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

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:TT, τ(t)t, tT, and lim t τ(t)=. We establish some new sufficient conditions for this equation.

MSC:34K11, 39A10, 39A99.

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:TT, τ(t)t, tT 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 supT=, 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)Δt= a b f(t)dt 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 ) ds>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 ds>1,
(1.6)

then every solution of (1.3) oscillates.

If T=Z, then σ(t)=t+1, μ(t)=1, f Δ (t)=Δf(t), a b f(t)Δt= 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)Δt=.
(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<α1. 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)0and t 0 τ α (t)q(t)Δt=
(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<α1 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=β1 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:TR, the (delta) derivative f Δ (t) at tT 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 sU. If the (delta) derivative f Δ (t) exists for all tT, 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 fog for a continuously differentiable function f:RR and a (delta) differentiable function g:TR

( f o g ) Δ = [ 0 1 f ( g + h μ g Δ ) d h ] g Δ .
(2.4)

For b,cT and a differentiable function f, the Cauchy integral of f Δ is defined by

b c f Δ (t)Δt=f(c)f(b)
(2.5)

and the improper integral is defined as

b f(t)Δt= lim c b c f(t)Δt.
(2.6)

Lemma 2.1 If A and B are nonnegative real numbers and λ>1, then

A λ λA B λ 1 +(λ1) 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 ] Δs=,
(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<β1, 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)Δs.

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 A0 and B0 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 ) ] Δs=,
(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 ) ] Δs

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),ts 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 tT, 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 A0 and B0 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 ] Δsw( 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)Δs,
(3.26)
q := lim inf t 1 t T t s α + 1 a ( s ) θ β ( s , g ( T ) ) q(s)Δs,
(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 0l1. We assume that

T θ β ( s , g ( T ) ) q(s)Δs<,T [ t 1 , ) T .
(3.30)

 □

Theorem 3.4 Let βα 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 ) ) Δs+β c 1 σ ( t ) ( w σ ( s ) ) α + 1 α a 1 α ( s ) Δs.
(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 ) ) Δs+β c 1 t α a ( t ) σ ( t ) ( w σ ( s ) ) α + 1 α a 1 α ( s ) Δs.
(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 andB:= 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 (tT) we get

T t s α + 1 w Δ ( s ) a ( s ) Δs T t s α + 1 q ( s ) θ β ( s , g ( T ) ) a ( s ) Δsβ c 1 T t s α + 1 ( w σ ( s ) ) α + 1 α a α + 1 α ( s ) Δs.

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 ) lϵ,st.

It follows that

σ(s)Ls,sT,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 ) Δs+ ( α β c 1 ) α L α ( α + 1 ) (tT).

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 ) Δs+ ( 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 =,0p1,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 ) ) Δt= t 0 ( t 2 ) Δ Δt=,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 .

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Correspondence to A. F. Güvenilir.

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AFG conceived of the study, and participated in its design and coordination. FN carried out the mathematical studies and participated in the sequence alignment and drafted the manuscript. All authors read and approved the final manuscript.

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Güvenilir, A.F., Nizigiyimana, F. Oscillation criteria for second-order quasi-linear delay dynamic equations on time scales. Adv Differ Equ 2014, 45 (2014). https://doi.org/10.1186/1687-1847-2014-45

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Keywords

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