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Sufficient criteria are established for the existence of periodic solutions for such equations, which generalize many known results for differential equations when the time scale is chosen

Volume 2010, Article ID 584375, 13 pages

doi:10.1155/2010/584375

Research Article

Existence of Periodic Solutions for

p-Laplacian Equations on Time Scales

Fengjuan Cao,1 Zhenlai Han,1, 2 and Shurong Sun1, 3

1 School of Science, University of Jinan, Jinan, Shandong 250022, China

2 School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China

3 Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla,

MO 65409-0020, USA

Correspondence should be addressed to Zhenlai Han,hanzhenlai@163.com

Received 30 July 2009; Revised 15 October 2009; Accepted 18 November 2009

Academic Editor: A Pankov

Copyrightq 2010 Fengjuan Cao 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 systematically explore the periodicity of Li´enard type p-Laplacian equations on time scales.

Sufficient criteria are established for the existence of periodic solutions for such equations, which generalize many known results for differential equations when the time scale is chosen as the set

of the real numbers The main method is based on the Mawhin’s continuation theorem

1 Introduction

In the past decades, periodic problems involving the scalar p-Laplacian were studied by many authors, especially for the second-order and three-order p-Laplacian differential equation,

see1 8 and the references therein Of the aforementioned works, Lu in 1 investigated the

existence of periodic solutions for a p-Laplacian Li´enard differential equation with a deviating

argument



ϕ p

yt fytyt  hyt gyt − τt et, 1.1

by Mawhin’s continuation theorem of coincidence degree theory3 The author obtained a new result for the existence of periodic solutions and investigated the relation between the

existence of periodic solutions and the deviating argument τt Cheung and Ren 4 studied

the existence of T-periodic solutions for a p-Laplacian Li´enard equation with a deviating

argument



ϕ p



xt fxtxt  gxt − τt et, 1.2

by Mawhin’s continuation theorem Two results for the existence of periodic solutions were obtained Such equations are derived from many fields, such as fluid mechanics and elastic mechanics

The theory of time scales has recently received a lot of attention since it has a tremendous potential for applications For example, it can be used to describe the behavior

of populations with hibernation periods The theory of time scales was initiated by Hilger

9 in his Ph.D thesis in 1990 in order to unify continuous and discrete analysis By choosing the time scale to be the set of real numbers, the result on dynamic equations yields a result concerning a corresponding ordinary differential equation, while choosing the time scale as the set of integers, the same result leads to a result for a corresponding difference equation Later, Bohner and Peterson systematically explore the theory of time scales and obtain many perfect results in10 and 11 Many examples are considered by the authors in these books But the research of periodic solutions on time scales has not got much attention, see

12–16 The methods usually used to explore the existence of periodic solutions on time scales are many fixed point theory, upper and lower solutions, Masseras theorem, and so on For example, Kaufmann and Raffoul in 12 use a fixed point theorem due to Krasnosel’ski

to show that the nonlinear neutral dynamic system with delay

xΔt −atx σ t  ctxΔt − k  qt, xt, xt − k, t ∈ T, 1.3

has a periodic solution Using the contraction mapping principle the authors show that the periodic solution is unique under a slightly more stringent inequality

The Mawhin’s continuation theorem has been extensively applied to explore the existence problem in ordinary differential difference equations but rarely applied to dynamic equations on general time scales In 13, Bohner et al introduce the Mawhin’s continuation theorem to explore the existence of periodic solutions in predator-prey and competition dynamic systems, where the authors established some suitable sufficient criteria

by defining some operators on time scales

In14, Li and Zhang have studied the periodic solutions for a periodic mutualism model

xΔt r1t



k1t  α1t expy

t − τ2



t, yt

1 expy

t − τ2



t, yt − exp{xt − σ1t, xt}



,

yΔt r2t



k2t  α2t expx

t − τ1



t, yt

1 exp{xt − τ1t, xt} − exp



y

t − σ2



t, yt

 1.4

on a time scaleT by employing Mawhin’s continuation theorem, and have obtained three sufficient criteria

Combining Brouwer’s fixed point theorem with Horn’s fixed point theorem, two classes of one-order linear dynamic equations on time scales

xΔt atxt  ht,

xΔt ft, x, with the initial condition xt0 x0,

1.5

are considered in15 by Liu and Li The authors presented some interesting properties of the exponential function on time scales and obtain a sufficient and necessary condition that

guarantees the existence of the periodic solutions of the equation xΔt atxt  ht.

