Asymptotic Behavior of Solutions of Higher-Order Dynamic Equations on Time Scales

doi:10.1155/2011/237219

Research Article

Asymptotic Behavior of Solutions of Higher-Order Dynamic Equations on Time Scales

Taixiang Sun,

Hongjian Xi,

and Xiaofeng Peng

1College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi 530004, China

2Department of Mathematics, Guangxi College of Finance and Economics, Nanning, Guangxi 530003, China

Correspondence should be addressed to Taixiang Sun,[email protected] Received 18 November 2010; Accepted 23 February 2011

Academic Editor: Abdelkader Boucherif

Copyrightq2011 Taixiang Sun et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We investigate the asymptotic behavior of solutions of the following higher-order dynamic equation x^Δnt ft, xt, x^Δt, . . . , x^Δn−1t 0, on an arbitrary time scale T, where the functionfis defined onT×Rⁿ. We give sufficient conditions under which every solutionxof this equation satisfies one of the following conditions:1 limt→ ∞x^Δn−1t 0;2 there exist constantsai 0 ≤ i ≤ n−1 witha0/0, such that limt→ ∞xt/_n−1

i0 aih_n−i−1t, t0 1, where hit, t0 0≤i≤n−1are as in Main Results.

1. Introduction

In this paper, we investigate the asymptotic behavior of solutions of the following higher- order dynamic equation

x^Δnt f

t, xt, x^Δt, . . . , x^Δn−1t

0, 1.1

on an arbitrary time scaleT, where the functionfis defined onT×Rⁿ.

Since we are interested in the asymptotic and oscillatory behavior of solutions near infinity, we assume that supT ∞, and define the time scale interval t0,∞_T {t ∈ T : t ≥ t₀}, where t₀ ∈ T. By a solution of 1.1, we mean a nontrivial real-valued function x ∈ Crd Tx,∞_T,R, Tx ≥ t0, which has the property that x^Δnt ∈ Crd Tx,∞_T,R and satisfies1.1on Tx,∞_T, whereC_rd is the space of rd-continuous functions. The solutions vanishing in some neighborhood of infinity will be excluded from our consideration.

A solutionxof1.1is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is called nonoscillatory.

The theory of time scales, which has recently received a lot of attention, was introduced by Hilger’s landmark paper 1 in order to create a theory that can unify continuous and discrete analysis. The cases when a time scale is equal to the real numbers or to the integers represent the classical theories of differential and of difference equations. Many other interesting time scales exist, and they give rise to many applicationssee 2. Not only the new theory of the so-called “dynamic equations” unifies the theories of differential equations and difference equations but also extends these classical cases to cases “in between,” for example, to the so-called q-difference equations when T q^N0, which has important applications in quantum theorysee 3.

On a time scaleT, the forward jump operator, the backward jump operator, and the graininess function are defined as

σt inf{s∈T :s > t}, ρt sup{s∈T :s < t}, μt σt−t, 1.2

respectively. We refer the reader to 2,4for further results on time scale calculus. Letp ∈ CrdT,Rwith 1μtpt/0, for allt ∈ T, then the delta exponential function ept, t0is defined as the unique solution of the initial value problem

y^Δpty,

yt0 1. 1.3

In recent years, there has been much research activity concerning the oscillation and nonoscillation of solutions of various equations on time scales, and we refer the reader to 5–18.

Recently, Erbe et al. 19–21considered the asymptotic behavior of solutions of the third-order dynamic equations

at

rtx^Δt_ΔΔ

ptfxt 0, x^ΔΔΔt ptxt 0,

at rtx^Δt_ΔγΔ

ft, xt 0,

1.4

respectively, and established some sufficient conditions for oscillation.

