EXISTENCE OF PERIODIC SOLUTIONS FOR HIGHER ORDER ...

TAIWANESE JOURNAL OF MATHEMATICS Vol. 16, No. 6, pp. 2259-2273, December 2012 This paper is available online at http://journal.taiwanmathsoc.org.tw

EXISTENCE OF PERIODIC SOLUTIONS FOR HIGHER ORDER DYNAMIC EQUATIONS ON TIME SCALES Sheng-Ping Wang Abstract. In this paper, we study a sufficient condition for the existence of a periodic solution for the nth order dynamic equations on time scales. The results are shown by the use of coincidence degree theory. The necessary a priori bounds are based on Wirtinger type inequality established as Lemma B.

1. INTRODUCTION The theory of dynamic systems on time scales T(closed subsets of the reals) provides a framework for dealing with both continuous and discrete dynamic systems simultaneously so as to bring out a new insight of subtle differences for these two types of systems. More and more integrated results spring up in recent years, for example, [3, 4, 10, 11, 13]. Qualitative properties including stability, oscillation theory and asymptotic behavior of the solutions are also widely discussed. The basic tool used in this article is the Mawhin’s coincidence degree theory [9], which can be directly applied to study the periodic boundary value problems. Many researchers have already focused on this topic for a long period of time and plenty of essential papers are worked out, see [2, 7, 8, 17, 23, 24]. Recently, they are generalized naturally on the so-called field, time scales. This consideration can be related with many interesting biological issues including predator-prey and competition dynamic systems, population models, or other mutualism models, for example [5, 6, 14, 25, 26, 27]. We also refer more detailed treatment to more references [1, 12, 15, 16, 19, 20, 21, 22]. However, one can see that, until now, there are relatively few conclusions for higher order dynamic systems. We briefly introduce the problem considered along the article. Let T be a time scale with period σ(T ), where 0, T ∈ T. The purpose is to consider the existence of solutions of the nonlinear periodic boundary value problem(n ≥ 2) (1.1)

n

n−1

xΔ = f (xΔ

, · · · , xΔ , x, t), t ∈ T,

Received March 29, 2012, accepted April 17, 2012. Communicated by Hari M. Srivastava. 2010 Mathematics Subject Classification: 34N05, 34B15. Key words and phrases: Periodic solution, Higher order dynamic equations, Time scales, Coincidence degree, Priori estimates, Wirtinger type inequality.

2259

2260 (1.2)

Sheng-Ping Wang i

i

xΔ (0) = xΔ (σ(T )), 0 ≤ i ≤ n − 1.

Throughout we assume that f is σn+1 -completely delta differentiable on D := Rn × [0, σ(T )]T (see Definition 2, [18]), its partial derivatives fi (1 ≤ i ≤ n) are continuous on D and f is σ(T )-periodic in t. This paper is organized as follows. In next section, some fundamental concepts and inequalities on time scales are exposed. Section 3 is devoted to derive the necessary a priori bounds which are based on Wirtinger type inequality (Lemma B). In Section 4 we develop our main existence result (that is, Theorem A) of solutions for the higher order periodic boundary value problem (1.1), (1.2). 2. PRELIMINARIES ON TIME SCALES In this section, we give some of the basics of the time scale theory and refer to [3, 4] for more details. Let T be a time scale which means any closed subset of R and the interval [a, b]T = {t ∈ T : a ≤ t ≤ b}. The embedding of T in R gives rise to the order and topological structure of the time scale in a canonical way. Definition 1. For t ∈ T, we define the forward and backward jump operators σ, ρ : T → T by σ(t) := inf{s ∈ T : s > t} and ρ(t) := sup{s ∈ T : s < t}. If t < sup T and σ(t) = t, then t is called right-dense (otherwise: right-scattered), and if t > inf T and ρ(t) = t, then t is called left-dense (otherwise: left-scattered). The graininess function μ : T → R+ is defined by μ(t) = σ(t) − t. We denote uσ (t) ≡ u(σ(t)) for t ∈ T and Tκ be the set of points of T except for a maximal element which is also left scattered. Definition 2. The mapping f : T → R is called rd-continuous if it is continuous at each right-dense or maximal point t ∈ T and if the left-sided limit exists at each left-dense points. The set of all rd-continuous functions f : T → R is denoted by Crd = Crd (T). Definition 3. Assume that f : T → R is a function and let t ∈ Tκ . Then we define to be the number(provided it exists) with the property that, for any given > 0, there is a neighborhood U of t such that f Δ (t)