In16, Bohner et al consider the system

xΔt G



t, exp

x

g1t, exp

x

g2t, , exp

x

g n t,

t

−∞ct, s exp{xs}Δs ,

1.6

easily verifiable sufficient criteria are established for the existence of periodic solutions of this class of nonautonomous scalar dynamic equations on time scales, the approach that authors used in this paper is based on Mawhin’s continuation theorem

In this paper, we consider the existence of periodic solutions for p-Laplacian equations

on a time scalesT

ϕ p

xΔt Δ fxtxΔt  gxt et, t ∈ T, 1.7

where p > 2 is a constant, ϕ p s |s| p−2 s, f, g ∈ CR, R, e ∈ CT, R, and e is a function with periodic ω > 0.T is a periodic time scale which has the subspace topology inherited from

the standard topology on R Sufficient criteria are established for the existence of periodic

solutions for such equations, which generalize many known results for differential equations when the time scales are chosen as the set of the real numbers The main method is based on the Mawhin’s continuation theorem

IfT R, 1.7 reduces to the differential equation



ϕ p



xt fxtxt  gxt et. 1.8

We will use Mawhin’s continuation theorem to study1.7

2 Preliminaries

In this section, we briefly give some basic definitions and lemmas on time scales which are used in what follows LetT be a time scale a nonempty closed subset of R The forward and backward jump operators σ, ρ : T → T and the graininess μ : T → R are defined, respectively, by

σt inf{s ∈ T : s > t}, ρt sup{s ∈ T : s < t}, μt σt − t. 2.1

We say that a point t ∈ T is left-dense if t > inf T and ρt t If t < sup T and σt t, then t is called right-dense A point t ∈ T is called left-scattered if ρt < t, while right-scattered if σt > t If T has a left-scattered maximum m, then we set T k T \ {m},

otherwise setTk T If T has a right-scattered minimum m, then set T k T \ {m}, otherwise

setTk T.

A function f : T → R is right-dense continuous rd-continuous provided that it is

continuous at right-dense point inT and its left side limits exist at left-dense points in T.

If f is continuous at each right-dense point and each left-dense point, then f is said to be

continuous function onT.

Definition 2.1see 10 Assume f : T → R is a function and let t ∈ T k We define fΔt to be

the numberif it exists with the property that for a given ε > 0, there exists a neighborhood

U of t such that



fσt − fs− fΔtσt − s 2.2

We call fΔt the delta derivative of f at t.

If f is continuous, then f is right-dense continuous, and if f is delta differentiable at t, then f is continuous at t.

Let f be right-dense continuous If FΔt ft, for all t ∈ T, then we define the delta

integral by

t

a

fsΔs Ft − Fa, for t, a ∈ T. 2.3

Definition 2.2see 12 We say that a time scale T is periodic if there is p > 0 such that if

t ∈ T, then t ± p ∈ T For T / R, the smallest positive p is called the period of the time scale Definition 2.3 see 12 Let T / R be a periodic time scale with period p We say that the function f : T → R is periodic with period ω if there exists a natural number n such that

ω np, ft  ω ft for all t ∈ T, and ω is the smallest number such that ft  ω ft If

T R, we say that f is periodic with period ω > 0 if ω is the smallest positive number such that f t  ω ft for all t ∈ T.

