Nonlinear oscillation of a class of second-order dynamic equations ...

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In this paper, we discuss the oscillation of certain second-order nonlinear dynamic ... Keywords: Nonlinear oscillation, Dynamic equation, Time scale, Riccati.
Applied Mathematical Sciences, Vol. 6, 2012, no. 60, 2957 - 2962

Nonlinear Oscillation of a Class of Second-Order Dynamic Equations on Time Scales Da-Xue Chen College of Science, Hunan Institute of Engineering Xiangtan 411104, P. R. China [email protected]

Abstract In this paper, we discuss the oscillation of certain second-order nonlinear dynamic equations on time scales. We establish some oscillation criteria for the equations by applying a generalized Riccati transformation technique. Our results improve and extend some known results in the recent literature. Mathematics Subject Classification: 39A10 Keywords: Nonlinear oscillation, Dynamic equation, Time scale, Riccati transformation

1 Introduction The paper is to deal with the oscillation of the following second-order nonlinear dynamic equation

( r (t ) | x

Δ

(t ) |α −1 x Δ (t ) ) + q(t ) | x(t ) |β −1 x(t ) = 0 Δ

(1.1)

on an arbitrary time scale T , where the following conditions are assumed to hold: (H1) α , β > 0 are constants and sup T = ∞ ; (H2) r and q are positive rdcontinuous functions defined on the time scale interval [t0 , ∞) . The theory of time scales, which has recently received a lot of attention, was introduced by Hilger in his Ph.D. Thesis [7] in order to unify continuous and discrete analysis. A time scale T is an arbitrary nonempty closed subset of the real numbers R (see [5]). Not only can the theory of dynamic equations on time scales unify the theories of differential equations and difference equations, but it

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Da-Xue Chen

is also able to extend these classical cases to cases ‘‘in between,” e.g., to the socalled q -difference equations. Dynamic equations on time scales have a lot of applications in population dynamics, quantum mechanics, electrical engineering, neural networks, heat transfer, combinatorics and so on. The book on the subject of time scales by Bohner and Peterson [5] summarize and organize much of time scale calculus and some applications. By a solution of (1.1), we mean a nontrivial real function x such that x ∈ Crd1 [t x , ∞) and r (t ) | x Δ (t ) |α −1 x Δ (t ) ∈ Crd1 [t x , ∞) for a certain t x ≥ t0 and satisfying (1.1) for t ≥ t x . Our attention is restricted to those solutions of (1.1) which exist on the half-line [t x , ∞) and satisfy sup{| x(t ) |: t > t*} > 0 for any t* ≥ t x . A solution x of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative; otherwise it is nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory. In the last decade, there has been increasing interest in obtaining sufficient conditions for the oscillation and nonoscillation of solutions of different classes of dynamic equations on time scales, and we refer the reader to the papers [1-4, 6, 810] and the references cited therein. Recently, Saker [8] established some oscillation criteria for the second-order half-linear dynamic equation

( r (t )( x

Δ

(t ))α ) + q(t ) xα (t ) = 0 Δ

(1.2)

on time scales, where α > 1 is an odd positive integer, and r and q satisfy (H2). Afterward, Hassan [10] also considered (1.2), where α is a quotient of odd positive integers, and obtained some sufficient conditions for the oscillation of the equation. Hassan [10] improved and extended the results of Saker [8]. Very recently, Grace et al. [9] studied the oscillation of the second-order nonlinear dynamic equation

( r (t )( x

Δ

(t ))α ) + q(t ) x β (t ) = 0 Δ

(1.3)

on time scales, where α , β are quotients of odd positive integers, and r and q satisfy (H2). Grace et al. [9] obtained some new oscillation results for (1.3) when α < β , α = β and α > β , respectively. It is clear that (1.2) and (1.3) are special cases of (1.1), and all the results of Saker [8], Hassan [10] and Grace et al. [9] can not be applied to (1.1) when α , β are not equal to quotients of odd positive integers. Therefore, it is of great interest to study the oscillation of (1.1) when α , β > 0 are constants. In this paper, we will establish some new oscillation criteria for (1.1) when α , β > 0 are constants. Our results extend and improve the results of Saker [8], Hassan [10] and Grace et al. [9]. We will need the following lemma to prove our main results.