|f (σ(t)) − f (s) − f Δ (t)|σ(t) − s|| ≤ |σ(t) − s| for all s ∈ U. We call f Δ (t) the delta derivative of f at t. If F Δ = f , then we define the Cauchy integral by s

r

f (τ )Δτ = F (s) − F (r) for r, s ∈ T.

Existence of Periodic Solutions for Higher Order DE on Time Scales

2261

Lemma 1. If t ∈ Tκ and f : T → R is delta differentiable at t, then f σ (t) = f (t) + μ(t)f Δ (t). Lemma 2. If f ∈ Crd and t ∈ Tκ , then

σ(t) t

f (s)Δs = μ(t)f (t).

Lemma 3. If f, g : T → R are delta differentiable at t ∈ Tκ , then (f g)Δ(t) = f Δ (t)g(t) + f σ (t)g Δ(t) = f (t)g Δ(t) + f Δ (t)g σ (t). Lemma 4. Every rd-continuous function has an antiderivative. In particular, if t t0 ∈ T, then F defined by F (t) := t0 f (τ )Δτ for all t ∈ T is an antiderivative of f , that is, F Δ = f . Definition 4. Let ω > 0. A time scale T is called ω-periodic if t+ω ∈ T whenever t ∈ T. A function p is said to be ω-periodic on T if p(t + ω) = p(t) for all t ∈ T. Next, two well-known conclusions, the Cauchy-Schwarz inequality and one type of Wirtinger’s inequality are stated. Theorem 1. Let a, b ∈ T. For rd-continuous f, g : [a, b]T → R we have b b b |f (t)g(t)|Δt ≤ |f (t)|2 Δt |g(t)|2Δt . a

a

a

1 . Then Theorem 2. Let M be positive and strictly monotone such that M ∈ Crd we have b b M (t)M σ (t) Δ Δ σ 2 (y (t))2 Δt |M (t)|(y (t)) Δt ≤ Ψ |M Δ (t)| a a 1 with y(a) = y(b) = 0, where for any y ∈ Crd

Ψ=

M (t) + sup σ t∈[a,b]T M (t)

μ(t)|M Δ (t)| M (t) 2 + sup . σ σ M (t) t∈[a,b]T t∈[a,b]T M (t) sup

3. A PRIORI ESTIMATES In this section, let x(t) be a σ(T )-periodic solution of the problem (1.1), (1.2) and we shall get the priori bounds for the derivatives of x(t) up to (n − 1)th order by imposing certain conditions on the partial derivatives of f . Such priori estimates will be used to construct a bounded open set in the next section. Firstly, we need two crucial lemmas showed as follows:

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Sheng-Ping Wang

1 . Then Lemma A. Let M be positive and strictly monotone such that M ∈ Crd we have b b M σ (t) Δ Δ 2 (y (t))2 Δt |M (t)|(y(t)) Δt ≤ Φ Δ (t)| |M a a 1 with y(a) = y(b) = 0, where for any y ∈ Crd 2 sup M σ (t) + sup μ(t)|M Δ (t)| + sup M σ (t) . Φ= t∈[a,b]T

t∈[a,b]T

t∈[a,b]T

Proof. We follow the steps of the proof of Theorem 2. For convenience we omit the argument (t) in the following arguments. Let b b Mσ Δ 2 Δ 2 (y ) Δt, M y Δt, V = J= Δ a a |M | α = sup M σ , β = sup μ|M Δ |. [a,b]T

