Qualitative Behavior Solution of the Dynamic Equation P. Mohankumar

Abbr:J.Comp.&Math.Sci.

2014, Vol.5(2): Pg.191-198

An International Open Free Access, Peer Reviewed Research Journal www.compmath-journal.org

Qualitative Behavior Solution of the Dynamic Equation

P. Mohankumar ¹ , A.K. Bhuvaneswari ² and A. Ramesh ³

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 .

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:

(H ₁ ): β > 0 is a quotient of odd positive integers;

(H ₂ ): τ δ , are fixed nonnegative constants such that the delay functions

( ) t t t

τ = − τ < ^and δ ( ) t = − t δ < t satisfy τ ( ) : t T → T _and

( ) : t T T

δ → _{for all} t ∈ T ^;

(H 3 ): q (t) and r t ( ) are real valued rd- continuous positive functions defined on T ^;

(H ₄ ): 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 ) ^∈ C _rd ^{′ }  t _y , ^∞ ) _and

( ^{− −} ) ^{∆ ∈ ′ }  ^∞ )

( ) ( ) ( ) ( _{rd y} ,

r t y t p t y t τ C t _,

0

t y ≥ t and satisfying equation (1.1) for all

y

t ≥ t . 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 assumed in the form [ 0 , ) [ 0 , )

T

t ∞ = t ∞ ^{∩ T.} T In this paper we obtain the oscillation criteria for equation (1.1) subject to the following two conditions:

1 ( )

s r s

∞

∆ = ∞

∫ (1.2) and

1 ( )

s r 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 in ⁸ under the condition (1.2).

We note that if ܶ = ℝ we have ( ) t t , ( ) t 0, f ( ) t f '( ). t

σ = µ = ^∆ = ^then

equation (1.1) becomes

( ) ^′

 + − ′  + − = ∈

 

 r t ( ) y t ( ) p t y t ( ) ( τ  q t y ( ) ^β ( t δ ) 0, t ℝ If T = ℕ we have

( ) n n 1, ( ) n 1, y ( ) n y n ( ). y(n 1) y(n)

σ = + µ = ^∆ = ∆ = + −

then equation (1.1) becomes

( )

( )

∆ r t ( ) ∆ y t ( ) + p t y t ( ) ( − τ + q t y t ( ) ^β ( − δ ) 0, = n ∈ ℕ If T = h ℕ,h >0 we have

y( ) y( ) ( ) , ( ) , (t) _h ( ). t h t

t t h t h y t

σ µ ^∆ + h −

= + = = ∆ =

then equation (1.1) becomes

( )

( )

∆ _h r t ( ) ∆ _h y t ( ) + p t y t ( ) ( − τ + q t y ( ) ^β ( t − δ ) = 0, t ∈ h

∆ r t ( ) ∆ y t ( ) + p t y t ( ) ( − τ + q t y ( ) ^β ( t − δ ) = 0, t ∈ h _{h ℕ}

If T = ݍ ^ℕ = ሼݐ: ݐ = ݍ ^௡ , ݊ ∈ ℕሽ, ݍ > 1 we have

y(q ) y( ) ( ) ( ), ( ) ( 1), (t) _q ( ). t t

t q t t q y t

σ µ ^∆ h −

= = − = ∆ =

then equation (1.1) becomes the second order q-neutral difference equations.

( )

( )

∆ _q r t ( ) ∆ _q y t ( ) + p t y t ( ) ( − τ + q t y ( ) ^β ( t − δ ) = 0, t ∈ q ሺݐ − ߜሻ = 0, ݐ ∈ ݍ ^ℕ

If T=ℕ ^ଶ = ሼݐ ^ଶ : ݐ ∈ ℕሽ, we have

( )

2

y ( 1 )

²

y( )

( ) 1 , ( ) 1 2 , (t) ( ).