Lemma 2.4 see 10 If a, b ∈ T, α, β ∈ R, and f, g ∈ CT, R, then

A1b

a αft  βgtΔt αb

a ftΔt  βb

a gtΔt;

A2 if ft ≥ 0 for all a ≤ t < b, thenb

a f tΔt ≥ 0;

A3 if |ft| ≤ gt on a, b : {t ∈ T : a ≤ t < b}, then |b

a ftΔt| ≤b

a gtΔt.

Lemma 2.5 H ¨older’s inequality 11 Let a, b ∈ T For rd-continuous functions f, g : a, b →

R, one has

b

a

ftgt

 b

a

ft p Δt

1/p b

a

gt q Δt

1/q

, 2.4

where p > 1 and q p/p − 1.

For convenience, we denote

κ min{0, ∞ ∩ T}, I ω κ, κ  ω ∩ T, g 1

ω

I ω

gsΔs 1

ω

κω

κ

gsΔs, 2.5

where g ∈ CT, R is an ω-periodic real function, that is, gt  ω gt for all t ∈ T.

Next, let us recall the continuation theorem in coincidence degree theory To do so, we introduce the following notations

Let X, Y be real Banach spaces, L : Dom L ⊂ X → Y a linear mapping, N : X → Y

a continuous mapping The mapping L will be called a Fredholm mapping of index zero if dimKer L codimIm L < ∞ and Im L is closed in Y If L is a Fredholm mapping of index zero and there exist continuous projections P : X → X, Q : Y → Y such that Im P Ker L, Im L Ker Q ImI −Q, then it follows that L| Dom L∩Ker P :I −PX → Im L is invertible We denote the inverse of that map by K P If Ω is an open bounded subset of X, the mapping N will be called L-compact on Ω if QNΩ is bounded and K P I − QN : Ω → X is compact Since

Im Q is isomorphic to Ker L, there exists an isomorphism J : Im Q → Ker L.

Lemma 2.6 continuation theorem Suppose that X and Y are two Banach spaces, and L :

Dom L ⊂ X → Y is a Fredholm operator of index 0 Furthermore, let Ω ⊂ X be an open bounded set and N : Ω → Y L-compact on Ω If

B1 Lx / λNx, for all x ∈ ∂Ω ∩ Dom L, λ ∈ 0, 1,

B2 Nx /∈ Im L, for all x ∈ ∂Ω ∩ Ker L,

B3 deg{JQN, Ω ∩ Ker L, 0} / 0, where J : Im Q → Ker L is an isomorphism,

then the equation Lx Nx has at least one solution in Ω ∩ Dom L.

Lemma 2.7 see 13 Let t1, t2∈ I ω and t ∈ T If g : T → R is ω-periodic, then

gt ≤ gt1 

κω

κ

Δs gt ≥ gt2 −

κω

κ

Δs 2.6

In order to use Mawhin’s continuation theorem to study the existence of ω-periodic

solutions for1.7, we consider the following system:

x1Δt ϕ q x2t |x2t| q−2 x2t,

xΔ2t −fx1tϕ q x2t − gx1t  et, 2.7

where 1 < q < 2 is a constant with 1/p  1/q 1 Clearly, if xt x1t, x2t is an

ω-periodic solution to2.7, then x1t must be an ω-periodic solution to 1.7 Thus, in order

to prove that1.7 has an ω-periodic solution, it suffices to show that 2.7 has an ω-periodic

solution

Now, we setΨω {u, v ∈ CT, R2 : ut  ω ut, vt  ω vt, for all t ∈ T}

with the normu, v max t∈I ω |ut|  max t∈I ω |vt|, for u, v ∈ Ψ ω It is easy to show that

Ψωis a Banach space when it is endowed with the above norm ·

Let

Ψω

0 {u, v ∈ Ψ ω : u 0, v 0},

Ψω

c u, v ∈ Ψ ω:ut, vt ≡ h1, h2 ∈ R2, for t∈ T. 2.8

Then it is easy to show thatΨω

c are both closed linear subspaces ofΨω We

claim thatΨω Ψω

0⊕Ψω

c , and dimΨ ω

c 2 Since for any u, v ∈ Ψ ω

0∩Ψω

c , we have ut, vt

h1, h2 ∈ R2, and

u 1

ω

κω

κ

usΔs h1 0, v 1

ω

κω

κ

vsΔs h2 0, 2.9

so we obtainu, v h1, h2 0, 0.