[a,b]T

Without loss of generality we assume that M Δ is of positive sign. Then we apply the Cauchy-Schwarz inequality(Theorem 1), Lemma 1, 3 and y(a) = y(b) = 0 to estimate b M Δ y 2 Δt J= a b b 2 Δ (M y ) Δt − M σ y Δ (y + y σ )Δt = a a b M σ y Δ (y + y σ )Δt =− a b M σ |y Δ ||2y + μy Δ |Δt ≤ a b b σ Δ M |y ||y|Δt + μM σ (y Δ )2 Δt ≤2 a a b b σ M μM Δ M σ Δ 2 Δ σ Δ |y (y ) Δt | M |M ||y|Δt + =2 |M Δ | |M Δ | a a b Mσ Δ 2 1 b σ Δ 2 1 2 |y | Δt M |M ||y| Δt 2 + βV ≤2 Δ| |M a √a ≤ 2α JV + βV. Therefore, by denoting H = VJ , we find that H 2 − 2αH − β ≤ 0, and solving for H ≥ 0 we obtain J = H 2 ≤ (α + α2 + β)2 = Φ V

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so that the proof is complete. 1 Lemma B. For any y ∈ Crd with y(a) = y(b), y Δ (a) = y Δ (b) and we have b

b

(y(t))2 Δt ≤ K 2

a

b a

y(t)Δt = 0,

(y Δ (t))2 Δt,

a

where K := σ(b) − a +

(σ(b) − a)2 + (σ(b) − a) sup μ(t). t∈[a,b]T

t Proof. Let f (t) = a y Δ (s)Δs, t ∈ [a, b]T, then f (a) = f (b) = 0. Choose M (t) = t − a and apply Lemma A to f (t), then we can conclude this lemma. In what follows, let T be a σ(T )-periodic time scale containing {0, T } and denote [0, σ(T )]T ≡ I. If h is a real-valued function which is bounded on the set W , weput |h|W = supz∈W |h(z)|. If h is square integrable over I, we denote σ(T ) |h(z)|2Δz. Let ||h|| = 0 Z = {z ∈ C(T) : z is σ(T )-periodic} with norm |z|Z = maxt∈I |z(t)| and n−1

X = {x ∈ C Δ

i

i

(T) : x is σ(T )-periodic with xΔ (0) = xΔ (σ(T )), 0 ≤ i ≤ n − 1} i

Δ with norm |x|X = Σn−1 i=0 maxt∈I |x (t)|. We see that Z and X are real Banach spaces. Our useful estimates are immediately listed as follows:

Lemma C. Let C = σ 2 (T ) +

[σ 2 (T )]2 + σ 2 (T ) sup μ(t). t∈I

When n ≥ 3, we assume that there exists a constant η > 0 such that f2 ≥ −η > −

(3.1) (3.2)

n−1

1 on D, CC1

C n−i−1 C1 |fn−i |D + C n−2 C1 |fn |D σ(T ) < 1 − ηCC1 ,

i=1,i=n−2

where C1 = C + supt∈I μ(t), are satisfied; when n = 2, assume that there exists a constant ζ < σ(T1 )2 such that (3.3)

f2 ≥ −ζ on D,

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Sheng-Ping Wang

|f1 |D
0 such that (3.5)

σ(T )

0

n−1

f (xΔ

(t), · · · , x(t), t)Δt = 0

for each x ∈ X such that inf |x(t)| ≥ R.

(3.6)

t∈T

Then there is a positive constant B such that i

||xΔ || ≤ Mi , 0 ≤ i ≤ n − 1,

(3.7) and

2 σ(T ) 1 + Mi , 0 ≤ i ≤ n − 2, |x (t)| ≤ C σ(T )

Δi

(3.8) where

Mi = C n−1−i B, for 1 ≤ i ≤ n − 2, M0 = σ(T ){R + C n−2 B σ(T )}. Furthermore, there exists a constant L such that n−1

|xΔ

(3.9) here N ∗ = 2

σ(T )L + √ B

σ(T )

(t)| ≤ N ∗ ,

, L is given as in (3.26).

Proof. Since f is σn+1 -completely delta differentiable on D := Rn × I, according to Theorem 9 [18], we rewrite (1.1) in the form (3.10)

n

xΔ =

n−1

i

n−1

xΔ Fn−i (xΔ

, · · · , xΔ, x, t) + g(t)

i=0

where Δn−1

Fn−i (x

Δ

, · · · , x , x, t) =

0

1

n−1

fn−i (ξxΔ

g(t) = f (0, 0, · · · , 0, t).

