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Oscillation criteria for second-order quasi-linear delay dynamic equations on time scales

Advances in Difference Equations20142014:45

https://doi.org/10.1186/1687-1847-2014-45

Received: 30 April 2013

Accepted: 10 January 2014

Published: 29 January 2014

Abstract

This paper is concerned with oscillations of the second-order delay nonlinear dynamic equation ( a ( t ) ( x Δ ( t ) ) α ) Δ + q ( t ) x β ( τ ( t ) ) = 0 on a time scale , where a and q are real-valued rd-continuous positive functions on , α and β are ratios of odd positive integers, τ : T T , τ ( t ) t , t T , and lim t τ ( t ) = . We establish some new sufficient conditions for this equation.

MSC:34K11, 39A10, 39A99.

Keywords

  • oscillation
  • delay quasi-linear dynamic equation
  • time scales

1 Introduction

The study of dynamic equations on time scales, which goes back to Hilger [1], provides a rapidly expanding body of literature where the main idea is to unify, extend, and generalise concepts from continuous, discrete and quantum calculus to arbitrary time scales analysis, where a time scale is simply any nonempty closed subset of the reals.

For detailed information regarding calculus on time scales, we refer the reader to Bohner and Peterson [2, 3].

Now we consider the second-order delay nonlinear dynamic equation
( a ( t ) ( x Δ ( t ) ) α ) Δ + q ( t ) x β ( τ ( t ) ) = 0
(1.1)

on an arbitrary time scale , where a and q are real-valued rd-continuous positive functions defined on ; α and β > 0 are ratios of odd positive integers; τ : T T , τ ( t ) t , t T and lim t τ ( t ) = .

We shall also consider the case
t 0 Δ s a 1 / α ( s ) = .
(1.2)

Since we are interested in the oscillatory and asymptotic behavior of solutions near infinity, we assume that sup T = , and we define the time scale interval [ t 0 , ) T by [ t 0 , ) T : = [ t 0 , ) T . By a solution of (1.1) we mean a nontrivial real-valued function x C rd 1 [ T x , ) , T x t 0 which has the property that a ( t ) ( x Δ ( t ) ) α C rd 1 [ T x , ) and satisfies (1.1) on [ T x , ) ; here C rd is the space of rd-continuous functions. The solutions vanishing in some neighborhood of infinity will be excluded from our consideration. A solution x of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is nonoscillatory.

If T = R then σ ( t ) = t , μ ( t ) = 0 , f Δ ( t ) = f ( t ) , a b f ( t ) Δ t = a b f ( t ) d t and when α = β (1.1) becomes the half-linear delay differential equation
( a ( t ) ( x ( t ) ) α ) + q ( t ) x α ( τ ( t ) ) = 0 .
(1.3)
When a ( t ) = 1 and α = 1 , (1.3) becomes
x ( t ) + q ( t ) x ( τ ( t ) ) = 0 .
(1.4)
Ohriska [4] proved that every solution of (1.4) oscillates if
lim sup t t t q ( s ) ( τ ( s ) s ) d s > 1
(1.5)
holds. Agarwal et al. [5] considered (1.3) and extended the condition (1.5) and proved that if
lim sup t t α t q ( s ) ( τ ( s ) s ) α d s > 1 ,
(1.6)

then every solution of (1.3) oscillates.

If T = Z , then σ ( t ) = t + 1 , μ ( t ) = 1 , f Δ ( t ) = Δ f ( t ) , a b f ( t ) Δ t = t = a b 1 f ( t ) , and (1.1) becomes the quasi-linear difference equation
Δ ( a ( t ) ( Δ x ( t ) ) α ) + q ( t ) x β ( τ ( t ) ) = 0 .
(1.7)
When α = β , (1.1) becomes the half-linear delay dynamic equation which has been considered by some authors and some oscillation and nonoscillation results have been obtained [2, 3, 510]. As a special case of (1.1) Agarwal et al. [11] considered the second-order delay dynamic equations on time scales
x Δ Δ ( t ) + q ( t ) x ( τ ( t ) ) = 0
(1.8)
and established some sufficient conditions for oscillations of (1.8) when
t 0 τ ( t ) q ( t ) Δ t = .
(1.9)
Saker [12] studied (1.8) to extend the result of Lomtatidze [13]. Recently Higgins [14] proved oscillation results for (1.8). Saker [15] examined oscillations for the half-linear dynamic equation
( a ( t ) ( x Δ ( t ) ) α ) Δ + q ( t ) x α ( t ) = 0
(1.10)
on time scales, where α > 1 is an odd positive integer and Agarwal et al. [6] and Grace et al. [10] studied oscillations for the same equation, (1.10), where α > 1 is the quotient of odd positive integers which cannot be applied when 0 < α 1 . Han et al. [16] and Hassan [17] solved this problem and improved Agarwal’s and Saker’s results. Erbe et al. [7] considered the half-linear delay dynamic equation
( a ( t ) ( x Δ ( t ) ) α ) Δ + q ( t ) x α ( τ ( t ) ) = 0
(1.11)
on time scales, where α > 1 is the quotient of odd positive integers and
a Δ ( t ) 0 and t 0 τ α ( t ) q ( t ) Δ t =
(1.12)
and utilised a Riccati transformation technique and established some oscillation criteria for (1.11). Erbe et al. [8] considered the half-linear delay dynamic equation (1.11) on time scales, where 0 < α 1 is the quotient of odd positive integers and established some sufficient conditions for oscillations when (1.12) holds. Han et al. [18] considered (1.11) and followed the proof that has been used in [15] and established some sufficient conditions for oscillations when a Δ ( t ) 0 . For oscillations of quasi-linear dynamic equations, Grace et al. [19] considered the equation
( a ( t ) ( x Δ ( t ) ) α ) Δ + q ( t ) x β ( t ) = 0 ;
(1.13)
and Saker and Grace [20] considered the delay equation
( a ( t ) ( x Δ ( t ) ) α ) Δ + q ( t ) x β ( τ ( t ) ) = 0
(1.14)