N

1 2

t t

t t t t y t

σ µ

^∆

t

 +  −

 

 

= + = + = ∆ =

+

then equation (1.1) becomes

( )

( )

∆ _N r t ( ) ∆ _N y t ( ) + p t y t ( ) ( − τ + q t y ( ) ^β ( t − δ ) = 0, t ∈ ሺݐ − ߜሻ = 0, ݐ ∈ ℕ ^ଶ

If ܶ = ሼݐ ௡ : ݊ ∈ ℕሽ, where { } ^t _n is the set if harmonic numbers defined by

0 0

1

0, 1 , ,

n n

k

t t n

= k

= = ∑ ∈ ^{ℕ ଴ ,} we have

( )

1

( ) , ( ) 1 , (t) y( ) 1 y( )

n n n 1 n n n

t t t y t t n t

σ = ₊ µ = n ^∆ = ∆ = + ∆

+

then equation (1.1) becomes

( )

( )

∆ t r t _n ( ) _n ∆ t y t _n ( ) _n + p t y t ( ) ( _{n n} − τ + q t y t ( ) ( _n ^β _n − δ ) 0, = t T ∈

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)

to be oscillatory or converge to zero. Our results include those of Jinfa ¹³ and I.

Kubiaczyk et al. ⁷ when _{T= ℕ} and ( ) p t is identically zero.

2. MAIN RESULT

First we consider the case, when condition (1.2) holds β ≥ 1 .To prove our main results, we will use the following lemma which called Keller’s chain rule.

Lemma 2.1 ³ . Let݂ ∶ ℝ → ℝ be continuously differentiable and suppose ݃ ∶ ܶ → ℝ is delta differentiable. Then ݂ 0 ∶ ܶ → ℝ is delta differentiable and the formula

( ^o ) ^{∆ = ∆} ∫ ^{1 '} ( ^{+ ∆} )

0

( ) ( ) ( ) ( ) ( )

f g t g t f g t h µ t g t dh

holds.

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

1

b ≥ and a positive number M

0

( ) 2 ( ) lim sup ( ) ( ) ( )

4 ( ) ( )

t

t t

K s C s

s s Q s s

s M t

α φ

φ β α

→ ∞

 

− ∆ = ∞

 

 

∫

(2.1) Where

( )

( )( ( ))

( ) ( )(1 ( )) , ( ) s s ( )

Q s q s p s ^β C s φ α _σ s

δ φ

α

∆ + ∆

= − − = + +

( ) ¹ ( ) ²

( ) .( ) ( ) ( )

K t = b t − δ ^{− β} α ^σ t r t − δ ^,

( ^{α ∆ ( ) t} ) ₊ ^{= max} { ^{α ∆ ( ),0 t} } ^and

( ^{φ ∆ ( ) t} ) ₊ ^{= max} { ^{φ ∆ ( ),0 t} }

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

Proof

Suppose to the contrary that y t ( ) is a nonoscillatory solution of equation (1.1).

Let t ₁ ≤ t ₀ be such that y t ≠ ( ) 0 for all t ≥ t 1 . Without loss of generality, we may assume that y t ( ) is an eventually positive solution of equation (1.1) with

( ) 0,

y t − θ > ^{where θ} = ^max { ^{τ δ ,} } ^for

all t ≥ t ₁ sufficiently large.

Set

( ) ( ) ( ) ( )

x t = y t + p t y t − τ (2.2) From equation (1.1) and (H ₂ ) we have

( ^{r t ( )(x ( )) ∆ t} ) ^{∆ = − q t y ( ) β ( t − δ ) < 0}

for all ^t ∈ [ ^t ₁ ^, ∞ ) _T (2.3) and this implies that ( ^{r t ( )(x ( )) ∆ t} ) ^{is an}

eventually decreasing function. We first show that ( ^{r t ( )(x ( )) ∆ t} ) is eventually nonnegative. If ( ^{r t ( )(x ( )) ∆ t} ) ^{< 0}

for t ≥ t ₂ ≥ t ₁ , we have

2 2

( )(x ( ) ( )(x ( ) c 0 r t ^∆ t ≤ r t ^∆ t = <

for some constant c _.

Hence c

(x ( ) ( ) t

r t

∆ ≤ ^.