, · · · , ξx, t)dξ,

2265

Existence of Periodic Solutions for Higher Order DE on Time Scales n−2 σ

Multiplying by xΔ have (3.11)

0

σ(T )

n−1

[xΔ

on both sides of (3.10) and integrating from 0 to σ(T ), we

]2 Δt = − −

n−1

σ(T ) 0 i=0σ(T )

n−2

xΔ

n−2 σ

xΔ

0

i

, · · · , x, t)Δt

(t)g(t)Δt.

Note that we also have σ(T ) n−2 n−1 xΔ σ (t)x(t)Fn (xΔ , · · · , x, t)Δt − 0 n−2 (3.12) ≤ |F | |x| σ(T )||xΔ σ ||, n D

n−1

xΔ Fn−i (xΔ

I

≤ ζ|x|2I σ(T ),

when n ≥ 3 when n = 2.

By integrating (1.1) over σ(T ) period, we then obtain

σ(T )

0

n−1

f (xΔ

(t), · · · , x(t), t)Δt = 0,

when x(t) is a possible σ(T )-periodic solution of (1.1). Hence, by means of (3.5), (3.6), this implies that there exists ξ ∈ I such that |x(ξ)| < R. For t ∈ I, we have

t

|x(t) − x(ξ)| ≤ ξ

Δ

|x (s)|Δs ≤

σ(T ) 0

|xΔ (s)|Δs ≤ ||xΔ || σ(T ).

Therefore, it follows from the above inequality and Lemma B(setting b = σ(T ), a = 0) that (3.13) |x|I ≤ R + ||xΔ || σ(T ), when n = 2, n−1 (3.14) ≤ R + C n−2 ||xΔ || σ(T ), when n ≥ 3. For the case n = 2, from (3.11), (3.12), we have ||xΔ ||2 ≤ ζ|x|2I + |F1 |D |x|I ||xΔ || σ(T ) + |x|I ||g|| σ(T ). Using (3.13), we obtain where

a||xΔ ||2 − b||xΔ|| − c ≤ 0,

a := 1 − ζσ(T )2 − |F1 |D σ(T ) > 0 by (3.4), 3 b := 2ζRσ(T ) 2 + |F1 |D R σ(T ) + ||g||σ(T ),

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Sheng-Ping Wang

c := ζR2 σ(T ) + R||g|| σ(T ). Hence, we have ||xΔ || ≤ S,

(3.15) here

√

b2 + 4ac . 2a For the case n ≥ 3, from (3.11), using (3.1), the Schwarz inequality (Theorem 1) and Wirtinger type inequality (Lemma B) established, we get σ(T ) n−2 n−1 Δn−1 2 || ≤ − xΔ σ xFn (xΔ , · · · , x, t)Δt ||x S=

b+

0

n−1

+

n−1

{C n−i + C n−i−1 sup μ(t)}||xΔ t∈I

i=1,i=n−2

Δn−1

+ η{C 2 + C sup μ(t)}||x

||2 |Fn−i |D n−1

||2 + ||g||{C + sup μ(t)}||xΔ

t∈I

||.

t∈I

Since, by (3.14), σ(T ) n−2 n−1 xΔ σ xFn (xΔ , · · · , x, t)Δt − 0 n−1 n−1 ≤ {R + C n−2 ||xΔ || σ(T )}|Fn |D {C + sup μ(t)}||xΔ || σ(T ), t∈I

one can immediately get ||x

Δn−1 2

|| ≤C

n−2

n−1

Δn−1 2

C1 ||x

|| |Fn |D σ(T ) +

n−1

C n−i−1 C1 ||xΔ

||2|Fn−i |D

i=1,i=n−2 Δn−1

+ ηCC1 ||x that is, 1−

n−1

||2 + C1 {||g|| + R|Fn |D

n−1

σ(T )}||xΔ

||,

C

n−i−1

C1 |Fn−i |D − C

i=1,i=n−2

≤ C1 {||g|| + R|Fn |D

n−2

n−1

C1 |Fn |D σ(T ) − ηCC1 ||xΔ

||

σ(T )}.