Nonlinear oscillation

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Lemma 1.1. (Bohner and Peterson [5], p. 32, Theorem 1.87) Let f : R → R be continuously differentiable and suppose g : T → R is delta differentiable. Then f o g : T → R is delta differentiable and satisfies

( f o g ) Δ (t ) =

{∫

1 0

}

f ' ( g (t ) + hμ (t ) g Δ (t ) ) dh g Δ (t ) ,

where μ (t ) := σ (t ) − t is the graininess function σ (t ) := inf{s ∈ Tt: s > t} is the forward jump operator on T .

on

T

,

here

2 Main Results Theorem 2.1. Suppose that (H1), (H2) and the following condition hold:



∞ t0

r −1/ α (t )Δt = ∞ .

(2.1)

Furthermore, assume that there exists a positive nondecreasing delta differentiable function ϕ such that for all T > t1 ≥ t0 , t

where u (t ) :=

(∫

lim sup ∫ ⎡⎣ q ( s )ϕ ( s ) − ϕ Δ ( s )uα ( s )v( s ) ⎤⎦ Δs = ∞ , T t →∞ t t1

r −1/ α ( s )Δs

)

(2.2)

−1

and

c1 is any positive constant , if α < β , ⎧c1 , ⎪ v(t ) := ⎨1, if α = β , ⎪c u β −α (t ), c is any positive constant, if α > β . 2 ⎩ 2

Then (1.1) is oscillatory. Proof. Suppose that x is a nonoscillatory solution of (1.1). Without loss of generality, we may assume that x is an eventually positive solution of (1.1). Then there exists t1 ≥ t0 such that x(t ) > 0 for t ∈ [t1 , ∞) . Therefore, from (1.1) we have (r (t ) | x Δ (t ) |α −1 x Δ (t )) Δ = − q (t ) x β (t ) < 0 for t ∈ [t1 , ∞) . It is easy to see that r (t ) | x Δ (t ) |α −1 x Δ (t ) is strictly decreasing on [t1 , ∞) and is eventually of one sign. We claim x Δ (t ) > 0 for t ∈ [t1 , ∞) . Assume on the contrary, then there exists t2 ≥ t1

such Δ

α −1

r (t ) | x (t ) |

that

x Δ (t2 ) ≤ 0

Δ

Δ

. α −1

x (t ) ≤ r (t3 ) | x (t3 ) | Δ

t3 > t 2

Take Δ

,

then

Δ

α −1

x (t3 ) := M < r (t2 ) | x (t2 ) | 1/ α

we

obtain

Δ

x (t2 ) ≤ 0 for

−1/ α

t ∈ [t3 , ∞) . Hence, we get x (t ) ≤ −(− M ) r (t ) for t ∈ [t3 , ∞) . Integrating both sides of the last inequality from t3 to t , we have t

x(t ) − x(t3 ) ≤ −(− M )1/ α ∫ r −1/ α ( s )Δs for t ∈ [t3 , ∞) . Letting t → ∞ and using (2.1), t3

we conclude lim x(t ) = −∞ . This contradicts the fact that x(t ) > 0 for t ∈ [t1 , ∞) . t →∞

Thus, we have x Δ (t ) > 0 for t ∈ [t1 , ∞) . Therefore, by Lemma 1.1 we obtain

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Da-Xue Chen

1

( x β (t )) Δ = β x Δ (t ) ∫ [ x(t ) + hμ (t ) x Δ (t )]β −1 dh > 0

w(t ) := r (t )( x Δ (t ))α ϕ (t ) / x β (t ) ( FG ) Δ = F Δ G + F σ G Δ

and

t ∈ [t1 , ∞) .

for

(F / G)

Δ

t ∈ [t1 , ∞)

for

0

Then

by

= F Δ / Gσ − FG Δ /(GGσ )

.

the

Let

formulas

for the delta

derivatives of the product FG and the quotient F / G of differentiable functions F and G , where σ is the forward jump operator on T , F σ := F o σ and

Gσ := G o σ , we get

wΔ = ( r ( x Δ )α ) ϕ / x β + ( r ( x Δ )α ) (ϕ / x β ) σ

Δ

Δ

= ( r ( x Δ )α ) ϕ / x β + ( r ( x Δ )α ) ⎡⎣ϕ Δ /( x β )σ − ϕ ( x β ) Δ / x β ( x β )σ ⎤⎦ . For t ≥ t1 , since ( x β (t )) Δ > 0 , t ≤ σ (t ) , x Δ (t ) > 0 and (r (t )( x Δ (t ))α ) Δ σ

