Cmjv05i02p0191 by Journal of Computer and Mathematical Sciences

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ISSN 0976-5727 (Print) ISSN 2319-8133 (Online) Abbr:J.Comp.&Math.Sci. 2014, Vol.5(2): Pg.191-198

Qualitative Behavior Solution of the Dynamic Equation P. Mohankumar1, A.K. Bhuvaneswari2 and A. Ramesh3 1

Professor of Mathematics, Aaarupadaiveedu Institute of Techonology, Vinayaka Missions University, Paiyanoor, Chengalpattu, Kancheepuram District, Tamilnadu, INDIA. 2 Assistant Professor of Mathematics, Aaarupadaiveedu Institute of Techonology, Vinayaka Missions University, Paiyanoor, Chengalpattu, Kancheepuram District, Tamilnadu, INDIA. 3 Sr. Lecturer in Mathematics, District Institute of Education and Training, Uthamacholapuram, Salem INDIA. (Received on: March 27, 2014) ABSTRACT In this paper we will establish the qualitative behavior solution of the dynamic equation of the form ∆ ∆

( r (t ) ( y(t ) + p(t )y (t − τ ) )

+ q(t )y β (t − δ ) = 0, t ∈ T

on the time Scale T. The method Riccati Transformation Keller’s chain rule Example is inserted to illustrate the result. 2010MSC: 74G55, 34N05. Keywords: Dynamic equation, Time Scale, Qualitative, Second order and Delay.

1. INTRODUCTION The theory of time scales, which provides new tools for exploring connections between the traditionally separated fields, has been developing rapidly and has received much attention. Dynamic equations can not only unify the theories of differential equation and difference

equations, but also extend these classical cases to cases “in between” and can be applied to other difference types of time scales. The theory of dynamic equations on time scales is an adequate mathematical apparatus for the stimulation of processes and phenomena observed to biotechnology, chemical technology economic, neural networks, physics, social science etc.1-3.

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Motivated by this observation, in this paper we are concerned with second order nonlinear dynamic equation on time scales.

( r (t )( y(t ) + p(t )y(t − τ ) ) ∆

∆

+ q(t )y β (t − δ ) = 0, t ∈ T

(1.1) Where T is a time scale. Throughout this paper we assume the following conditions without further mention: (H1): β > 0 is a quotient of odd positive integers; (H2): τ , δ are fixed nonnegative constants such that the delay functions τ (t ) = t − τ < t and δ (t ) = t − δ < t satisfy τ (t ) : T → T and δ (t ) : T → T for all t ∈ T ; (H3): q (t) and r (t ) are real valued rdcontinuous positive functions defined on T ; (H4): p(t ) is a positive and rd-continuous function on T such that 0 ≤ p(t ) < 1 . By a solution of equation (1.1), we mean a nontrivial real- valued function which has the properties ( y (t ) + p(t )y (t − τ ) ∈ Crd′ t y , ∞ and

)

∆

r (t ) ( y (t ) − p(t )y (t − τ ) ∈ Crd′ t y , ∞ ) , t y ≥ t0 and satisfying equation (1.1) for all t ≥ t y . A solution y (t ) of equation (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative, that is if for every b > a there exists t > b such that y (t ) = 0 or y (t )y (σ (t )) < 0 ; otherwise it is called nonoscillatory. Since we are interested in qualitative behavior of solutions, we will suppose that the time scale T under considerations is not bounded

above and therefore the time scale is T assumed in the form [t0 , ∞ )T = [t0 , ∞ ) ∩T. In this paper we obtain the oscillation criteria for equation (1.1) subject to the following two conditions: ∞

∫ r (s )∆s = ∞

(1.2)

and ∞

∫ r (s )∆s < ∞

(1.3)

When p(t ) ≡ 0 , equation (1.1) reduces to the following equations

(

r (t ) ( y ( t ) )

∆

)

∆

+ q(t )y β (t − δ ) = 0, t ∈ T

(1.4) In the super linear case, when β ≥ 1 , the oscillation of the solutions of equation (1.1) was discussed by Saker in8 under the condition (1.2). We note that if ܶ = ℝ we have σ (t ) = t , µ (t ) = 0, f ∆ (t ) = f '(t ). then equation (1.1) becomes ′ β  ′  r (t ) ( y (t ) + p(t )y (t − τ )  + q(t )y (t − δ ) = 0, t ∈ ℝ  