Karpuz 22studied the asymptotic nature of all bounded solutions of the following higher-order nonlinear forced neutral dynamic equation

xt Atxαt^Δnf

t, x

βt

, x

γt

ϕt. 1.5

Chen 23 derived some sufficient conditions for the oscillation and asymptotic behavior of thenth-order nonlinear neutral delay dynamic equations

atΨxt

xt ptxτt_Δ^n−1α−1xt ptxτt^Δn−1 _γΔ

λFt, xδt 0,

1.6

on an arbitrary time scaleT. Motivated by the above studies, in this paper, we study1.1and give sufficient conditions under which every solutionxof1.1satisfies one of the following conditions:1lim_{t→ ∞}x^Δn−1t 0;2there exist constantsa_i 0≤i≤n−1witha₀/0, such that lim_{t→ ∞}xt/_n−1

i0 aih_n−i−1t, t0 1, wherehit, t0 0≤i≤n−1are as in Section2.

2. Main Results

Letkbe a nonnegative integer ands, t∈T, then we define a sequence of functionshkt, sas follows:

hkt, s

⎧⎪

⎪⎨

⎪⎪

⎩

1 ifk0,

_t

s

h_k−1τ, sΔτ ifk≥1.

2.1

To obtain our main results, we need the following lemmas.

Lemma 2.1. Letnbe a positive integer, then there existsT_n> t₀, such that

h_k1t, t0−hkt, t0≥1 fort≥Tn, 0≤k≤n−1. 2.2

Proof. We will prove the above by induction. First, ifk0, then we takeT₁≥t₀2. Thus,

h₁t, t0−h₀t, t0 t−t₀−1≥1 fort≥T₁. 2.3

Next, we assume that there existsTm > t0, such thath_k1t, t0−hkt, t0 ≥ 1 fort ≥ Tmand 0≤k≤mwith 0≤m < n−1, then

h_m1t, t0−hmt, t0

_t

t0

hmτ, t0−h_m−1τ, t0Δτ

_Tm

t0

hmτ, t0−h_m−1τ, t0Δτ _t

Tm

hmτ, t0−h_m−1τ, t0Δτ

≥ _Tm

t0

hmτ, t0−h_m−1τ, t0Δτ _t

Tm

Δτ

_Tm

t0

hmτ, t0−h_m−1τ, t0Δτt−T_m,

2.4

from which it follows that there existsT_m1> Tm, such thath_k1t, t0−hkt, t0≥1 fort≥T_m1 and 0≤k≤m1. The proof is completed.

Lemma 2.2see 24. Letp∈CrdT, 0,∞, then

1 _t

t0

psΔs≤ept, t0≤e

_t

t0psΔs. 2.5

Lemma 2.3see 2. Lety, p∈CrdT, 0,∞andA∈ 0,∞, then

yt≤A

_t

t0

yτpτΔτ, ∀t∈T 2.6

implies

yt≤Ae_pt, t0, ∀t∈T. 2.7

Lemma 2.4see 2. Letnbe a positive integer. Suppose thatxisntimes differentiable onT. Let α∈T^κn−1andt∈T, then

xt ⁿ⁻¹

k0

hkt, αx^Δkα _ρⁿ⁻¹_t

α

h_n−1t, στx^ΔnτΔτ. 2.8

Lemma 2.5see 2. Assume thatf andg are differentiable onT with lim_{t→ ∞}gt ∞. If there existsT > t0, such that

gt>0, g^Δt>0, ∀t≥T, 2.9

then

t→ ∞lim f^Δt

g^Δt r or ∞implies lim

t→ ∞

ft

gt r or∞. 2.10

Lemma 2.6see 23. Letxbe defined on t0,∞_T, andxt>0 withx^Δnt≤0 fort≥t₀and not eventually zero. Ifxis bounded, then

1lim_{t→ ∞}x^Δit 0 for 1≤i≤n−1,

2 −1ⁱ¹x^Δn−it>0 for allt≥t0and 1≤i≤n−1.

Now, one states and proves the main results.