Now integrating from t to t ₂ we obtain

2

2

( ) ( ) 1 ,

( )

t

t

x t x t c s

≤ + ∫ r s ∆

Which implied by (H ₁ ) that x t → −∞ ( ) . this contradicts that fact the x t > ( ) 0 ^{for all}

t ≥ t 1

Hence ( ^{r t ( )(x ( )) ∆ t} ) ^is eventually nonnegative. Thus we see that there is some

t 1 such that

( )

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

for t ≥ t ₁ (2.4 A) From (2.2), (2.4) and (2.4 A) we have

then , for t ≥ t ₂ = t ₁ + δ ^{we have}

( )

( ) 1 ( ) ( )

y t − δ ≥ − 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 ) ) β x β ( t − δ ) ≤ 0}

for all ^t ∈ [ ^t ₁ ^, ∞ ) _T (2.5) Define

( )( ( )) ( ) ( )

( )

r t x t

w t t

x ^β t α

δ

∆

=

−

for [ 2 , )

T

t ∈ t ∞ Then w t > ( ) 0 , and using Lemma (2.1) we have

( ) ^{( ) ( )} ( )

( ) ( )( (t)) ( )( (t))

( ) ( )

t t

w t r t x r t x

x t x t

σ

β β

α α

δ δ

∆ ∆

∆ ∆   ∆

=   −   + −

( )

( ) ( ) ( ) ( ) 1 ( ) ( )

y t = x t − p t y t − τ ≥ − p t x t

( ) ( ) ^{( ) ( ) ( )} ( ^{( )} )

( ) ( ) ( )( (t)) ( )( (t))

( ) ( ) ( ( ) )

x t t t x t

w t t r t x r t x

x t x t x t

β β

σ

β β β

δ α α δ

α

δ δ σ δ

∆ ∆

∆ ∆ ∆ ∆

 − − − 

 

= +

 

− − −

 

( ^{( )} ) ^{( ) ( )(} ( ^(t)) ) ( ^{( )} )

( ) ( )Q(t) ( )

( ) ( ) ( ( ) )

t t r t x x t

w t t w t

t x t x t

σ β

σ

σ β β

α α δ

α

α δ σ δ

∆ ∆ ∆

∆ +

 − 

 

≤ − + −

 − − 

  (2.6)

The Keller’s chain rule yields that

( )

( )

1 1

0

1

( ) (1 ) ( )

( ) ( )

x t hx h x dhx t

x t x t

β

β σ

β

β

β

−

∆ ∆

− ∆

 

=  + − 

≥

∫

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

( ^{x β ( t − δ )} ) ^{∆ = β} ( ^{x β − 1 ( t − δ )} )( ^{x ∆ ( t − δ )} )

Also from (2.4) we have for t ≥ t ₂ ,

( ^{( ) ( )} )

( )

( )

r t x t x t

r t

σ

δ

δ

∆

∆ − ≥

−

Substitute the last inequality in (2.6), we obtain

( ^{( )} ) ^{( ) ( )(} ( ^(t)) ) ₁

( ) ( )Q(t) ( ) ( )

( ) ( ) ( ) ( ( ) )

t t r t x

w t t w t x t

t r t x t x t

σ

σ β

σ β β

α βα

α δ

α δ δ σ δ

∆ ∆

∆ + −

 

 

≤ − + − −

 − − − 

 

Since x ^∆ ( ) t ≥ 0 ^{we have x} ( ^{σ ( ) t} − ^δ ) ≥ ^{x t (} − ^{δ )} and this implies that

( ) ( ) ₁

2

( ) ( ) ( )( (t))

( ) ( )Q(t) ( ) ( )

( ) ( ) ( ( ) )

t t r t x

w t t w t x t

t r t x t

σ

σ β

σ β

α βα

α δ

α δ σ δ

∆ ∆

∆ + −

 

 

≤ − + − −

 − − 

  (2.7)

From (2.5) and (2.7), we obtain

( ) ^{( )} ( )

( ) ( )

2

2 1

( ) ( ) ( ( ) ( )

( ) ( )Q(t) ( )

( ) ( ) ( ) ( ) ( (t)

t t r t w t

w t t w t

t r t t x t x

σ σ

σ σ β

α βα σ

α α δ α δ σ

∆

∆ +

− ∆

 

 

≤ − + −

 − − 

  ^(2.8)

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

[ )

2 1 ,

T

t ∈ t ∞ such that ^{r t ( )} ( ^{x ∆ ( ) t} ) ^≤ _M ¹

for some positive constant M ^and

[ )

2 1 ,

T

t ∈ t ∞ ^{. Hence}

( ¹ ) ( ^{( )} )

( )

M r t

x t

σ

∆ σ ≥ (2.9) Further from (2.4) we have

0

0 0 0

( ) ( ) ( ) ( )( ),

t

t

x t − x t = ∫ x ^∆ s ∆ ≤ s x ^∆ t t − t and thus there exists a ^T ∈ [ ^t ₀ ^, ∞ ) _T ^a

suitable constant b ≥ 1 such that x t ( ) ≤ bt

for [ )

.