on an arbitrary time scale where τ ( t ) t or τ ( t ) t . The special case of (1.14) where 0 < α = β 1 has been studied in [9] by Erbe et al.

In this paper, we establish some new sufficient conditions for oscillations of (1.1).

2 Preliminary result

For a function f : T R , the (delta) derivative f Δ ( t ) at t T is defined to be the number (if it exists) such that for all ϵ > 0 there is a neighborhood U of t with
| f ( σ ( t ) ) f ( s ) f Δ ( σ ( t ) s ) | ϵ | σ ( t ) s |
(2.1)
for all s U . If the (delta) derivative f Δ ( t ) exists for all t T , then we say that f is (delta) differentiable on . For two (delta) differentiable functions f and g, the derivative of the product fg and the quotient f / g (where g g σ 0 ) are given as in [[2], Theorem 1.20]
( f g ) Δ = f Δ g + f σ g Δ = f g Δ + f Δ g σ ,
(2.2)
( f g ) Δ = f Δ g f g Δ g g σ
(2.3)
as well as of the chain rule [[2], Theorem 1.90] for the derivative of the composite function f o g for a continuously differentiable function f : R R and a (delta) differentiable function g : T R
( f o g ) Δ = [ 0 1 f ( g + h μ g Δ ) d h ] g Δ .
(2.4)
For b , c T and a differentiable function f, the Cauchy integral of f Δ is defined by
b c f Δ ( t ) Δ t = f ( c ) f ( b )
(2.5)
and the improper integral is defined as
b f ( t ) Δ t = lim c b c f ( t ) Δ t .
(2.6)
Lemma 2.1 If A and B are nonnegative real numbers and λ > 1 , then
A λ λ A B λ 1 + ( λ 1 ) B λ 0 ,

where the equality holds if and only if A = B .

3 The main results

In this section, by employing a Riccati transformation technique, we establish oscillation criteria for (1.1). To prove our main result, we will use the formula
( x α ( t ) ) Δ = α { 0 1 [ h x σ ( t ) + ( 1 h ) x ( t ) ] α 1 d h } x Δ ( t ) ,
(3.1)

which is a simple consequence of the Pötzsche chain rule [2].

In the following theorems, we assume that (1.2) holds.

Theorem 3.1 Assume that there exists a positive nondecreasing delta differentiable function ξ ( t ) such that, for all sufficiently large T t 0 , and for τ ( t ) > g ( T ) , we have
lim sup t t 0 t [ θ β ( s , g ( T ) ) ξ ( s ) q ( s ) ( α β ) α a ( s ) ( ξ Δ ( s ) ) α + 1 ( α + 1 ) α + 1 ( ξ ( s ) δ 1 ( s ) ) α ] Δ s = ,
(3.2)
where ( ξ Δ ( s ) ) + : = max { 0 , ξ Δ ( s ) } ,
δ 1 ( t ) = { c 1  is any positive constant , if  β > α , 1 , if  β = α , c 2 ( η σ ( t ) ) α β α , c 2  is any positive constant , if  β < α ,
(3.3)
and
θ ( t , g ( T ) ) = g ( T ) τ ( t ) Δ s a 1 α ( s ) g ( T ) t Δ s a 1 α ( s ) , η ( t ) = ( t 1 t Δ s a 1 α ( s ) ) 1 .
(3.4)

Then every solution of (1.1) is oscillatory on [ t 0 , ) T .