Note that the term in the bracket of the above inequality is positive by the hypothesis (3.2). Thus we have n−1 (3.16) ||xΔ || ≤ K{||g|| + R|Fn |D σ(T )},

2267


where K = C1−1 −

n−1

C n−i−1 |Fn−i |D − C n−2 |Fn |D σ(T ) − ηC

−1

.

i=1,i=n−2

Combing (3.15) and (3.16), we have for n ≥ 2, we have n−1

||xΔ

(3.17)

|| ≤ B,

where

σ(T )]}.

B = max{S, K[||g|| + R|Fn |D

By (3.17) and the Wirtinger type inequality (Lemma B) again, we have i

n−1

||xΔ || ≤ C n−1−i ||xΔ

(3.18)

|| ≤ Mi

for 1 ≤ i ≤ n − 1 and by (3.13), (3.14), ||x|| ≤ M0 . Also, (3.8) can derived from (3.18). For t ∈ I, 0 ≤ i ≤ n − 2, we have Δi

Δi

(3.19) |x (t) − x (0)| ≤

t 0

i+1

|xΔ

i+1

(s)|Δs ≤ ||xΔ

|| σ(T ) ≤ C −1 σ(T )Mi .

From the inequality (3.19), we have for 0 ≤ i ≤ n − 2, i

i

i

i

|xΔ (t)| ≤ |xΔ (0)| + C −1

(3.20)

σ(T )Mi ,

and |xΔ (t)| ≥ |xΔ (0)| − C −1

(3.21) i

If |xΔ (0)| − C −1

σ(T )Mi ≤ 0, it follows immediately that i

|xΔ | ≤ 2C −1

(3.22) i

If |xΔ (0)| − C −1 and get (3.23)

σ(T )Mi .

σ(T )Mi on I.

σ(T )Mi > 0, we integrate (3.21) on both sides from 0 to σ(T ) i

i

||xΔ || ≥ {|xΔ (0)| − C −1

σ(T )Mi } σ(T ).

From (3.18) and (3.23), we have (3.24)

Δi

|x (0)| ≤

1 + σ(T )

σ(T ) Mi . C

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Sheng-Ping Wang

Hence, it follows from (3.20) that 2 σ(T ) 1 Mi , t ∈ I. + |x (t)| ≤ C σ(T )

Δi

(3.25)

Combining (3.22) and (3.25), we obtain (3.8). On the other hand, rewrite (1.1) in the alternative form x

Δn

n−1

=

i

n−1

xΔ Gn−i (xΔ

, · · · , xΔ, x, t) + g(t),

i=0

where for 0 ≤ i ≤ n − 1, Δn−1

Gn−i (x

Δ

, · · · , x , x, t) =

1

0

i

i−1

fn−i (0, · · · , 0, ξxΔ , xΔ

, · · · , x, t)dξ.

Note that for any t1 , t2 ∈ I, we have |x

Δn−1

Δn−1

(t1 )−x

(t2 )| ≤

t2

n

|xΔ (s)|Δs

t1 n−1

t2

≤

i=0

t1

i

n−1

|xΔ Gn−i (xΔ

, · · · , xΔ, x, t)|Δt+

t2

|g(t)|Δt.

t1

By the Schwarz inequality we have, for n = 2, n−1

|xΔ

n−1

(t1 ) − xΔ

(t2 )| ≤

√

t1 − t2 {|f1 |D M1 + M0 |f2 |E + ||g||}.

or for n ≥ 3, n−1

|xΔ

n−1

(t1 ) − xΔ ≤

(t2 )| n−1

√ t1 − t2 {

|fn−i |D Mi + Mn−2 |f2 |E + |fn |D M0 + ||g||},

i=1,i=n−2

that is,

n−1

|xΔ

n−1

(t1 ) − xΔ

√ (t2 )| ≤ L t1 − t2 ,

where (3.26)

L = max (1 − ζ)C n−2 B + |f2 |E R σ(T ) + C n−2 Bσ(T ) + ||g||, 1 − ηC)B + |f2 |E CB + |fn |D R σ(T ) + C n−2 Bσ(T ) + ||g|| ( C1


and E=

{0}×Πn−2 i=0

2269

2 σ(T ) 2 σ(T ) 1 1 + + −( )Mn−2−i , ( )Mn−2−i ×I. C C σ(T ) σ(T )

In particular, for t ∈ I, we obtain n−1

|xΔ

n−1

(t) − xΔ

(0)| ≤ L σ(T ).