, T

t ∈ T ∞ Hence

1 ( ) 1 ( ) 1

x ^{− β} t − δ ≤ b ^{− β} t − δ ^{− β}

for ^t ∈ [ ^T ₁ ^, ∞ ) _{T .} (2.10) Where T ₁ = T δ + . Now substitute (2.9) and (2.10) in (2.8), we have

( )

( ) 1 ( ) 2 ( ) ²

( ) ( )

( ) ( )Q(t) ( ) ( )

( ) b.( ) ( ) ( )

t M t

w t t w t w t

t t t r t

σ σ

σ β σ

α β α

α α δ α δ

∆

∆ +

−

 

 

≤ − + −

 − − 

  _(2.11)

Multiplying (2.11) by φ ( ) s and integrating from t ₂ to t , (t ≥ t ) ₂ we have

( )

( ) ( ) ( )

2 2 2

2

2 1 2

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )

( ) ( ) ( )

.( ) ( ) ( )

t t t

t t t

t

t

s

s s Q s s s w s s s w s s

s

M s

s w s s

b s s r s

σ σ

σ

β σ

α

φ α φ φ

α φ β α

δ α δ

∆

∆ +

−

∆ ≤ − ∆ + ∆

− ∆

− −

∫ ∫ ∫

∫

(2.12)

Using integration by parts, we obtain

[ ] ( )

( )

2

2 2

2

2 2

( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

t t

t t

t t

t

t

s w s s t w t s w s s

t w t s w s s

σ

σ

φ φ φ

φ φ

∆ ∆

∆ +

∆ ≤ − ∆

≤ − − ∆

∫ ∫

∫

(2.13)

From (2.12) and (2.13), we have

( )

( )

( ) ( ) ( )

2 2 2

2

2 2

2 1 2

( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( )

( ) ( ) ( ( ))

.( ) ( ) ( )

t t t

t t t

t

t

s

s s Q s s w t t s w s s s w s s

s

M s

s w s s

b s s r s

σ σ

σ

σ

β σ

α

φ α φ φ φ

α

φ β α σ

δ α δ

∆ + ∆

+

−

∆ ≤ + ∆ + ∆

− ∆

− −

∫ ∫ ∫

∫

(2.14)

From (2.14) becomes

2 2

2

2

2 2

2

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) 2 ( ) ( )

( ) ( ) 4 ( ) ( )

t t

t t

t

t

K s C s

M s

s s Q s s w t t s w s

K s s M s

K s C s s M s s

β α σ

φ α φ φ

φ β α

φ β α

 

∆ ≤ +   −  

 

+ ∆

∫ ∫

∫

Hence

2

2

2 2

( ) ( )

lim ( ) ( ) ( ) ( ) ( )

4 ( ) ( )

t

t t

K s C s

s s Q s s w t t

s M s

φ α φ

φ β α

→ ∞

 

− ∆ ≤ < ∞

 

 

∫

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 Leighton- Wintner Theorem such that

( )

0

lim sup ( ) 1 ( )

t

t

t

q s p s δ ^β s

→∞ ∫ − − ∆ = ∞ ^(2.18)

Then every solution of equation (1.1) oscillates on [ ^t ₀ ^, ∞ ) _{T .}

3. EXAMPLE

Consider the dynamic equation of the form

[ )

5

1 1 3

( ) ( ) ( ) 0, 1,

2 1 ^T

y t y t y t t

t

τ δ

∆ ∆

   

+ − + − = ∈ ∞

   

    +

 

3.1

τ _and δ are non-negative constants such that t τ

− ^and t − δ ∈ T . In the example 5 / 3, ( ) r t 1/ ( t 1), ( ) p t 1/ 2.

β = = + = ^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

( )

0 0

5

3 5

3

1 1

lim sup ( ) 1 ( ) lim sup

2 1

t t

t t

t t

q s p s s s

s δ

→∞ − − ∆ = →∞ ∆ = ∞

∫ ∫ +

Hence every solution of equation (3.1) is oscillatory.

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