Proof Let x be a nonoscillatory solution of (1.1) on [ t 0 , ) T . Then, without loss of generality, there is a t 1 [ t 0 , ) T , sufficiently large, such that x ( t ) > 0 and x ( τ ( t ) ) > 0 on [ t 1 , ) T . Since a ( x Δ ) α is a strictly decreasing function, it is of one sign. We claim that x Δ > 0 on [ t 1 , ) T . If not, then there is a t 2 [ t 1 , ) T such that x Δ ( t 2 ) < 0 . Using the fact a ( x Δ ) α is strictly decreasing, we get
x ( t ) x ( t 2 ) + [ a ( t 2 ) ( x Δ ( t 2 ) ) α ] 1 / α t 2 t Δ s a 1 / α ( s ) , as  t ,
which implies that x ( t ) is eventually negative. This is a contradiction. Hence x Δ > 0 on [ t 1 , ) T . Consider the generalized Riccati substitution
w ( t ) = ξ ( t ) a ( t ) ( x Δ ( t ) ) α x β ( t ) ,
(3.5)
then w ( t ) > 0 . By the product rule and then the quotient rule, we have
w Δ ( t ) = ( ξ ( t ) x β ( t ) ) ( a ( t ) ( x Δ ( t ) ) α ) Δ + ( a ( t ) ( x Δ ( t ) ) α ) σ ( ξ ( t ) x β ( t ) ) Δ = q ( t ) ξ ( t ) ( x ( τ ( t ) ) x ( t ) ) β + ξ Δ ( t ) ( a ( t ) ( x Δ ( t ) ) α ) σ x β ( σ ( t ) ) ξ ( t ) ( a ( t ) ( x Δ ( t ) ) α ) σ ( x β ( t ) ) Δ x β ( t ) x β ( σ ( t ) ) .
(3.6)
Using the fact that x ( t ) is increasing and a ( x Δ ) α is decreasing, we have
x ( t ) x ( τ ( t ) ) = τ ( t ) t ( a ( s ) ( x Δ ( s ) ) α ) a 1 α ( s ) 1 α Δ s τ ( t ) t ( a ( τ ( t ) ) ( x Δ ( τ ( t ) ) ) α ) a 1 α ( s ) 1 α Δ s = a 1 α ( τ ( t ) ) x Δ ( τ ( t ) ) τ ( t ) t Δ s a 1 α ( s )
and so
x ( t ) x ( τ ( t ) ) 1 + a 1 α ( τ ( t ) ) x Δ ( τ ( t ) ) x ( τ ( t ) ) τ ( t ) t Δ s a 1 α ( s )
(3.7)
for t t 1 . Also, we see that
x ( τ ( t ) ) > x ( τ ( t ) ) x ( g ( T ) ) = g ( T ) τ ( t ) ( a ( s ) ( x Δ ( s ) ) α ) a 1 α ( s ) 1 α Δ s a 1 α ( τ ( t ) ) x Δ ( τ ( t ) ) g ( T ) τ ( t ) Δ s a 1 α ( s ) , a 1 α ( τ ( t ) ) x Δ ( τ ( t ) ) x ( τ ( t ) ) ( g ( T ) τ ( t ) Δ s a 1 α ( s ) ) 1 .
(3.8)
Therefore, (3.7) and (3.8) imply that
x ( τ ( t ) ) x ( t ) g ( T ) τ ( t ) Δ s a 1 α ( s ) g ( T ) t Δ s a 1 α ( s ) = θ ( t , g ( T ) ) .
(3.9)
In view of (3.1), we get
( x β ( t ) ) Δ = β x Δ ( t ) 0 1 [ ( x ( t ) + h μ ( t ) x Δ ( t ) ) ] β 1 d h { β ( x σ ( t ) ) β 1 x Δ ( t ) , 0 < β 1 , β ( x ( t ) ) β 1 x Δ ( t ) , β > 1 .
(3.10)
From (3.9), (3.10), the fact that x ( t ) is an increasing function and the definition of w ( t ) , if 0 < β 1 , we have
w Δ ( t ) q ( t ) ξ ( t ) θ β ( t , g ( T ) ) + ξ Δ ( t ) w σ ( t ) ξ σ ( t ) β ξ ( t ) ( a ( t ) ( x Δ ( t ) ) α ) σ ( x σ ( t ) ) β + 1 x Δ ( t ) = q ( t ) ξ ( t ) θ β ( t , g ( T ) ) + ξ Δ ( t ) w σ ( t ) ξ σ ( t ) β ξ ( t ) w σ ( t ) ξ σ ( t ) x Δ ( t ) x σ ( t ) ,
whereas, if β > 1 , we have
w Δ ( t ) q ( t ) ξ ( t ) θ β ( t , g ( T ) ) + ξ Δ ( t ) w σ ( t ) ξ σ ( t ) β ξ ( t ) ( a ( t ) ( x Δ ( t ) ) α ) σ ( x σ ( t ) ) β + 1 x Δ ( t ) = q ( t ) ξ ( t ) θ β ( t , g ( T ) ) + ξ Δ ( t ) w σ ( t ) ξ σ ( t ) β ξ ( t ) w σ ( t ) ξ σ ( t ) x Δ ( t ) x σ ( t ) .
Using the fact that a ( t ) ( x Δ ( t ) ) α is strictly decreasing on [ t 1 , ) T , we find
x Δ ( t ) x σ ( t ) ( ( a ( t ) ( x Δ ( t ) ) α ) σ ) 1 α a 1 α ( t ) ( x σ ( t ) ) β α ( x σ ( t ) ) β α α = ( w σ ( t ) ξ σ ( t ) ) 1 α ( x σ ( t ) ) β α α a 1 α ( t ) .
Thus for β > 0 , we have
w Δ ( t ) q ( t ) ξ ( t ) θ β ( t , g ( T ) ) + ξ Δ ( t ) w σ ( t ) ξ σ ( t ) β ξ ( t ) a 1 α ( t ) ( w σ ( t ) ξ σ ( t ) ) α + 1 α ( x σ ( t ) ) β α α .
(3.11)