Following the calculations as we did from (3.19) − (3.25), we get n−1

|xΔ

(t)| ≤ N ∗ .

4. EXISTENCE THEOREM In this section we are going to show the existence of periodic solutions for the problem (1.1), (1.2) via coincidence degree theory associated with Lemma C. Our proof utilizes a continuation theorem of Mawhin [9] which we state here for the reader’s convenience. Let X and Z be real Banach spaces and let L : domL ⊂ X → Z be a linear Fredholm mapping, that is, imL(the range of L) is closed and the dimension of kerL(the kernel of L) and codimension of imL are finite. Let Ω be a bounded open subset of X and let ¯ ⊂X →Z N :Ω ¯ that is, if P : X → X be a (not necessarily linear) mapping which is L-compact on Ω, and Q : Z → Z denote bounded linear projections such that imP = kerL, imL = kerQ and if KP.Q = (L | kerP ∩ domL)−1 (I − Q), ¯ → Z is continuous, QN (Ω) ¯ is bounded where I is the identity map on Z, then QN : Ω ¯ → X is completely continuous. The following continuation theorem and KP,QN : Ω has been established. Theorem 3. Let the above assumptions hold and let indL(= dim kerL − codim imL) = 0. Further assume (1) for each λ ∈ (0, 1) and each x ∈ domL ∩ ∂Ω, Lx = λN x, (2) for each x ∈ kerL ∩ ∂Ω, N x ∈ / imL, that is, QN x = 0, (3) dB (JQN | kerL, Ω ∩ kerL, 0) = 0, where dB denotes the Brouwer degree and J : imQ → kerL is any isomorphism.

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Sheng-Ping Wang

¯ such that Lx = N x. Then there exists at least one x ∈ domL ∩ Ω With the aid of Theorem 3, we shall now construct our main result as follows. Theorem A. Suppose that (3.1) − (3.2) (or (3.3) − (3.4) as n = 2) and (3.5) hold. Assume that

σ(T )

(4.1) 0

where R0 = { √ 1

f (0, · · · , 0, R0, t)Δt

σ(T )

+

2

√

σ(T ) }M0 C

0

σ(T )

f (0, · · · , 0, −R0, t)Δt < 0,

+ 1. Then the problem (1.1), (1.2) has a σ(T )-

periodic solution.

n

Proof. Define L : domL ⊂ X → Z by Lx = xΔ , where domL = {x ∈ X : n n−1 x ∈ C Δ (T)}. Define N : X → Z by N x(t) = f (xΔ (t), · · · , xΔ (t), x(t), t). We see that x ∈ kerL if and only if x(t) = x(0) for all t ∈ I if and only if σ(T ) 1 x(s)Δs, for all t ∈ I. Hence dim kerL = 1. And we also note that x(t) = σ(T ) 0 n z ∈ Z is in imL if and only if there exists a solution x ∈ X satisfying xΔ = z(t) if σ(T ) σ(T ) z(s)Δs = 0. Thus we have imL = {z ∈ Z : 0 z(s)Δs = 0}. It and only if 0 is obvious that imL is closed in Z. Note that dim(cokerL) = dim(Z/imL) = 1. Indeed, let [z1 ] and [z2 ] be two σ(T ) zi (s)Δs = 0, i = 1, 2. Hence equivalent classes in cokerL other than imL; then 0 there exists a real constant c = 0 such that z1 − cz2 ∈ imL. Thus dim(Z/imL) = 1. Therefore, indL = 0. Define 2 σ(T )Mi Mi + + 1, Ω = {x ∈ X : |x |I < C σ(T ) Δi

