + 1 γ ( t ) ρ γ + 1 γ ( t ) .
(17)
In the second case γ > 1 . Applying the Keller chain rule, we have
( z γ ( t ) ) Δ = γ 0 1 [ h z σ + ( 1 h ) z ] γ 1 z Δ ( t ) d h γ ( z ( t ) ) γ 1 z Δ ( t ) ,
(18)
in the view of (18), Lemma 2.2, Lemma 2.3, and (9), we have
w Δ ( t ) ρ σ ( t ) q 2 ( t ) ( ϕ 2 ( t ) σ ( t ) ) γ + ( ρ Δ ( t ) ) + ρ ( t ) w ( t ) w Δ ( t ) γ ρ σ ( t ) r ( t ) ( [ z ( t ) ] Δ Δ ) γ z Δ ( t ) z γ ( t ) z γ + 1 ( t ) z γ σ ( t ) , w Δ ( t ) ρ σ ( t ) q 2 ( t ) ( ϕ 2 ( t ) σ ( t ) ) γ + ( ρ Δ ( t ) ) + ρ ( t ) w ( t ) w Δ ( t ) γ ρ σ ( t ) ( t σ ( t ) ) γ R ( t , t ) w γ + 1 γ ( t ) ρ γ + 1 γ ( t ) .
(19)
By (17), (19), and the definition of b ( t ) and β ( t ) , we have, for γ > 0 ,
w Δ ( t ) ρ σ ( t ) q 2 ( t ) ( ϕ 2 ( t ) σ ( t ) ) γ + ( ρ Δ ( t ) ) + ρ ( t ) w ( t ) γ ρ σ ( t ) β ( t ) R ( t , t ) w λ ( t ) ρ λ ( t ) ,
(20)
where λ : = γ + 1 γ . Define A 0 and B 0 by
A λ : = γ ρ σ ( t ) β ( t ) R ( t , t ) w λ ( t ) ρ λ ( t ) , B λ 1 : = ρ Δ ( t ) λ ( γ ρ σ ( t ) β ( t ) R ( t , t ) ) 1 λ .
Then using the inequality [15]
λ A B λ 1 A λ ( λ 1 ) B λ ,
(21)
which yields
( ρ Δ ( t ) ) + ρ ( t ) w ( t ) γ ρ σ ( t ) β ( t ) R ( t , t ) w λ ( t ) ρ λ ( t ) ( ( ρ Δ ( t ) ) + ) γ + 1 ( γ + 1 ) γ + 1 ( β ( t ) ρ σ ( t ) R ( t , t ) ) γ .
From this last inequality and (20), we find
w Δ ( t ) ρ σ ( t ) q 2 ( t ) ( ϕ 2 ( t ) σ ( t ) ) γ + ( ( ρ Δ ( t ) ) + ) γ + 1 ( γ + 1 ) γ + 1 ( β ( t ) ρ σ ( t ) R ( t , t ) ) γ .
Integrating both sides from T to t, we get
T t [ ρ σ ( s ) q 2 ( s ) ( ϕ 2 ( s ) σ ( s ) ) γ ( ( ρ Δ ( s ) ) + ) γ + 1 ( γ + 1 ) γ + 1 ( β ( s ) ρ σ ( s ) R ( s , t ) ) γ ] Δ s w ( T ) w ( t ) w ( T ) ,

which contradicts assumption (13). This completes the proof of Theorem 3.1. □

Remark 3.1 From Theorem 3.1, we can obtain different conditions for oscillation of (1) with different choices of ρ ( t ) .

Remark 3.2 The conclusion of Theorem 3.1 remains intact if assumption (13) is replaced by the two conditions
lim sup t T t ρ σ ( s ) q 2 ( s ) ( ϕ 2 ( s ) σ ( s ) ) γ Δ s = , lim sup t T t ( ( ρ Δ ( s ) ) + ) γ + 1 ( γ + 1 ) γ + 1 ( β ( s ) ρ σ ( s ) ψ ( s , t ) ) γ Δ s < .

For example, let ρ ( t ) = t . Now Theorem 3.1 yields the following results.