Take X Y Ψ ω Define

L : Dom L x x1, x2 ∈ C1

T, R2

: xt  ω xt, xΔt  ω xΔt⊂ X → Y,

2.10 by

Lxt xΔt



xΔ1t

xΔ2t , 2.11 and N : X → Y, by

Nxt



ϕ q x2t

−fx1tϕ q x2t − gx1t  et . 2.12 Define the operator P : X → X and Q : Y → Y by

P x P



x1

x2



x1

x2

, Qy Q



y1

y2



y1

y2 , x ∈ X, y ∈ Y. 2.13

It is easy to see that2.7 can be converted to the abstract equation Lx Nx.

Then Ker L Ψω

c , Im L Ψω

0, and dimKer L 2 codim Im L Since Ψ ω

0 is closed

inΨω , it follows that L is a Fredholm mapping of index zero It is not difficult to show that

P and Q are continuous projections such that Im P Ker L and Im L Ker Q ImI − Q.

Furthermore, the generalized inverseto L P  K P : Im L → Ker P ∩ Dom L exists and is given

by

K P



x1

x2

⎛

⎝X1− X1

X2− X2

⎞

⎠, where X i t

t

κ

x i sΔs, i 1, 2. 2.14

Since for every x ∈ Ker P ∩ Dom L, we have

K P Lxt K P



xΔ1t

xΔ2t

⎛

⎜

⎜

⎜

t

κ

xΔ1sΔs − 1

ω

κω

κ

t

κ

xΔ1sΔsΔt

t

κ

xΔ2sΔs − 1

ω

κω

κ

t

κ

xΔ2sΔsΔt

⎞

⎟

⎟

⎟

⎛

⎜

⎜

⎜

x1t − x1κ − 1

ω

κω

κ

x1t − x1κΔt

x2t − x2κ − 1

ω

κω

κ

x2t − x2κΔt

⎞

⎟

⎟

⎟

⎛

⎜

⎜

⎜

x1t − 1 ω

κω

κ

x1tΔt

x2t − 1 ω

κω

κ

x2tΔt

⎞

⎟

⎟

⎟,

2.15

from the definition of P and the condition that x ∈ Ker P ∩ Dom L, then 1/ωκω κ x1tΔt

1/ωκω κ x2tΔt 0 Thus, we get K P Lxt xt Similarly, we can prove that LK P xt xt, for every xt ∈ Im L So the operator K P is well defined Thus,

QN



x1

x2

⎛

⎜

⎝

1

ω

κω

κ

ϕ q x2sΔs

1

ω

κω

κ



−fx1sϕ q x2s − gx1s  esΔs

⎞

⎟

⎠. 2.16

Denote Nx1 N1, Nx2 N2 We have

K P I − QN



x1

x2

⎛

⎜

⎜

⎜

t

κ



N1s − 1

ω

κω

κ

N1rΔr



Δs − 1 ω

κω

κ

t

κ



N1s − 1

ω

κω

κ

N1rΔr



ΔsΔt

t

κ



N2s − 1

ω

κω

κ

N2rΔr



Δs − 1 ω

κω

κ

t

κ



N2s − 1

ω

κω

κ

N2rΔr



ΔsΔt

⎞

⎟

⎟

⎟.

2.17

Clearly, QN and K P I − QN are continuous Since X is a Banach space, it is easy to show that K P I − QNΩ is a compact for any open bounded set Ω ⊂ X Moreover, QNΩ

is bounded Thus, N is L-compact on Ω.