n−1

0 ≤ i ≤ n − 2, |xΔ

|I < N ∗ + 1},

where Mi and N ∗ are given in the Lemma C. Then Ω is a bounded open subset of X ¯ ∩ domL = ∅. Note that N is continuous on Ω ¯ and N (Ω) ¯ is bounded in Z. such that Ω σ(T ) 1 x(s)Δs. Then Q is a continuous projection Define Q : Z → Z by Qx = σ(T ) 0 with imL = im(I −Q), imQ = kerL. Define T : imL → X to be a right inverse of L so that LT z = z for every z ∈ imL and P T = 0, where P : X → X is some projection with imP = kerL. By Arzela-Ascoli theorem, we see that KP,Q N ≡ T (I − Q)N is ¯ completely continuous on Ω. Next we claim that Lx = λN x for every x ∈ ∂Ω ∩ domL and λ ∈ (0, 1). Suppose n n−1 not; then there exists a function x(t) satisfying xΔ = λf (xΔ , · · · , xΔ, x, t) with i i n−1 xΔ (0) = xΔ (σ(T )), 0 ≤ i ≤ n − 1 for some λ ∈ (0, 1) and (xΔ (t), · · · , xΔ (t),


2271

x(t), t) ∈ G, where G = {(x

Δn−1

2 Mi + , · · · , x , x, t) : |x |I ≤ σ(T ) Δi

Δ

n−1

0 ≤ i ≤ n − 2, |xΔ

σ(T )Mi + 1, C

|I ≤ N ∗ + 1, t ∈ I}

n−1

and (xΔ (t0 ), · · · , xΔ (t0 ), x(t0 ), t0 ) ∈ ∂G for some t0 ∈ I. This is impossible, i since by arguments similar to those in the Lemma C, one can see that |xΔ |I ≤ √ n−1 )Mi √Mi + 2 σ(T , 0 ≤ i ≤ n − 2 and |xΔ |I ≤ N ∗ . C σ(T )

2

√

We also see that QN a = 0 for every a ∈ ∂Ω ∩ kerL. Indeed, a(t) = ±( √Mi

σ(T )

σ(T )Mi C

+

+ 1) = ±R0 and by the hypothesis (4.1), 1 QN a = σ(T )

σ(T ) 0

f (0, · · · , 0, ±R0, t)Δt = 0.

Finally we claim dB (JQN | kerL, Ω ∩ kerL, 0) = 0. Here we take J to be an identity operator in Z since imQ = kerL. From the hypothesis (4.1) we see that dB (QN | kerL,Ω ∩ kerL, 0) σ(T ) 1 f (0, · · · , 0, •, t)Δt, (−R0, R0), 0 = 0. = dB σ(T ) 0 ¯ Therefore the problem By Theorem 3, Lx = N x has at least one solution x ∈ Ω. (1.1), (1.2) has a periodic solution. REFERENCES 1. D. R. Anderson, Multiple periodic solutions for a second-order problem on periodic time scales, Nonlinear Anal., 60 (2005), 101-115. 2. J. Bebernes, R. Gaines and K. Schmitt, Existence of periodic solutions for third and fourth order ordinary differential equations via coincidence degree, Ann. Soc. Sci. Brux., Ser. I., 88 (1974), 25-36. 3. M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkha¨ user, Boston, 2001. 4. M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkha¨ user, Boston, 2003. 5. M. Bohner, M. Fan and J. Zhang, Existence of periodic solutions in predator-prey and competition dynamic systems, Nonlinear Anal. Real World Appl., 7 (2006), 1193-1204.

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24. G. Q. Wang and S. S. Cheng, A priori bounds for periodic solutions of a delay Rayleigh equation, Appl. Math. Lett., 12 (1999), 41-44. 25. J. Zhanga, M. Fana and H. Zhuc, Periodic solution of single population models on time scales, Math. Comput. Modelling, 52 (2010), 515-521. 26. K. Zhuang and Z. Wen, Periodic solutions for a delayed population model on time scales, Int. J. Comput. Math. Sci., 4(3) (2010), 166-168. 27. K. Zhuang and Z. Wen, Periodic solutions for a delay model of plankton allelopathy on time scales, Electron. J. Qual. Theory Differ. Equ., 28 (2011), 1-7. Sheng-Ping Wang Holistic Education Center Cardinal Tien College of Healthcare and Management No. 11, Zhongxing Road Sanxing Township Yilan County 26646, Taiwan E-mail: [email protected]