We consider the following three cases:

Case (i). β > α .

In this case, since x Δ ( t ) > 0 , there exists t 2 t 1 such that x σ ( t ) x ( t ) c > 0 . This implies that ( x σ ( t ) ) β α α c 1 , where c 1 = c β α α .

Case (ii). β = α .

In this case, we see that ( x σ ( t ) ) β α α = 1 .

Case (iii). β < α .

Then there exists a positive constant b : = a ( t 1 ) ( x Δ ( t 1 ) ) α . Using the decreasing of a ( x Δ ) α , we have
x Δ ( t ) b 1 α a 1 α ( t ) for  t t 1 .
Integrating this inequality from t 1 to t, we have
x ( t ) x ( t 1 ) + b 1 α t 1 t a 1 α ( s ) Δ s .
Thus, there exist a constant b 1 > 0 and t 2 t 1 such that
x ( t ) b 1 η 1 ( t ) for  t t 2
and hence
( x σ ( t ) ) β α α c 2 ( η σ ) α β α ( t ) for  t t 2 ,

where c 2 = b 1 β α α .

By (3.3), we get
w Δ ( t ) q ( t ) ξ ( t ) θ β ( t , g ( T ) ) + ξ Δ ( t ) w σ ( t ) ξ σ ( t ) β a 1 α ( t ) ξ ( t ) ( w σ ( t ) ξ σ ( t ) ) α + 1 α δ 1 ( t )
(3.12)
for t t 2 . Defining A 0 and B 0 by
A = ( β δ 1 ( t ) ξ ( t ) ) α α + 1 a 1 α + 1 ( t ) w σ ( t ) ξ σ ( t ) , B = ( α α + 1 ) α ( ξ Δ ( t ) ξ σ ( t ) ) α [ ( β δ 1 ( t ) ξ ( t ) ) α α + 1 ξ σ ( t ) a 1 α + 1 ( t ) ] α
and using Lemma 2.1 we get
β δ 1 ( t ) ξ ( t ) a 1 α ( t ) ( w σ ( t ) ξ σ ( t ) ) α + 1 α ξ Δ ( t ) w σ ( t ) ξ σ ( t ) ( α β ) α a ( t ) ( ξ Δ ( t ) ) α + 1 ( α + 1 ) α + 1 ( δ 1 ( t ) ξ ( t ) ) α ,
where λ = α + 1 α . Therefore, by (3.12), we have
w Δ ( t ) q ( t ) ξ ( t ) θ β ( t , g ( T ) ) + ( α β ) α a ( t ) ( ξ Δ ( t ) ) α + 1 ( α + 1 ) α + 1 ( δ 1 ( t ) ξ ( t ) ) α .
(3.13)
Integrating (3.13) from t 2 to t, we get as t
w ( t ) w ( t 2 ) t 2 t [ q ( s ) ξ ( s ) θ β ( s , g ( T ) ) ( α β ) α a ( s ) ( ξ Δ ( s ) ) α + 1 ( α + 1 ) α + 1 ( δ 1 ( s ) ξ ( s ) ) α ] Δ s = ,

which contradicts (3.5). □

Theorem 3.2 Assume that there exists a positive nondecreasing delta differentiable function ξ ( t ) such that, for all sufficiently large T t 0 , and for τ ( t ) > g ( T ) , we have
lim sup t t 0 t [ q ( s ) ξ ( s ) θ β ( s , g ( T ) ) δ 2 ( s ) η α ( s ) ξ Δ ( s ) ] Δ s = ,
(3.14)
where
δ 2 ( t ) = { c 1  is any positive constant , if  β > α , 1 , if  β = α , c 2 η β α ( t ) , c 2  is any positive constant , if  β < α ,
(3.15)

and θ is as in Theorem 3.1. Then (1.1) is oscillatory on [ t 0 , ) T .