Δ

= − q(t ) x β (t ) < 0 , we have wΔ < ( r ( x Δ )α ) ϕ / x β + ( r ( x Δ )α ) ϕ Δ /( x β )σ = −qϕ + ( r ( x Δ )α ) ϕ Δ /( x β )σ σ

Δ

σ

≤ − qϕ + ϕ Δ r ( x Δ )α / x β . Since

x(t ) = x(t1 ) + ∫ x Δ ( s )Δs = x(t1 ) + ∫ r −1/ α ( s ) ( r ( s )( x Δ ( s ))α ) t

t

t1

t1

1/ α

(2.3)

Δs

t

≥ r1/ α (t ) x Δ (t ) ∫ r −1/ α ( s )Δs for t ≥ t1 ,

(x

we find

Δ

(t ) / x(t ) ) ≤ r −1 (t ) α

(∫ r t

t1

−1/ α

( s ) Δs

)

t1

−α

:= uα (t ) / r (t ) for t > t1 . Thus,

from (2.3) we obtain wΔ < −qϕ + ϕ Δ uα xα − β on (t1 , ∞) . Next, we consider the following three cases: Case (i). Let α < β . For t ∈ [t1 , ∞) , since x(t ) ≥ x(t1 ) > 0 , we have

(2.4)

xα − β (t ) ≤ ( x(t1 ))α − β := c1 . Case (ii). Let α = β . Then, for t ∈ [t1 , ∞) we get

(2.5)

(2.6) xα − β (t ) = 1 . Case (iii). Let α > β . Since r (t )( x Δ (t ))α ≤ r (t1 )( x Δ (t1 ))α := b for t ∈ [t1 , ∞) , we obtain x Δ (t ) ≤ b1/ α r −1/ α (t ) for t ∈ [t1 , ∞) . Integrating both sides of the last t

inequality from t1 to t , we have x(t ) ≤ x(t1 ) + b1/ α ∫ r −1/ α ( s )Δs for t ∈ [t1 , ∞) . t1

Therefore,

there

exist

a

constant

b1 > 0

and

t4 > t1

such

that

t

x(t ) ≤ b1 ∫ r −1/ α ( s )Δs := b1u −1 (t ) for t ∈ [t4 , ∞) . Hence, for t ∈ [t4 , ∞) we get t1

α −β

xα − β (t ) ≤ b1 u β −α (t ) := c2u β −α (t ) .

(2.7)

Nonlinear oscillation

Thus,

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t ≥ t4

for

,

from

(2.4)-(2.7)

it

follows

that

α

w (t ) < −q (t )ϕ (t ) + ϕ (t )u (t )v(t ) . Integrating both sides of the last inequality Δ

Δ

from t4 to t , we obtain



t t4

⎡⎣ q( s )ϕ ( s ) − ϕ Δ ( s )uα ( s )v( s ) ⎤⎦ Δs ≤ w(t4 ) − w(t ) < w(t4 ) t

for t ≥ t4 . Hence, we get lim sup ∫ ⎡⎣ q ( s)ϕ ( s ) − ϕ Δ ( s)u α ( s)v( s ) ⎤⎦ Δs ≤ w(t4 ) < ∞ , t4 t →∞ which contradicts (2.2). The proof is complete. Theorem 2.2. Suppose that (H1), (H2) and



∞ t0

r −1/ α (t )Δt < ∞ hold.

Furthermore, assume that there exists a positive nondecreasing delta differentiable function ϕ such that for all T > t1 ≥ t0 , (2.2) and the following condition hold:



∞ T

(

z

r −1 ( z ) ∫ ψ β ( s )q ( s )Δs T

)

1/ α

Δz = ∞ ,

(2.8)



where ψ (t ) := ∫ r −1/ α ( s )Δs . Then (1.1) is oscillatory. t

Proof. Assume that x is a nonoscillatory solution of (1.1). Without loss of generality, we may assume that x is an eventually positive solution of (1.1). Then there exists t1 ≥ t0 such that x(t ) > 0 for t ∈ [t1 , ∞) . Therefore, there are two cases

for the sign of x Δ (t ) . The proof when x Δ (t ) is eventually positive is similar to that of Theorem 2.1 and hence is omitted. Next, assume that x Δ (t ) is eventually negative. Then there exists t2 ≥ t1 such x Δ (t ) < 0

that

( r (t )(− x

Δ

for

t ∈ [t2 , ∞)

.