Theorem 2.7. Assume that there existst₁> t₀, such that the functionft, u0, . . . , u_n−1satisfies ft, u0, . . . , u_n−1≤ⁿ⁻¹

i0

pit|ui|, ∀t, u0, . . . , u_n−1∈ t1,∞_T×Rⁿ, 2.11

wherepit 0≤i≤n−1are nonnegative functions on t1,∞_Tand

t→ ∞lime_qt, t1<∞, 2.12

withqt _n−1

i0 p_ith_n−i−1t, t0 t≥t₁, then every solutionxof1.1satisfies one of the following conditions:

1lim_{t→ ∞}x^Δn−1t 0,

2there exist constantsa_i 0≤i≤n−1witha₀/0, such that

t→ ∞lim _n−1 xt

i0 aih_n−i−1t, t0 1. 2.13

Proof. Letxbe a solution of1.1, then it follows from Lemma2.4that for 0≤m≤n−1,

x^Δmt ^n−m−1

k0

h_kt, t1x^Δkmt1

_ρn−m−1t t1

h_n−m−1t, στx^ΔnτΔτ fort≥t₁. 2.14

By2.11and Lemma2.1, we see that there existsT > t₁, such that fort≥Tand 0≤m≤n−1, x^Δmt≤h_n−m−1t, t0

_n−m−1

k0

x^Δkmt1 _t

t1

n−1 i0

p_iτx^ΔiτΔτ

. 2.15

Then we obtain

x^Δmt≤h_n−m−1t, t0Ft fort≥T, 0≤m≤n−1, 2.16

where

Ft A

_t

T n−1

i0

p_iτx^ΔiτΔτ, 2.17

with

A max

0≤m≤n−1

_n−m−1

k0

x^Δkmt1

_T

t1

n−1

i0

piτx^ΔiτΔτ. 2.18

Using2.16and2.17, it follows that

Ft≤A

_t

T n−1

i0

piτhn−i−1τ, t0FτΔτ fort≥T. 2.19

By Lemma2.3, we have

Ft≤Aeqt, T ∀t≥T, 2.20

withqt _n−1

i0 p_ith_n−i−1t, t0. Hence from2.12, there exists a finite constantc >0, such thatFt≤cfort≥T. Thus, inequality2.20implies that

x^Δmt≤h_n−m−1t, t0c fort≥T, 0≤m≤n−1. 2.21

By1.1, we see that ift≥T, then

x^Δn−1t x^Δn−1T− _t

T

f

τ, xτ, x^Δτ, . . . , x^Δn−1τ

Δτ. 2.22

Since condition2.12and Lemma2.2implies that

t→ ∞lim _t

T n−1

i0

p_iτh_n−i−1τ, t0Δτ <∞, 2.23

we find from 2.11 and 2.21 that the sum in 2.22 converges as t → ∞. Therefore, lim_{t→ ∞}x^Δn−1texists and is a finite number. Let lim_{t→ ∞}x^Δn−1t a0. Ifa0/0, then it follows from Lemma2.5that

t→ ∞lim xt

h_n−1t, t0 lim

t→ ∞x^Δn−1t a₀, 2.24

andxhas the desired asymptotic property. The proof is completed.

Theorem 2.8. Assume that there exist functions p_i : t0,∞_T → 0,∞ 0 ≤ i ≤ n, and nondecreasing continuous functionsg_i:0,∞ → 0,∞ 0≤i≤n−1, andt₁> t₀such that

ft, u0, . . . , u_n−1≤ⁿ⁻¹

i0

p_itgi

|ui| h_n−i−1t, t0

p_nt fort≥t₁, 2.25

with

_∞

t1

pitΔtPi<∞ for 0≤i≤n, _∞

ε

_n−1ds

i0 g_is∞ for anyε >0,

2.26

then every solutionxof1.1satisfies one of the following conditions:

1lim_{t→ ∞}x^Δn−1t 0,

2there exist constantsa_i 0≤i≤n−1witha₀/0 such that

t→ ∞lim _n−1 xt

i0 a_ih_n−i−1t, t0 1. 2.27

Proof. Letxbe a solution of1.1, then it follows from Lemma2.4that for 0≤m≤n−1,

x^Δmt ^n−m−1

k0

h_kt, t1x^Δkmt1

_ρ^n−m−1_t

t1

h_n−m−1t, στx^ΔnτΔτ fort≥t₁. 2.28

By Lemma2.1and2.25, we see that there existsT > t₁, such that fort≥Tand 0≤m≤n−1, x^Δmt≤h_n−m−1t, t0