Proof Let x be a nonoscillatory solution of (1.1) on [ t 0 , ) T . Then, without loss of generality, there is a t 1 [ t 0 , ) T , sufficiently large, such that x ( t ) > 0 and x ( τ ( t ) ) > 0 on [ t 1 , ) . Proceeding as in the proof of Theorem 3.1, we obtain (3.11). Therefore
w Δ ( t ) q ( t ) ξ ( t ) θ β ( t , g ( T ) ) + ξ Δ ( t ) w σ ( t ) ξ σ ( t ) .
(3.16)
Using the definition of w ( t ) , it follows from (3.16) that
w Δ ( t ) q ( t ) ξ ( t ) θ β ( t , g ( T ) ) + ξ Δ ( t ) ( a ( t ) ( x Δ ( t ) ) α ) σ x β ( σ ( t ) ) q ( t ) ξ ( t ) θ β ( t , g ( T ) ) + ξ Δ ( t ) a ( t ) ( x Δ ( t ) ) α x β ( t ) = q ( t ) ξ ( t ) θ β ( t , g ( T ) ) + ξ Δ ( t ) a ( t ) ( x Δ ( t ) x ( t ) ) α x α β ( t ) .
(3.17)
Now, from
x ( t ) = x ( t 1 ) + t 1 t ( a ( s ) ( x Δ ( s ) ) α ) a 1 α ( s ) 1 α Δ s a 1 α ( t ) x Δ ( t ) t 1 t Δ s a 1 α ( s )
it follows that
( x Δ ( t ) x ( t ) ) α a 1 ( t ) ( t 1 t Δ s a 1 α ( s ) ) α = η α ( t ) a ( t ) .
(3.18)
Using (3.18) in (3.17), we have
w Δ ( t ) q ( t ) ξ ( t ) θ β ( t , g ( T ) ) + ξ Δ ( t ) ( η ( t ) ) α x α β ( t ) .
(3.19)

Next as in the proof of Theorem 3.1, we consider the Cases (i), (ii) and (iii).

Case (i). β > α .

In this case, since x Δ ( t ) > 0 , there exists t 2 t 1 such that x σ ( t ) x ( t ) c > 0 . This implies that x α β ( t ) c 1 , where c 1 = c α β .

Case (ii). β = α .

In this case, we see that x α β ( t ) = 1 .

Case (iii). β < α .

Proceeding as in the proof of Theorem 3.1, there exist a constant b 1 > 0 and t 2 t 1 such that
x ( t ) b 1 η 1 ( t ) for  t t 2
and hence
x α β ( t ) c 2 η β α ( t ) for  t t 2 ,

where c 2 = b 1 α β .

Using these three cases in (3.19) and the definition of δ 2 ( t ) , we get
w Δ ( t ) q ( t ) ξ ( t ) θ β ( t , g ( T ) ) + δ 2 ( t ) ( η ( t ) ) α ξ Δ ( t )
for t t 2 . Integrating the above inequality from t 2 to t, we have
0 < w ( t ) w ( t 2 ) t 2 t [ q ( s ) ξ ( s ) θ β ( s , g ( T ) ) δ 2 ( s ) η α ( s ) ξ Δ ( s ) ] Δ s

which gives a contradiction using (3.14). □

We next state and prove a Philos-type oscillation criterion for (1.1).

Theorem 3.3 Assume that there exist functions H and h such that for each fixed t, H ( t , s ) and h ( t , s ) are rd-continuous with respect to s on D = { ( t , s ) , t s t 0 } such that
H ( t , t ) = 0 , t t 0 , H ( t , s ) > 0 , t > s t 0
(3.20)
and H has a non-positive continuous Δ-partial derivative H Δ s ( t , s ) with respect to the second variable, and that it satisfies
H Δ s ( t , s ) H ( t , s ) ξ Δ ( s ) ξ σ ( s ) = h ( t , s ) ξ σ ( s ) ( H ( t , s ) ) α α + 1
(3.21)
and for all sufficiently large T t 0 , and for t T , τ ( t ) > g ( T ) , we have
lim sup t 1 H ( t , T ) × T t [ q ( s ) ξ ( s ) θ β ( s , g ( T ) ) H ( t , s ) ( α β ) α ( h ( t , s ) ) α + 1 a ( s ) ( α + 1 ) α + 1 ξ α ( s ) ( δ 1 ( s ) ) α ] Δ s = ,
(3.22)

where ξ ( s ) is a positive Δ-differentiable function and h ( t , s ) : = max { 0 , h ( t , s ) } . Then every solution of (1.1) is oscillatory on [ t 0 , ) T .