If T = ℕ we have σ (n ) = n + 1, µ(n ) = 1, y ∆ (n ) = ∆y (n ). = y(n+ 1) − y(n)

then equation (1.1) becomes ∆ ( r (t )∆ ( y (t ) + p(t )y(t − τ ) ) + q(t )y β (t − δ ) = 0, n ∈ ℕ

If T = h ℕ,h >0 we have σ (t ) = t + h, µ (t ) = h, y ∆ (t) = ∆ h (t ). =

y(t + h) − y(t ) h

then equation (1.1) becomes

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p(t )y (t − τ

P. Mohankumar, et al., J. Comp. & Math. Sci. Vol.5 (2), 191-198 (2014)

193

oscillatory zero. Our ∆ h ( r (t )∆ h ( y (t ) + p(t )y (t − τ ) ) + q (t )y β (t −to δ )be = 0, t ∈ h or converge to 13 results include those of Jinfa and I. 7 + q(t )y β (t − δ ) = 0, t ∈ hh ℕ Kubiaczyk et al. when T= ℕ and p(t ) ℕ ௡ If T = ‫ = ݍ‬ሼ‫ݐ‬: ‫ ݍ = ݐ‬, ݊ ∈ ℕሽ, ‫ > ݍ‬1 is identically zero. we have σ (t ) = q(t ), µ (t ) = (q − 1), y ∆ (t) = ∆ q (t ). =

y(q t ) − y(t ) h

2. MAIN RESULT

then equation (1.1) becomes the second order q-neutral difference equations. ∆ q ( r (t )∆q ( y (t ) + p(t )y (t − τ ) )

First we consider the case, when condition (1.2) holds β ≥ 1 .To prove our main results, we will use the following + q (t )y β (t −lemma δ ) = 0,which t ∈ qcalled Keller’s chain rule.

ሺ‫ ݐ‬− ߜሻ = 0, ‫ ݍ ∈ ݐ‬ℕ ଶ

If T=ℕ = σ (t ) =

(

ሼ‫ ݐ‬ଶ

: ‫ ∈ ݐ‬ℕሽ, we have

y  2 t + 1 , µ (t ) = 1 + 2 t , y ∆ (t) = ∆ N (t ). = 

)

(

2 t + 1  − y(t )  1+ 2 t

)

then equation (1.1) becomes

Lemma 2.13. Let݂ ∶ ℝ → ℝ be continuously differentiable and suppose ݃ ∶ ܶ → ℝ is delta differentiable. Then ݂ 0 ∶ ܶ → ℝ is delta differentiable and the formula

(f o g )

∆

(t ) = g (t )∫ f ' g (t ) + hµ (t )g ∆ (t ) dh

∆ N ( r (t )∆ N ( y (t ) + p(t )y (t − τ ) ) + q (t )y β (t −holds. δ ) = 0, t ∈ ሺ‫ ݐ‬− ߜሻ = 0, ‫ ∈ ݐ‬ℕଶ If ܶ = ሼ‫ݐ‬௡ : ݊ ∈ ℕሽ, where {t n } is the set if harmonic numbers defined by n 1 t0 = 0, tn = ∑ , n ∈ ℕ 0଴,, k =1 k

we have σ (t n ) = t n +1, µ (tn ) =

1 , y ∆ (t) = ∆tn y(t n ) = ( n + 1) ∆ y(t n ) n +1

then equation (1.1) becomes

(

∆

)

Now we state and prove our main results: Theorem 2.2. Assume that condition (1.2) holds. Furthermore assume that there exist positive rd-continuous delta differentiable functions α (t ) and φ (t ) such that for every b ≥ 1 and a positive number M t  K (s )C 2 (s )  lim sup ∫ α (s )φ (s )Q(s ) −  ∆s = ∞ t →∞ 4φ (s )β Mα (t )  t0 

(2.1) ∆tn ( r (tn )∆tn ( y(tn ) + p(tn )y(tn −τ ) ) + q(tn )y (tn −δ ) = 0, t ∈T β

This paper is organized as follows: we present some oscillation solutions of equations (1.1) when (1.2) holds. When (1.3) holds, we also establish some conditions which are sufficient for solutions of equations (1.1)