⎡

⎣^n−m−1

k0

x^Δkmt1 _t

t1

⎡

⎣ⁿ⁻¹

i0

p_iτgi

⎛

⎝ x^Δiτ h_n−i−1τ, t0

⎞

⎠p_nτ

⎤

⎦Δτ

⎤

⎦. 2.29

Then, we obtain

x^Δmt≤h_n−m−1t, t0Ft, fort≥T, 0≤m≤n−1, 2.30

where

Ft A

_t

T n−1

i0

piτgi

⎛

⎝

x^Δiτ h_n−i−1τ, t0

⎞

⎠Δτ, 2.31

with

A max

0≤m≤n−1

_n−m−1

k0

x^Δkmt1

_T

t1

n−1 i0

piτgi

⎛

⎝

x^Δiτ h_n−i−1τ, t0

⎞

⎠ΔτPn. 2.32

Using2.30and2.31, it follows that

Ft≤A

_t

T n−1

i0

piτgiFτΔτ fort≥T. 2.33

Write

ut A

_t

T

n−1 i0

p_iτgiFτΔτ fort≥T, 2.34

G y

_y

A

_n−1ds

i0 g_is, 2.35

then

Gut^Δu^Δt ₁

0

Ghut 1−hu^σtdh

#_n−1

i0

pitgiFt

$ ₁

0

_n−1 dh

i0 gihut 1−hu^σt

≤ _n−1

i0 pitgiut _n−1

i0 giut

≤ⁿ⁻¹

i0

p_it,

2.36

from which it follows that

Gut≤GuT

_t

T n−1

i0

p_iτΔτ ≤GuT ⁿ⁻¹

i0

P_i. 2.37

Since limy→ ∞Gy ∞andGyis strictly increasing, there exists a constantc >0, such that ut≤cfort≥T. By2.30,2.33, and2.34, we have

x^Δmt≤h_n−m−1t, t0c fort≥T, 0≤m≤n−1. 2.38

It follows from1.1that ift≥T, then

x^Δn−1t x^Δn−1T− _t

T

f

τ, xτ, x^Δτ, . . . , x^Δn−1τ

Δτ. 2.39

Since2.38and condition2.25implies that _t

T

f

τ, xτ, x^Δτ, . . . , x^Δn−1τΔτ

≤ _t

T

⎡

⎣ⁿ⁻¹

i0

p_iτgi

⎛

⎝ x^Δiτ h_n−i−1τ, t0

⎞

⎠p_nτ

⎤

⎦Δτ

≤ⁿ⁻¹

i0

P_ig_ic P_n M <∞,

2.40

we see that the sum in2.39converges ast → ∞. Therefore, limt→ ∞x^Δn−1texists and is a finite number. Let lim_{t→ ∞}x^Δn−1t a₀. Ifa₀/0, then it follows from Lemma2.5that

t→ ∞lim xt

h_n−1t, t0 lim

t→ ∞x^Δn−1t a0, 2.41

andxhas the desired asymptotic property. The proof is completed.

Theorem 2.9. Assume that there exist positive functionsp: t0,∞_T → 0,∞, and nondecreasing continuous functionsg_i:0,∞ → 0,∞ 0≤i≤n−1, andt₁> t₀, such that

ft, u0, . . . , u_n−1≤pt%ⁿ⁻¹

i0

gi

|ui| h_n−i−1t, t0

fort≥t1, 2.42

with

_∞

t1

ptΔtP <∞, _∞

ε

&_n−1ds

i0gis ∞, for anyε >0,

2.43

then every solutionxof1.1satisfies one of the following conditions:

1lim_{t→ ∞}x^Δn−1t 0,

2there exist constantsai 0≤i≤n−1witha0/0, such that

t→ ∞lim _n−1 xt

i0 a_ih_n−i−1t, t0 1. 2.44

Proof. Arguing as in the proof of Theorem2.8, we see that there existsT > t1, such that for t≥T and 0≤m≤n−1,

x^Δmt≤h_n−m−1t, t0

⎡

⎣^n−m−1

k0

x^Δkmt1 _t

t1

%n−1 i0

pτgi

⎛

⎝ x^Δiτ h_n−i−1τ, t0

⎞

⎠Δτ

⎤

⎦, 2.45

from which we obtain

x^Δmt≤h_n−m−1t, t0Ft fort≥T, 0≤m≤n−1, 2.46

where

Ft A

_t

T

%n−1 i0

pτgi

⎛

⎝

x^Δiτ h_n−i−1τ, t0

⎞

⎠, 2.47

A max

0≤m≤n−1

_n−m−1

k0

x^Δkmt0

_T

t1

%n−1 i0

pτgi

⎛

⎝ x^Δiτ h_n−i−1τ, t0

⎞

⎠. 2.48

Using2.46and2.47, it follows that

Ft≤A

_t

T

%n−1 i0

pτgiFτΔτ fort≥T. 2.49

Write

ut A

_t

T

%n−1 i0

pτgiFτΔτ fort≥T, 2.50

G y

_y

A

&_n−1ds

i0g_is, 2.51

then

Gut^Δu^Δt ₁

0

Ghut 1−hu^σtdh

#_n−1

%

i0

ptgiFt

$ ₁

0

&_n−1 dh

i0g_ihut 1−hu^σt

≤

&_n−1

i0ptgiut

&_n−1

i0g_iut pt,

2.52

from which it follows that

Gut≤GuT

_t

T

pτΔτ ≤GuT P. 2.53

The rest of the proof is similar to that of Theorem2.8, and the details are omitted. The proof is completed.

Theorem 2.10. Assume that the functionft, u0, . . . , u_n−1satisfies

1ft, u0, . . . , u_n−1 ptFu0, . . . , u_n−1for allt, u0, . . . , u_n−1∈ t0,∞_T×Rⁿ, 2pt≥0 fort≥t0and_∞

t0 h_n−1τ, t0pτΔτ∞,

3u₀Fu0, . . . , u_n−1 > 0 foru₀/0 andFu0, . . . , u_n−1is continuous atu0,0, . . . ,0with u₀/0,

then (1) ifnis even, then every bounded solution of 1.1is oscillatory; (2) ifnis odd, then every bounded solution xt of 1.1 is either oscillatory or tends monotonically to zero together with x^Δit 1≤i≤n−1.

Proof. Assume that 1.1 has a nonoscillatory solution x on t0,∞, then, without loss of generality, there is at1 ≥ t0, sufficiently large, such that xt > 0 for t ≥ t1. It follows from 1.1thatx^Δnt≤0 fort≥t₁and not eventually zero. By Lemma2.6, we have

t→ ∞limx^Δit 0, for 1≤i≤n−1,

−1ⁱ¹x^Δn−it>0 ∀t≥t₁, 1≤i≤n−1,

2.54

andxtis eventually monotone. Alsox^Δt>0 fort≥t₁ifnis even andx^Δt<0 fort≥t₁ ifnis odd. Sincextis bounded, we find lim_{t→ ∞}xt c≥0. Furthermore, ifnis even, then c >0.

We claim thatc0. If not, then there existst₂> t₁, such that

F

xt, x^Δt, . . . , x^Δn−1t

> Fc,0, . . . ,0

2 >0 fort≥t2, 2.55 sinceFis continuous atc,0, . . . ,0by the condition3. From1.1and2.55, we have

x^Δnt ptFc,0, . . . ,0

2 ≤0, fort≥t₂. 2.56

Multiplying the above inequality byh_n−1t, t0, and integrating fromt₂tot, we obtain _t