Thus,

from

(1.1)

we

have

(t ))α ) = q(t ) x β (t ) > 0 for t ∈ [t2 , ∞) , which implies that r (t )(− x Δ (t ))α Δ

is strictly increasing on [t2 , ∞) . Hence, we have r ( s)( − x Δ ( s))α ≥ r (t )(− x Δ (t ))α for s ≥ t ≥ t2 . Then for s ≥ t ≥ t2 we conclude − x Δ ( s ) ≥ r −1/ α ( s )r1/ α (t )(− x Δ (t )) . Integrating both sides of the last inequality from t ≥ t2 to z ≥ t and letting z → ∞ , for t ∈ [t2 , ∞) we get x(t ) ≥

(∫

∞ t

)

r −1/ α ( s)Δs r1/ α (t )(− x Δ (t ))

:= ψ (t )r1/ α (t )(− x Δ (t )) ≥ ψ (t )r1/ α (t2 )(− x Δ (t2 )) := cψ (t ),

c := − r1/ α (t2 ) x Δ (t2 ) > 0

where

( r (t )(− x

Δ

.

Thus,

from

(1.1)

we

obtain

(t ))α ) = q(t ) x β (t ) ≥ c βψ β (t )q (t ) for t ∈ [t2 , ∞) . Integrating both sides Δ

of the last inequality from t2 to t , for t ∈ [t2 , ∞) we have t

t

t2

t2

r (t )(− x Δ (t ))α ≥ r (t2 )(− x Δ (t2 ))α + c β ∫ ψ β ( s )q( s )Δs > c β ∫ ψ β ( s )q ( s )Δs .

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Da-Xue Chen

(

Δ

−1

Hence, we obtain − x (t ) > r (t )c both

sides

of

x(t ) ≤ x(t2 ) − c β / α ∫

the

t t2

(r

−1

last z

β



t t2

ψ ( s ) q ( s ) Δs β

inequality

( z ) ∫ ψ β ( s ) q ( s ) Δs t2

)

1/ α

from

)

1/ α

t2

for t ∈ [t2 , ∞) . Integrating to

t

,

we

find

Δz for t ∈ [t2 , ∞) . Letting t → ∞

and using (2.8), we see lim x(t ) = −∞ . This contradicts the fact that x(t ) > 0 for t →∞

t ∈ [t1 , ∞) . The proof is complete. Acknowledgements

This work was supported by the Natural Science Foundation of Hunan Province of P. R. China (Grant No. 11JJ3010).

References [1] [2]

[3]

[4]

[5] [6]

[7] [8] [9]

[10]

A. K. Tripathy, Some ocillation results for second order nonlinear dynamic equations of neutral type, Nonlinear Anal., 71 (2009), 1727-1735. D.-X. Chen, Oscillation of second-order Emden–Fowler neutral delay dynamic equations on time scales, Math. Comput. Modelling, 51 (2010), 1221-1229. D.-X. Chen, Oscillation and asymptotic behavior for nth-order nonlinear neutral delay dynamic equations on time scales, Acta Appl. Math., 109 (2010), 703-719. L. Erbe, J. Baoguo and A. Peterson, Oscillation of nth order superlinear dynamic equations on time scales, Rocky Mountain J. Math., 41 (2011), 471491. M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Application, Birkhäuser, Boston, 2001. Q. Yang and Z. Xu, Oscillation criteria for second order quasilinear neutral delay differential equations on time scales, Comput. Math. Appl., 62 (2011), 3682-3691. S. Hilger, Analysis on measure chains–a unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18-56. S.H. Saker, Oscillation criteria of second-order half-linear dynamic equations on time scales, J. Comput. Appl. Math., 177 (2005), 375-387. S.R. Grace, R.P. Agarwal, B. Kaymakçalan and W. Sae-Jie, On the oscillation of certain second order nonlinear dynamic equations, Math. Comput. Modelling, 50 (2009), 273-286. T.S. Hassan, Oscillation criteria for half-linear dynamic equations on time scales, J. Math. Anal. Appl., 345 (2008), 176-185.

Received: December, 2011