Corollary 3.1 Assume that (H1)-(H5), (5), and (10) hold. If
lim sup t T t [ σ ( s ) q 2 ( s ) ( ϕ 2 ( s ) σ ( s ) ) γ 1 ( γ + 1 ) γ + 1 ( β ( s ) σ ( s ) R ( s , t ) ) γ ] Δ s =
(22)

holds, then every solution (1) is either oscillatory or lim t x ( t ) = 0 .

For example, let ρ ( t ) = 1 . Now Theorem 3.1 yields the following results.

Corollary 3.2 Assume that (H1)-(H5), (5), and (10) hold. If
lim sup t T t q 2 ( s ) ( ϕ 2 ( s ) σ ( s ) ) γ Δ s = ,
(23)

then every solution (1) is either oscillatory or lim t x ( t ) = 0 .

Theorem 3.2 Assume that (2), (5), and (10) hold. Furthermore, suppose that there exist functions H , h C r d ( D , R ) , where D ( t , s ) : t s t 0 such that
H ( t , t ) = 0 , t 0 , H ( t , s ) > 0 , t > s t 0 ,
and H has a nonpositive continuous Δ-partial derivative H Δ s ( t , s ) with respect to the second variable and satisfies
H Δ s ( σ ( t ) , s ) + H ( σ ( t ) , σ ( s ) ) ρ Δ ( s ) ρ ( s ) = h ( t , s ) ρ ( s ) H ( σ ( t ) , σ ( s ) ) γ γ + 1 ,
(24)
and for all sufficiently large T 1 [ t 0 , ) T , there is a T > T 1 such that
lim sup t 1 H ( σ ( t ) , T ) T σ ( t ) K ( t , s ) = ,
(25)
where ρ is a positive Δ-differentiable function and
K ( t , s ) = H ( σ ( t ) , σ ( s ) ) ρ σ ( s ) q 2 ( s ) ( ϕ 2 ( s ) σ ( s ) ) γ ( h ( t , s ) ) γ + 1 ( γ + 1 ) γ + 1 ( β ( s ) ρ σ ( s ) R ( s , T 1 ) ) γ Δ s = .

Then every solution of (1) is either oscillatory or tends to zero.