Where Q(s ) = q(s )(1 − p(s − δ ))β , C(s ) = 1− β

K (t ) = ( b.(t − δ ) )

(α (t )) (φ (t ) ) ∆

∆

{ = max {φ

(α

φ (s )(α ∆ (s ))+ + φ ∆ (s ) ασ

(

(t ) r (t − δ ) ,

} (t ),0}

= max α ∆ (t ),0 ∆

)

and

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)

194

P. Mohankumar, et al., J. Comp. & Math. Sci. Vol.5 (2), 191-198 (2014)

holds. Then every solutions of equation (1.1) oscillates on [t 0 , ∞ )T .

∆

Hence (x (t ) ≤

c . r (t )

Now integrating from t 2 to t we obtain Proof

Suppose to the contrary that y (t ) is a nonoscillatory solution of equation (1.1). Let t1 ≤ t0 be such that y (t ) ≠ 0 for all

t ≥ t1 . Without loss of generality, we may assume that y (t ) is an eventually positive solution

equation (1.1) with y (t − θ ) > 0, where θ = max {τ ,δ } for

all t ≥ t1 sufficiently large. Set

∆

(t ))

)

(2.2)

for all t ∈ [ t1, ∞ )T

( r (t )(x

∆

)

(t )) is an

eventually decreasing function. We first show that r (t )(x ∆ (t )) is eventually

)

(

∆

)

(

w (t ) = r (t )( x (t))

)

eventually

nonnegative. Thus we see that there is some t1 such that

)

∆

≥0

for t ≥ t1 (2.4 A) From (2.2), (2.4) and (2.4 A) we have

y ( t − δ ) ≥ (1 − p(t − δ ) ) x ( t − δ ) . Using the last inequality in (2.3), we have

( r (t )(x

∆

(t ))

)

∆

+ q (t ) (1 − p(t − δ )) x β (t − δ ) ≤ 0

for all t ∈ [ t1, ∞ )T

w (t ) = α (t )

r (t )(x ∆ (t ) ≤ r (t 2 )(x ∆ (t 2 ) = c < 0 for some constant c .

)

(2.5)

r (t )( x ∆ (t )) for t ∈ [t 2 , ∞ )T x β (t − δ ) Then w (t ) > 0 , and using Lemma (2.1)

for t ≥ t 2 ≥ t1 , we have

∆

(t ))

Define

nonnegative. If r (t )(x (t )) < 0

∆

then , for t ≥ t 2 = t1 + δ we have (2.3)

(

( r (t )(x

Hence

y (t ) = x (t ) − p(t )y (t − τ ) ≥ (1 − p(t ) ) x (t )

= −q(t )y β (t − δ ) < 0

and this implies that

t ≥ t1

(

From equation (1.1) and (H2) we have

( r (t )(x

Which implied by (H1) that x (t ) → −∞. this contradicts that fact the x (t ) > 0 for all

x (t ) > 0, x ∆ (t ) ≥ 0, r (t )( x ∆ (t )) > 0 and r (t )( x ∆ (t ))

x (t ) = y (t ) + p(t )y (t − τ )

∆

1 ∆s, r (s ) t2

x ( t ) ≤ x (t 2 ) + c ∫

we have

∆

 α (t )  α (t ) ∆  x β (t − δ )  + x β (t − δ ) r (t )( x (t))  

(

)

∆

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P. Mohankumar, et al., J. Comp. & Math. Sci. Vol.5 (2), 191-198 (2014)

195

 x β (t − δ )α ∆ (t ) − α (t ) x β (t − δ ) ∆   + r (t )( x ∆ (t))    x β (t − δ )x β (σ (t ) − δ )   σ ∆  α (t ) r (t )( x ∆ (t)) x β (t − δ )  α ∆ (t ) ∆ σ +  w (t ) ≤ −α (t )Q(t) + w (t ) −    α σ (t ) x β (t − δ )x β (σ (t ) − δ ) (2.6)   α (t ) w ∆ (t ) = β r (t )( x ∆ (t)) x (t − δ )

(

∆

(

) (

)

(

)(

)

The Keller’s chain rule yields that

(t )

)

∆

β −1

= β ∫  hx σ + (1 − h )x 

dhx ∆ (t )

≥ β ( x(t ) )

β −1

x ∆ (t )