Proof Let x be a nonoscillatory solution of (1.1) on [ t 0 , ) T . Then, without loss of generality, there is a t 1 [ t 0 , ) T , sufficiently large, such that x ( t ) > 0 and x ( τ ( t ) ) > 0 on [ t 1 , ) . Again we define w ( t ) function as in the proof of Theorem 3.1. Then, proceeding as in the proof of Theorem 3.1, we obtain (3.12). Multiplying both sides of inequality (3.12), with t replaced by s, by H ( t , s ) and integrating with respect to s from t 2 to t, we get
t 2 t H ( t , s ) q ( s ) ξ ( s ) θ β ( s , g ( T ) ) Δ s t 2 t H ( t , s ) w Δ ( s ) Δ s + t 2 t H ( t , s ) ξ Δ ( s ) w σ ( s ) ξ σ ( s ) Δ s β t 2 t H ( t , s ) a 1 α ( s ) ξ ( s ) ( w σ ( s ) ξ σ ( s ) ) α + 1 α δ 1 ( s ) Δ s .
(3.23)
Integrating (3.23) by parts and using (3.20) and (3.21), we obtain
t 2 t H ( t , s ) θ β ( s , g ( T ) ) q ( s ) ξ ( s ) Δ s H ( t , t 2 ) w ( t 2 ) + t 2 t h ( t , s ) ξ σ ( s ) ( H ( t , s ) ) α α + 1 w σ ( s ) Δ s β t 2 t H ( t , s ) a 1 α ( s ) ξ ( s ) ( w σ ( s ) ξ σ ( s ) ) α + 1 α δ 1 ( s ) Δ s .
(3.24)
Again, defining A 0 and B 0 by
A = [ β H ( t , s ) δ 1 ( s ) ξ ( s ) a 1 / α ( s ) ] α α + 1 w σ ( s ) ξ σ ( s ) , B = ( h ( t , s ) ) α a α α + 1 ( s ) ( α + 1 α ) α ( β δ 1 ( s ) ξ ( s ) ) α 2 α + 1
and using Lemma 2.1 where λ = α + 1 α , we get
β H ( t , s ) ξ ( s ) δ 1 ( s ) ( w σ ( s ) ) α + 1 α a 1 / α ( s ) ( ξ σ ( s ) ) α + 1 α h ( t , s ) ξ σ ( s ) ( H ( t , s ) ) α α + 1 w σ ( s ) 1 α ( h ( t , s ) ) α + 1 a ( s ) β α ( α + 1 α ) α + 1 ξ α ( s ) ( δ 1 ( s ) ) α .
(3.25)
From (3.24) and (3.25), we have
1 H ( t , t 2 ) t 2 t [ H ( t , s ) θ β ( s , g ( T ) ) q ( s ) ξ ( s ) ( α β ) α ( h ( t , s ) ) α + 1 a ( s ) ( α + 1 ) α + 1 ξ α ( s ) ( δ 1 ( s ) ) α ] Δ s w ( t 2 ) ,

which contradicts assumption (3.22).

Now we introduce the following notation for T [ t 1 , ) T . For all sufficiently large T such that g ( T ) < τ ( T ) ,
p : = lim inf t t α a ( t ) σ ( t ) θ β ( s , g ( T ) ) q ( s ) Δ s ,
(3.26)
q : = lim inf t 1 t T t s α + 1 a ( s ) θ β ( s , g ( T ) ) q ( s ) Δ s ,
(3.27)
r : = lim inf t t α w σ ( t ) a ( t ) ,
(3.28)
R : = lim sup t t α w σ ( t ) a ( t ) .
(3.29)
Assume that l = lim inf t t σ ( t ) . Note that 0 l 1 . We assume that
T θ β ( s , g ( T ) ) q ( s ) Δ s < , T [ t 1 , ) T .
(3.30)

 □

Theorem 3.4 Let β α and assume a ( t ) is a delta differentiable function such that a Δ ( t ) 0 and (3.30) holds. Furthermore, assume l > 0 and
p > ( α 2 β c 1 ) α ( α + 1 ) α + 1 l α 2
(3.31)
or
p + q > ( α β c 1 ) α 1 l α ( α + 1 )
(3.32)

for all large T. Then every solution of (1.1) is oscillatory on [ t 0 , ) T .