t2

h_n−1τ, t0x^ΔnτΔτ _t

t2

h_n−1τ, t0pτFc,0, . . . ,0

2 Δτ ≤0, fort≥t₂. 2.57

Since

_t

t2

h_n−1τ, t0x^ΔnτΔτ≥ ⁿ

i1

−1ⁱ¹h_n−iτ, t0x^Δn−iτ

t

t2

≥ⁿ

i1

−1ⁱh_n−it2, t₀x^Δn−it2 −1ⁿ¹xt,

2.58

we get

A −1ⁿ¹xt

_t

t2

h_n−1τ, t0pτFc,0, . . . ,0

2 Δτ≤0, fort≥t₂, 2.59

whereA_n

i1−1ⁱh_n−it2, t0x^Δn−it2. Thus,_∞

t2 h_n−1τ, t0pτΔτ <∞sincextis bounded, which gives a contradiction to the condition2. The proof is completed.

3. Examples

Example 3.1. Consider the following higher-order dynamic equation:

x^Δnt ⁿ⁻¹

i0

1 t^βi

x^Δit

h_n−i−1t, t0 0, 3.1

wheret≥t1> t0 >0 andβi>10≤i≤n−1. Letpit 1/ t^βih_n−i−1t, t0 0≤i≤n−1and

ft, u0, . . . , u_n−1

n−1

i0

1 t^βi

ui

h_n−i−1t, t0, 3.2

then we have

ft, u0, . . . , u_n−1≤ⁿ⁻¹

i0

p_it|ui|, ∀t, u0, . . . , u_n−1∈ t1,∞_T×Rⁿ,

eⁿ⁻¹

i0pithn−i−1t, t1 eⁿ⁻¹

i01/t^βit, t1 ≤e

t t1

_n−1

i01/τ^βiΔτ <∞,

3.3

by Example 5.60 in 4. Thus, it follows from Theorem 2.7 that if x is a solution of 3.1 with lim_{t→ ∞}x^Δn−1t/0, then there exist constantsa_i 0 ≤ i ≤ n−1with a₀/0, such that lim_{t→ ∞}xt/_n−1

i0 aih_n−i−1t, t0 1.

Example 3.2. Consider the following higher-order dynamic equation:

x^Δnt ⁿ⁻¹

i0

1 t^βi

# x^Δit h_n−i−1t, t0

$αi

1

t^βn 0, 3.4

wheret > t0>0,αi∈0,1 0≤i≤n−1, andβi>10≤i≤n. Letgiu u^αi 0≤i≤n−1, p_it 1/t^βi 0≤i≤n, and

ft, u0, . . . , u_n−1

n−1

i0

1 t^βi

ui

h_n−i−1t, t0 _αi

1

t^βn. 3.5

It is easy to verify thatft, u0, . . . , u_n−1satisfies the conditions of Theorem2.8. Thus, it follows that ifxis a solution of3.4with limt→ ∞x^Δn−1t/0, then there exist constantsai 0 ≤ i ≤ n−1witha₀/0, such that lim_{t→ ∞}xt/_n−1

i0 a_ih_n−i−1t, t0 1.

Example 3.3. Consider the following higher-order dynamic equation:

x^Δnt 1 t^β

%n−1 i0

# x^Δit h_n−i−1t, t0

$αi

0, 3.6

wheret > t₀>0, α_i∈0,1 0≤i≤n−1with 0<_n−1

i0 α_i<1 andβ >1. Letg_iu u^αi 0≤ i≤n−1, pt 1/t^β, and

ft, u0, . . . , u_n−1

%n−1 i0

1 t^β

u_i h_n−i−1t, t0

_αi

. 3.7

It is easy to verify thatft, u0, . . . , u_n−1satisfies the conditions of Theorem2.9. Thus, it follows that ifxis a solution of3.6with lim_{t→ ∞}x^Δn−1t/0, then there exist constantsa_i 0 ≤ i ≤ n−1witha0/0, such that limt→ ∞xt/_n−1

i0 aih_n−i−1t, t0 1.

Acknowledgment

This paper was supported by NSFCno. 10861002and NSFGno. 2010GXNSFA013106, no.

2011GXNSFA018135and IPGGEno. 105931003060.

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