Then , for t ∈ [t 2 , ∞ )T sufficiently large, we have

(t − δ )

)

∆

(

)(

= β x β −1(t − δ ) x ∆ (t − δ )

)

Also from (2.4) we have for t ≥ t 2 , ∆

x∆

( r (t )x (t )) (t − δ ) ≥ r (t − δ )

Substitute the last inequality in (2.6), we obtain σ ∆   βα ( t ) r ( t )( x (t)) +   x β −1(t − δ ) w w (t ) − σ β β  r (t − δ ) x (t − δ ) x (σ (t ) − δ )  α (t )   ∆ Since x (t ) ≥ 0 we have x (σ (t ) − δ ) ≥ x (t − δ ) and this implies that ∆

w∆

(α (t ) ≤ −α (t )Q(t) +

∆

(t )

)

(

)

 βα (t ) r (t )( x ∆ (t)) σ  +  x β −1(t − δ ) w σ (t ) −  σ 2β  α (t ) r (t − δ ) x (σ (t ) − δ )   

(α (t ) ≤ −α (t )Q(t) +

∆

(t )

)

(

)

(2.7)

From (2.5) and (2.7), we obtain

∆

2 σ  βα ( t ) r ( σ ( t ) w ( t ) ( ) + w (t ) −   r (t − δ ) α σ (t ) 2 x 1− β (t − δ ) x ∆ (σ (t) α σ (t ) 

(α (t ) ≤ −α (t )Q(t) +

∆

(t )

)

(

)

(

Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)

)

   

(2.8)

196

P. Mohankumar, et al., J. Comp. & Math. Sci. Vol.5 (2), 191-198 (2014)

Now from equation (2.4), we see that r (t ) x ∆ (t ) is positive and non increasing

(

)

function on T , and therefore there exists a

t2 ∈ [t1, ∞ )T for

some

1 such that r (t ) x (t ) ≤ M positive constant M and

(

)

∆

t2 ∈ [t1, ∞ )T . Hence (2.9)

Further from (2.4) we have

(2.11)

and thus there exists a T ∈ [t0 , ∞ )T

suitable constant b ≥ 1 such that x (t ) ≤ bt for t ∈ [T , ∞ )T .

for t ∈ [T1, ∞ )T .

(2.10)

Where T1 = T + δ . Now substitute (2.9) and (2.10) in (2.8), we have

  β Mα (t ) σ   w σ (t ) − w ( t ) 2 σ 1 β − σ  ( b.(t − δ )) α (t ) α (t ) r (t − δ )  

(α

∆

(t )

)

(

φ (s )

and

integrating

∫ φ (s )α (s )Q(s )∆s ≤ − ∫ φ (s )w (s )∆s + ∫ φ (s ) ∆

x1− β (t − δ ) ≤ b1− β (t − δ )1−β

1 ≥ M r (σ (t ) ) x (σ (t ) )

Multiplying

x (t ) − x (t 0 ) = ∫ x ∆ (s )∆s ≤ x ∆ (t 0 )(t − t0 ),

Hence

∆

w ∆ (t ) ≤ −α (t )Q(t) +

(α

β Mα (s )

t2 to t, (t ≥ t 2 )

from

(s )

α (s )

− ∫ φ (s )

∆

)

(2.11) we

have

(s )∆s (2.12)

1− β

( b.(s − δ )) (α σ (s ))

r (s − δ )

)

( s ) ∆s

Using integration by parts, we obtain t

∫ φ (s )w t2

t ∆

∆

(s )∆s ≤ [φ (t )w (t )]t − ∫ (φ (s )) w σ (s )∆s 2

(2.13)

≤ −φ (t 2 )w (t 2 ) − ∫ φ (s ) +w (s )∆s

(

∆

)

From (2.12) and (2.13), we have Journal of Computer and Mathematical Sciences Vol. 5, Issue 2, 30 April, 2014 Pages (123 - 257)

P. Mohankumar, et al., J. Comp. & Math. Sci. Vol.5 (2), 191-198 (2014) t

∫ φ (s )α (s )Q(s )∆s ≤ w (t2 )φ (t2 ) + ∫ t2

(α φ (s )

(s )

)w +

t σ

α σ (s )

− ∫ φ (s )