Proof Let x be a nonoscillatory solution of (1.1). Without loss of generality we assume that there is a t 1 [ t 0 , ) T , sufficiently large, such that x ( t ) > 0 and x ( τ ( t ) ) > 0 on [ t 1 , ) . Again we define w ( t ) as in the proof of Theorem 3.1 by putting ξ ( t ) = 1 and δ ( t ) = c 1 . Proceeding as in the proof of Theorem 3.1, we obtain from (3.12)
w Δ ( t ) q ( t ) θ β ( t , g ( T ) ) β c 1 ( w σ ( t ) ) α + 1 α a 1 α ( t ) .
(3.33)
First, we assume that inequality (3.31) holds. It follows from (3.5) and a ( t ) ( x Δ ( t ) ) α being strictly decreasing that
x ( t ) = x ( t 0 ) + t 0 t ( a ( s ) ( x Δ ( s ) ) α ) a 1 α ( s ) 1 α Δ s x ( t 0 ) + t 0 t ( a ( t ) ( x Δ ( t ) ) α ) a 1 α ( s ) 1 α Δ s a 1 α ( t ) x Δ ( t ) t 0 t Δ s a 1 α ( s )
and it follows that
a ( t ) ( x Δ ( t ) x ( t ) ) α ( t 0 t Δ s a 1 α ( s ) ) α
and hence
w ( t ) = a ( t ) ( x Δ ( t ) x ( t ) ) α x α β ( t ) c 1 ( t 0 t Δ s a 1 α ( s ) ) α .
(3.34)
Using (1.2), we have lim t w ( t ) = 0 . Integrating (3.33) from σ ( t ) to ∞ and using lim t w ( t ) = 0 , we have
w σ ( t ) σ ( t ) q ( s ) θ β ( s , g ( T ) ) Δ s + β c 1 σ ( t ) ( w σ ( s ) ) α + 1 α a 1 α ( s ) Δ s .
(3.35)
Multiplying (3.35) by t α a ( t ) we get
t α a ( t ) w σ ( t ) t α a ( t ) σ ( t ) q ( s ) θ β ( s , g ( T ) ) Δ s + β c 1 t α a ( t ) σ ( t ) ( w σ ( s ) ) α + 1 α a 1 α ( s ) Δ s .
(3.36)
Let 0 < ϵ < l . Then by the definition of p and r we can pick t 1 [ T , ) T sufficiently large, so we have
t α w σ ( t ) a ( t ) ( p ϵ ) + β c 1 t α a ( t ) σ ( t ) s ( w σ ( s ) ) 1 α s α w σ ( s ) a ( s ) a 1 α ( s ) a ( s ) s α + 1 Δ s ( p ϵ ) + β c 1 ( r ϵ ) α + 1 α t α a ( t ) σ ( t ) a ( s ) s α + 1 Δ s ( p ϵ ) + β c 1 α ( r ϵ ) α + 1 α t α σ ( t ) α Δ s s α + 1 ;
(3.37)
for σ ( t ) [ t 1 , ) T . Using the Pötzsche chain rule we get
( 1 s α ) Δ = α 0 1 1 [ s + μ h ] α + 1 d h 0 1 ( α s α + 1 ) d h = α s α + 1 .
(3.38)
Then from (3.37) and (3.38), we have t [ t 0 , ) T , and
t α w σ ( t ) a ( t ) ( p ϵ ) + β c 1 α ( r ϵ ) α + 1 α ( t σ ( t ) ) α .
Taking the lim inf of both sides as t we get
r p ϵ + β c 1 α ( r ϵ ) α + 1 α l α .
Since ϵ > 0 is arbitrary, we get
p r β c 1 α ( r ) λ l α ,
(3.39)
where λ = α + 1 α . By using Lemma 2.1, with
A : = ( β c 1 α ) α α + 1 r l α 2 α + 1 and B : = 1 ( α + 1 α ) α ( β c 1 α ) α 2 α + 1 l α 3 α + 1
we get
β c 1 α ( r ) α + 1 α l α r ( α 2 β c 1 ) α ( α + 1 ) α + 1 l α 2 .
(3.40)
It follows from (3.39) and (3.40) that
p ( α 2 β c 1 ) α ( α + 1 ) α + 1 l α 2 ,
which contradicts (3.31). Next, we assume that (3.32) holds. Multiplying both sides of (3.33) by t α + 1 a ( t ) , and integrating from T to t ( t T ) we get
T t s α + 1 w Δ ( s ) a ( s ) Δ s T t s α + 1 q ( s ) θ β ( s , g ( T ) ) a ( s ) Δ s β c 1 T t s α + 1 ( w σ ( s ) ) α + 1 α a α + 1 α ( s ) Δ s .
Using integration by parts, the quotient rule and applying the Pötzsche chain rule, we obtain
t α + 1 w ( t ) a ( t ) T α + 1 w ( T ) a ( T ) + ( α + 1 ) T t ( σ ( s ) ) a ( s ) α w σ ( s ) Δ s T t s α + 1 q ( s ) θ β ( s , g ( T ) ) a ( s ) Δ s β c 1 T t ( s α w σ ( s ) a ( s ) ) α + 1 α Δ s .
Let 0 < ϵ < l be given. Then using the definition of l, we can assume without loss of generality that T is sufficiently large so that
s σ ( s ) l ϵ , s t .
It follows that
σ ( s ) L s , s T , where  L : = 1 l ϵ > 0 .
Therefore
t α + 1 w ( t ) a ( t ) T α + 1 w ( T ) a ( T ) T t s α + 1 q ( s ) θ β ( s , g ( T ) ) a ( s ) Δ s + T t [ ( α + 1 ) L α s α w σ ( s ) a ( s ) β c 1 ( s α w σ ( s ) a ( s ) ) α + 1 α ] Δ s .
(3.41)
It follows that
t α + 1 w ( t ) a ( t ) T α + 1 w ( T ) a ( T ) T t s α + 1 q ( s ) θ β ( s , g ( T ) ) a ( s ) Δ s + T t [ ( α + 1 ) L α u ( s ) β c 1 u λ ( s ) ] Δ s ,
where u ( s ) = s α w σ ( s ) a ( s ) and λ = α + 1 α . Using the inequality of Lemma 2.1 with
A λ = β c 1 u λ ( s ) and B λ 1 = α ( β c 1 ) 1 λ L α
we get
β c 1 u λ ( s ) ( α + 1 ) L α u ( s ) ( λ 1 ) ( α ( β c 1 ) 1 λ L α ) λ λ 1 = 1 α ( α ( β c 1 ) 1 λ L α ) α + 1 = ( α β c 1 ) α L α ( α + 1 ) .
(3.42)
Then we have
t α + 1 w ( t ) a ( t ) T α + 1 w ( T ) a ( T ) T t s α + 1 q ( s ) θ β ( s , g ( T ) ) a ( s ) Δ s + ( α β c 1 ) α L α ( α + 1 ) ( t T ) .
Since w Δ ( t ) 0 and dividing the last inequality by t we have
t α w σ ( t ) a ( t ) T α + 1 w ( T ) t a ( T ) 1 t T t s α + 1 q ( s ) θ β ( s , g ( T ) ) a ( s ) Δ s + ( α β c 1 ) α L α ( α + 1 ) ( 1 T t ) .
Taking the lim sup of both sides as t we obtain
R q + ( α β c 1 ) α L α ( α + 1 ) .
Since 0 < ϵ < l is arbitrary, we get
R q + ( α β c 1 ) α 1 l α ( α + 1 ) .
Using (3.39), we have
p r β c 1 α ( r ) α + 1 α l α r R q + ( α β c 1 ) α 1 l α ( α + 1 ) .
Therefore
p + q ( α β c 1 ) α 1 l α ( α + 1 ) ,

