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

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 ) ] ) Δξ=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.

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 ) ] ) Δξ=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 Δt=;
(2)

(H2) p(t,η) C r d ([ t 0 ,)×[a,b],R), 0p(t) a b p(t,η)ΔηP<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)tand 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)tand lim t min ξ [ c , d ] ϕ(t,ξ)=;

(H5) the function f C r d (T,R) is assumed to satisfy uf(u)>0 and there exists a positive rd-continuous function δ(t) on T such that f ( u ) u γ δ, for u0.

Define the function by

z(t)=x(t)+ a b p(t,η)x [ τ ( t , η ) ] Δη.
(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 ) ] ) Δξ=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 ) ) ) Δξ=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:TT satisfies τ(t)t, and τ(t) as t, f(x) C r d (T×R,R) is assumed to satisfy uf(t,u)>0, for u0, and there exists a function p on T such that f ( t , u ) u γ p(t)>0, for u0.

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)dh,

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 ) ] ) Δξ<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 Δs.

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 ΔuΔv=
(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)Δξ= q 1 (t) ( z ( ϕ 1 ( t ) ) ) γ ,
(7)

where q 1 (t)= K γ δ c d q(t,ξ)Δξ, 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 Δs.

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 Δu.

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 ΔuΔv.

Because (7) and the last inequality contradict (5), we have I=0. Since 0x(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)Δs=.
(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))>dϕ(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)Δξ, 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 ) ) Δs0.

This implies that

r( t 2 ) ( z Δ Δ ( t 2 ) ) γ t 2 t q 2 (s) ( z ( ϕ 2 ( s ) ) ) γ Δs d γ t 2 t q 2 (s) ϕ 2 γ (s)Δs,

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 ] Δs=.
(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)Δξ, 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<γ1 and γ>1. In the first case 0<γ1. 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)dhγ ( 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)dhγ ( 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 A0 and B0 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]

λA B λ 1 A λ (λ1) 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 ] Δsw(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 ] Δs=
(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 Δs=,
(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):ts 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 Δs=.

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)Δs=H ( σ ( t ) , T ) w(T) T σ ( t ) H Δ s ( σ ( t ) , s ) w(s)Δs.

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]

λA B λ 1 A λ (λ1) 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(tξ)Δξ=0,
(27)

where γ=1, r(t)=1, τ(t,η)=tη, ϕ(t,ξ)=tξ, δ=1, q 2 (t)= β t ϕ 2 ( t ) , p(t,η)= e t , q(t,ξ)=βt/( t 2 tξ) ( 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 ) ] ) Δξ=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 Δs= lim sup t T t β s Δs=.

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

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Correspondence to Mehmet Tamer Şenel.

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Şenel, M.T., Utku, N. Oscillation criteria for third-order neutral dynamic equations with continuously distributed delay. Adv Differ Equ 2014, 220 (2014). https://doi.org/10.1186/1687-1847-2014-220

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Keywords

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