∆

(s )∆s + ∫ φ ∆ (s ) w σ (s )∆s

( b.(s − δ ))

(

)

β Mα (s ) 1− β

)

197

α ( s ) r (s − δ )

(2.14) σ

)

(σ (s )) ∆s

From (2.14) becomes

 K (s ) C (s )  β Mα (s ) σ ∫t φ (s )α (s )Q(s )∆s ≤ w (t2 )φ (t2 ) +t∫  φ (s ) K (s ) w (s ) − 2 φ (s )β Mα (s )   2 2  t

+∫ t2

K (s )C 2 (s ) ∆s 4φ (s )β Mα (s )

Hence

 K (s )C 2 (s )  lim ∫ φ (s )α (s )Q(s ) −  ∆s ≤ w (t 2 )φ (t 2 ) < ∞ t →∞ 4φ (s )β Mα (s )  t2  t

This is contradiction to the condition (2.1). This Complete the proof. Corollary 2.3 Assume condition (1.2) holds. Let α (t ) = 1 Now Theorem 2.2 yields the following well known result LeightonWintner Theorem such that

lim sup ∫ q(s ) (1 − p(s − δ ) ) t →∞

τ and δ are non-negative constants such that t − τ and t − δ ∈ T . In the example β = 5 / 3, r (t ) = 1/ (t + 1), p(t ) = 1/ 2. It is easy to see that assumptions (H1) –(H4) hold. Choose φ (s ) = 1 and α (s ) = 1, then from corollary 2.3 we have

∆s = ∞ (2.18)

Then every solution of equation (1.1) oscillates on [t0 , ∞ )T . 3. EXAMPLE Consider the dynamic equation of the form ∆

∆ 5  1 1   y 3 (t − δ ) = 0, t ∈ [1, ∞ )T   y (t ) + y (t − τ )   +   t +1 2   

3.1

lim sup ∫ q(s ) (1 − p(s − δ ) ) 3 ∆s = lim sup t →∞

t →∞

1 2

∫ s + 1 ∆s = ∞

5 3 t0

Hence every solution of equation (3.1) is oscillatory. REFERENCES 1. R. P. Agarwal, M. Bohner and A. Peterson, Dynamic equations on time scales: A survey. J. Comp. Appl .Math., Special issue on dynamic equations on time scales, edited by R.P.Agarwal, M.Bohner and D.O’Regan (Preprint in Ulmer Seminare 5), 141, 1–26 (2002).

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2. D. K. Anderson and A. Zafer, Nonlinear oscillation of second order dynamic equations on time scales, Appl. Math. Let., 22, 1591-1597 (2009). 3. M. Bohner and A. Peterson, Dynamic Equations on Time Scales, An Introduction with Applications, Birkhuser, Boston, (2001). 4. S. R. Grace, R. Agarwal, M. Bohner and D. O’Regan, Oscillation of second order strongly superlinear and strongly sublinear dynamic equations, Comm. Nonl. Sci Num. Sim., 14,3463-3471 (2009). 5. M. Huang andW. Feng, Oscillation for forced second order nonlinear dynamic equations on time scales, Elect.J. Diff. Eqn., 145, 1–8 (2005). 6. C. Jinfa, Kamenev type oscillation criteria for delay difference equations, Acta. Math. Sci., 27, 574–580 (2007). 7. I. Kubiczyk, S. H. Saker and J. Morchalo, Kamenev type oscillation

criteria for sublinear delay difference equations, Indian J. Pure Appl. Math., 34, 1273–1284 (2003). 8. S. H. Saker, Oscillation of second order nonlinear neutral delay dynamic equations, J. Comp. Appl. Math., 187, 123–141 (2006). 9. P. Mohankumar and A. Ramesh, Oscillatory Behaviour of The Solution of The Third Order Nonlinear Neutral Delay Difference Equation, International Journal of Engineering Research & Technology (IJERT) ISSN: 2278-0181 Vol. 2 Issue 7,No.1164-1168 July – (2013). 10. B. Selvaraj, P. Mohankumar and A. Ramesh, On The Oscillatory Behavior of The Solutions to Second Order Nonlinear Difference Equations, International Journal of Mathematics and Statistics Invention (IJMSI) EISSN: 2321 – 4767 P-ISSN: 2321 - 4759 Vol., Issue1, PP-19-21 Aug.( 2013).

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