which contradicts (3.32). □

4 Examples

In this section, we give some examples to illustrate our main results. To obtain the conditions for oscillations, we will use the following fact:
t 0 Δ t t p = , 0 p 1 , t [ t 0 , ) T .
Example 4.1 Consider the second-order delay half-linear dynamic equation on times scales
( 1 ( t + σ ( t ) ) 3 ( x Δ ( t ) ) 3 ) Δ + ( t + 1 ) 3 t 2 x 3 ( t ) = 0 , t [ 1 , ) T ,
(4.1)
where a ( t ) = 1 ( t + σ ( t ) ) 3 , q ( t ) = ( t + 1 ) 3 t 2 , τ ( t ) = t and α = β = 3 . We see that
t 0 Δ s a 1 / α ( s ) = t 0 ( t + σ ( t ) ) Δ t = t 0 ( t 2 ) Δ Δ t = , t [ t 0 , ) T .
Taking T t 0 such that g ( T ) = 1 , we get t > t 0 , τ ( t ) > 1 . Then
θ ( t , g ( T ) ) = 1 t 1 / 2 Δ s a 1 / α ( s ) 1 t Δ s a 1 / α ( s ) = t 1 t 2 1 = 1 t + 1 .
Then, taking ξ ( t ) = t by Theorem 3.1, we have
lim sup t T t [ θ α ( s , 1 ) ξ ( s ) q ( s ) a ( s ) ( ξ Δ ( s ) + ) α + 1 ( α + 1 ) α + 1 ξ α ( s ) ] Δ s = lim sup t 1 t [ 1 ( s + 1 ) 3 s ( s + 1 ) 3 s 2 1 4 4 s 3 ( s + σ ( s ) ) 3 ] Δ s lim sup t 1 t [ 1 s 1 4 4 s 3 ( 2 s ) 3 ] Δ s = , t [ 1 , ) T .

Then (4.1) is oscillatory.

Example 4.2 Consider the second-order delay half-linear dynamic equation on times scales
( 1 ( t + σ ( t ) ) 3 ( x Δ ( t ) ) 3 ) Δ + ( T 2 t 2 1 ) 3 t 2 ( T 2 t 1 ) 3 x 3 ( t ) = 0 , t [ 1 , ) T ,
(4.2)

where T is sufficiently large, a ( t ) = 1 ( t + σ ( t ) ) 3 , q ( t ) = ( T 2 t 2 1 ) 3 t 2 ( T 2 t 1 ) 3 , α = β = 3 and τ ( t ) = t 1 / 2 .

If we choose g ( T ) = 1 T , we get
θ ( t , 1 T ) = 1 T t 1 / 2 Δ s a 1 / α ( s ) 1 T t Δ s a 1 / α ( s ) = t 1 T 2 t 2 1 T 2 = T 2 t 1 T 2 t 2 1 .
When ξ ( t ) = t by Theorem 3.1, we have
lim sup t 1 t [ ( T 2 s 1 T 2 s 2 1 ) 3 ( T 2 s 2 1 ) 3 s 2 ( T 2 s 1 ) 3 s 1 4 4 s 3 ( s + σ ( s ) ) 3 ] Δ s lim sup t 1 t [ 1 s 1 4 4 s 3 ( 2 s ) 3 ] Δ s = .

Then (4.2) is oscillatory on [ 1 , ) T .

Declarations

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Sciences, Ankara University, Tandogan, Turkey

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© Güvenilir and Nizigiyimana; 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 cited.

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