3 Main Results

In this section, we present our main results

Theorem 3.1 Suppose that there exist positive constants d1and d2such that the following conditions hold:

i uσtuΔtfut < 0, |uσt| > d1, t ∈ T,

ii uσtgut − et < 0, |uσt| > d2, t ∈ T,

then1.7 has at least one ω-periodic solution.

Proof Consider the equation Lx λNx, λ ∈ 0, 1, where L and N are defined by the second

section LetΩ1 {x ∈ X : Lx λNx, λ ∈ 0, 1}.

If x



x1t

x2t

∈ Ω1, then we have

xΔ1t λϕ q x2t,

xΔ2t −fx1txΔ

1t − λgx1t  λet. 3.1

From the first equation of 3.1, we obtain x2t ϕ p 1/λxΔ

1t, and then by

substituting it into the second equation of3.1, we get



ϕ p

 1

λ x

Δ

1t

Δ

−fx1txΔ

1t − λgx1t  λet. 3.2

Integrating both sides of3.2 from κ to κ  ω, noting that x1κ x1κ  ω, xΔ

1κ

xΔ1κ  ω, and applyingLemma 2.4, we have

κω

κ

f x1txΔ

1tΔt −

κω

κ



gx1t − etΔt, 3.3

that is,

κω

κ

fx1txΔ

1t  gx1t − etΔt 0. 3.4

There must exist ξ ∈ I ωsuch that

fx1ξxΔ

1ξ  gx1ξ − eξ ≥ 0. 3.5

From conditionsi and ii, when xσξ > max{d1, d2}, we have fx1ξxΔ

1ξ < 0, and gx1ξ − eξ < 0, which contradicts to 3.5 Consequently xσξ ≤ max{d1, d2} Similarly, there must exist η ∈ I ωsuch that

f

x1



η

xΔ1

η

 gx1



η

− eη

Then we have xση ≥ − max{d1, d2} ApplyingLemma 2.7, we get

− max{d1, d2} −

κω

κ

Δ

1s 1t ≤ max{d1, d2} 

κω

κ

Δ

1s 3.7

Let d max{d1, d2} Then 3.7 equals to the following inequality:

|x1t| ≤ d 

κω

κ

Δ

Let E1 {t ∈ I ω:|x1t| ≤ d}, E2 {t ∈ I ω:|x1t| > d}.

Consider the second equation of3.1 and 3.8, then we have

κω

κ

xΔ1tx2tΔt −

κω

κ

x1σtxΔ

2tΔt

κω

κ

fx1txΔ

1tx1σtΔt  λ

κω

κ

x1σtg x1t − etΔt

≤

κω

κ

fx1t Δ

1t 1σt|Δt  λ

E1

x1σtgx1t − etΔt

 λ

E2

x1σtgx1t − etΔt

≤ sup

t∈I ω

f x1t



d

κω

κ

Δ

1t

 κω

κ

Δ

1t

 λ

E1

x1σtgx1t − etΔt

≤ sup

t∈I ω

f x1t

 κω

κ

Δ

1t

2

 d sup

t∈I ω

fx1t

κω

κ

Δ

1t

 λ

E1

x1σtgx1t − etΔt.

.

3.9 ApplyingLemma 2.5, we obtain that

1

λ p−1

κω

κ

Δ

1t p Δt ≤ ω sup

t∈I ω

fx1t

κω

κ

Δ

1t 2Δt  d sup

t∈I ω

fx1t

κω

κ

Δ

1t

 λ



d κω

κ

Δ

1t

 κω

κ

gx1t − et

≤ Q1

κω

κ

Δ

1t2Δt  Q2

κω

κ

Δ

1t

 λdω sup

t∈I ω

gx1t − et

≤ Q1

κω

κ

Δ

1t2Δt  Q2

κω

κ

Δ

.

3.10

solutions for such equations, which generalize many known results for differential equations when the time. .. ω:|x1t| > d}.

Consider the second equation of 3.1 and 3.8, then we have

κω...

Lemma 2.5 H ¨older’s inequality 11 Let a, b ∈ T For rd-continuous functions f, g :