Proof Suppose that x ( t ) is a non-oscillatory solution of (1) and z ( t ) is defined as in (3). Without loss of generality, we may assume that there is a t 1 [ t 0 , ) T sufficiently large so that the conclusions of Lemma 2.1 hold and (24) holds for t 2 > t 1 . If case (1) of Lemma 2.1 holds then proceeding as in the proof of Theorem 3.1, we see that (20) holds for t > t 2 . Multiplying both sides of (20) by H ( σ ( t ) , σ ( s ) ) and integrating from T to σ ( t ) , we get
T σ ( t ) H ( σ ( t ) , σ ( s ) ) ρ σ ( s ) q 2 ( s ) ( ϕ 2 ( s ) σ ( s ) ) γ Δ s T σ ( t ) H ( σ ( t ) , σ ( s ) ) w Δ ( s ) Δ s + T σ ( t ) H ( σ ( t ) , σ ( s ) ) ρ Δ ( s ) ρ ( s ) w ( s ) Δ s T σ ( t ) H ( σ ( t ) , σ ( s ) ) γ ρ σ ( s ) β ( s ) R ( s , T 1 ) w λ ( s ) ρ λ ( s ) Δ s ( λ = γ + 1 γ ) .
(26)
Integrating by parts and using H ( t , t ) = 0 , we obtain
T σ ( t ) H ( σ ( t ) , σ ( s ) ) w Δ ( s ) Δ s = H ( σ ( t ) , T ) w ( T ) T σ ( t ) H Δ s ( σ ( t ) , s ) w ( s ) Δ s .
It then follows from (26) that
T σ ( t ) H ( σ ( t ) , σ ( s ) ) ρ σ ( s ) q 2 ( s ) ( ϕ 2 ( s ) σ ( s ) ) γ Δ s H ( σ ( t ) , T ) w ( T ) + T σ ( t ) H Δ s ( σ ( t ) , s ) w ( s ) Δ s + T σ ( t ) H ( σ ( t ) , σ ( s ) ) ρ Δ ( s ) ρ ( s ) w ( s ) Δ s T σ ( t ) H ( σ ( t ) , σ ( s ) ) γ ρ σ ( s ) β ( s ) R ( s , T 1 ) w λ ( s ) ρ λ ( s ) Δ s , T σ ( t ) H ( σ ( t ) , σ ( s ) ) ρ σ ( s ) q 2 ( s ) ( ϕ 2 ( s ) σ ( s ) ) γ Δ s H ( σ ( t ) , T ) w ( T ) + [ T σ ( t ) H Δ s ( σ ( t ) , s ) + H ( σ ( t ) , σ ( s ) ) ρ Δ ( s ) ρ ( s ) ] w ( s ) Δ s T σ ( t ) H ( σ ( t ) , σ ( s ) ) γ ρ σ ( s ) β ( s ) R ( s , T 1 ) w λ ( s ) ρ λ ( s ) Δ s .
It then follows from (24) that
T σ ( t ) H ( σ ( t ) , σ ( s ) ) ρ σ ( s ) q 2 ( s ) ( ϕ 2 ( s ) σ ( s ) ) γ Δ s H ( σ ( t ) , T ) w ( T ) + T σ ( t ) [ h ( t , s ) ρ ( s ) H ( σ ( t ) , σ ( s ) ) γ γ + 1 ] w ( s ) Δ s T σ ( t ) H ( σ ( t ) , σ ( s ) ) γ ρ σ ( s ) β ( s ) R ( s , T 1 ) w λ ( s ) ρ λ ( s ) Δ s H ( σ ( t ) , T ) w ( T ) + T σ ( t ) [ h ( t , s ) ρ ( s ) H ( σ ( t ) , σ ( s ) ) γ γ + 1 ] w ( s ) Δ s T σ ( t ) H ( σ ( t ) , σ ( s ) ) γ ρ σ ( s ) β ( s ) R ( s , T 1 ) w λ ( s ) ρ λ ( s ) Δ s .
Therefore, as in Theorem 3.1, by letting
A λ : = H ( σ ( t ) , σ ( s ) ) γ ρ σ ( t ) β ( t ) R ( t , T 1 ) w λ ( t ) ρ λ ( t ) , B λ 1 : = h ( t , s ) λ ( γ ρ σ ( t ) β ( t ) R ( t , T 1 ) ) 1 λ .
Then using the inequality [15]
λ A B λ 1 A λ ( λ 1 ) B λ .
We have
T σ ( t ) [ h ( t , s ) ρ ( s ) H ( σ ( t ) , σ ( s ) ) γ γ + 1 ] w ( s ) Δ s T σ ( t ) H ( σ ( t ) , σ ( s ) ) γ ρ σ ( s ) β ( s ) R ( s , T 1 ) w λ ( s ) ρ λ ( s ) Δ s = T σ ( t ) ( h ( t , s ) ) γ + 1 ( γ + 1 ) γ + 1 ( β ( s ) ρ σ ( s ) R ( t , T 1 ) ) γ Δ s , T σ ( t ) H ( σ ( t ) , σ ( s ) ) ρ σ ( s ) q 2 ( s ) ( ϕ 2 ( s ) σ ( s ) ) γ Δ s H ( σ ( t ) , T ) w ( T ) + T σ ( t ) ( h ( t , s ) ) γ + 1 ( γ + 1 ) γ + 1 ( β ( s ) ρ σ ( s ) R ( t , T 1 ) ) γ Δ s .
Then for T > T 1 we have
T σ ( t ) [ H ( σ ( t ) , σ ( s ) ) ρ σ ( s ) q 2 ( s ) ( ϕ 2 ( s ) σ ( s ) ) γ ( h ( t , s ) ) γ + 1 ( γ + 1 ) γ + 1 ( β ( s ) ρ σ ( s ) R ( s , T 1 ) ) γ ] Δ s H ( σ ( t ) , T ) w ( T ) ,
and this implies that
1 H ( σ ( t ) , T ) T σ ( t ) [ H ( σ ( t ) , σ ( s ) ) ρ σ ( s ) q 2 ( s ) ( ϕ 2 ( s ) σ ( s ) ) γ ( h ( t , s ) ) γ + 1 ( γ + 1 ) γ + 1 ( β ( s ) ρ σ ( s ) R ( s , T 1 ) ) γ ] Δ s < w ( T ) ,

for all large T, which contradicts (25). This completes the proof of Theorem 3.2. □

Remark 3.3 The conclusion of Theorem 3.2 remains intact if assumption (25) is replaced by the two conditions
lim sup t 1 H ( σ ( t ) , T ) T σ ( t ) H ( σ ( t ) , σ ( s ) ) ρ σ ( s ) q 2 ( s ) ( ϕ 2 ( s ) σ ( s ) ) γ Δ s = , lim inf t 1 H ( σ ( t ) , T ) T σ ( t ) ( h ( t , s ) ) γ + 1 ( γ + 1 ) γ + 1 ( β ( s ) ρ σ ( s ) R ( s , T 1 ) ) γ Δ s < .
Remark 3.4 Define w as (14), we also get
w Δ ( t ) = r σ ( t ) ( z Δ Δ ( t ) ) γ σ [ ρ ( t ) z γ ( t ) ] Δ + ρ ( t ) z γ [ r ( t ) ( z Δ Δ ( t ) ) γ ] Δ ,

similar to the proofs of Theorem 3.1, we can obtain different results. We leave the details to the reader.

Example 3.1 Consider the following third-order neutral dynamic equation t [ t 0 , ) T :
( x ( t ) + a b e t x ( t η ) Δ η ) Δ Δ Δ + c d β t ( t 2 t ξ ) ( t 2 t ξ ) σ x ( t ξ ) Δ ξ = 0 ,
(27)

where γ = 1 , r ( t ) = 1 , τ ( t , η ) = t η , ϕ ( t , ξ ) = t ξ , δ = 1 , q 2 ( t ) = β t ϕ 2 ( t ) , p ( t , η ) = e t , q ( t , ξ ) = β t / ( t 2 t ξ ) ( t 2 t ξ ) σ .

It is clear that condition (2), (5), and (10) hold. Therefore, by Theorem 3.1, picking ρ ( t ) = t , we have
lim sup t T t [ ρ σ ( s ) q 2 ( s ) ( ϕ 2 ( s ) σ ( s ) ) γ ( ( ρ Δ ( s ) ) + ) γ + 1 ( γ + 1 ) γ + 1 ( β ( s ) ρ σ ( s ) R ( s , t ) ) γ ] Δ s = lim sup t T t [ β s 1 ( γ + 1 ) ( γ + 1 ) s ( s t ) ] Δ s = .

Hence, by Theorem 3.1 every solution of (27) is oscillatory or tends to zero if β > 0 .

Example 3.2 Consider the following third-order neutral dynamic equation t [ t 0 , ) T :
[ 1 t ( [ x ( t ) + a b 1 2 x [ τ ( t 2 ) ] Δ η ] Δ Δ ) 3 ] Δ + c d q ( t , ξ ) f ( x [ ϕ ( t 2 ) ] ) Δ ξ = 0 ,
(28)

where γ = 3 , r ( t ) = 1 t , τ ( t , η ) = t 2 , ϕ ( t , ξ ) = t 2 , δ = 1 , q 2 ( t ) = β t σ 3 ( s ) ϕ 2 3 ( t ) , p ( t , η ) = 1 2 .

It is clear that condition (2), (5), and (10) hold. Therefore, by Theorem 3.1, picking ρ ( t ) = 1 , we have
lim sup t T t q 2 ( s ) ( ϕ 2 ( s ) σ ( s ) ) 3 Δ s = lim sup t T t β s Δ s = .

Hence, by Theorem 3.1 every solution of (28) is oscillatory or tends to zero if β > 0 .

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Sciences, Erciyes University, Kayseri, 38039, Turkey
(2)
Institute of Sciences, Erciyes University, Kayseri, 38039, Turkey

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© Şenel and Utku; licensee Springer. 2014

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 credited.

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