一個反應擴散方程系統之行波解的穩定性研究

行政院國家科學委員會專題研究計畫 成果報告

一個反應擴散方程系統之行波解的穩定性研究

研究成果報告(精簡版)

計 畫 類 別 ： 個別型 計 畫 編 號 ： NSC 98-2115-M-004-001- 執 行 期 間 ： 98 年 08 月 01 日至 99 年 07 月 31 日 執 行 單 位 ： 國立政治大學應用數學學系 計 畫 主 持 人 ： 符聖珍 計畫參與人員： 碩士班研究生-兼任助理人員：李宣緯 處 理 方 式 ： 本計畫可公開查詢

中 華 民 國 99 年 10 月 20 日

Existence of solutions to PBVPs for first-order

impulsive dynamic equations on time scales ?

Sheng-Chen Fu and Yi-Chang Liang

Department of Mathematical Sciences, National Chengchi University, 64, S-2 Zhi-nan Road, Taipei 116, Taiwan

Abstract

In this paper we are concernd with periodic boundary value problems for first-order impulsive dynamic equations on time scales. By using Schaefer’s theorem and Banach’s fixed point theorem we acquire some new existence results.

Key words: Time scale; Periodic boundary value problem; Impulsive dynamic equation

MSC: 34N05

1 Introduction

The theory of dynamic equations on time scales has received a lot of at-tention since it can not only unify, extend, and generalize the theories of differential equations and difference equations but also have various practical applications. For more details about this theory, we refer the readers to [1], [2], and [3]. One of the important research trends is the investigation of impulsive dynamic equations on time scales. Recently, some researchers have focused their attention on periodic boundary value problems (PBVPs for short) for first-order impulsive dynamic equations. For example, Geng, Xu, and Zhu [4] applied the method of upper and lower solutions coupled with monotone it-erative techniques to derive the existence of extremal solutions and Wang [5]

? This work is partially supported by National Science Council of the Republic of China under the contract 98-2115-M-004-001.

Corresponding author: Sheng-Chen Fu.

Email addresses: fu@nccu.edu.tw (Sheng-Chen Fu); 96751003@nccu.edu.tw (Yi-Chang Liang).

used the Guo-Krasnoselskii fixed point theorem to obtain some existence cri-teria for positive solutions. However, to the best of the authors’ knowledge, there is no existence criteria for (not necessarily positive) solutions to PBVPs for first-order impulsive dynamic equations on time scales so far.

Let T be a time scale, i.e., a nonempty closed subset of R, and let 0, T ∈ T. Throughout this paper, [0, T ]T represents an interval on T, i.e., [0, T ]T =

[0, T ] ∩ T. Other types of intervals on T can be represented by a similar way. Let J = [0, σ(T )]_T. Motivated by [6], [7], and the above works, in this paper, we are concerned with the existence of solutions to the following PBVPs for first-order impulsive dynamic equations on T

x∆+ p(t)xσ = f (t, x), t ∈ [0, T ]_T, t 6= tk, k = 1, . . . , m, (1)

x(tk+) − x(tk−) = Ik(x(tk−)), k = 1, . . . , m, (2)

x(0) = x(σ(T )), (3)

where f ∈ C(J × R, R), Ik ∈ C(R, R), p : J → [0, ∞) is rd-continuous and

regressive with p ≡/ 0, and the points tk, k = 1, . . . , m, are right-dense in T

such that 0 < t1 < · · · < tm < T . For convenience, we shall refer to (1)-(2)-(3)

as (NP).

When Ik(x) ≡ 0 for all k = 1, . . . , m, the problem (NP) can be reduced to

the following PBVPs with no impulse effects

x∆+ p(t)xσ = f (t, x), t ∈ [0, T ]_T, x(0) = x(σ(T )),

which has been investigated by several researchers; see for example, [8], [9], [10], [11], and the references cited therein.

PBVPs for first-order impulsive differential equations and difference equa-tions (i.e., the cases T = R and T = Z) have been studied; see for example, [12], [13], [6], [14], [15], [16], [17], [7], [18] for T = R and [19] for T = Z.

Let J0 = [0, t1]T, Jk = (tk, tk+1]T for k = 1, . . . , m − 1, and Jm = (tm, σ(T )]T

and let

P C = {x : J → R| xk ∈ C(Jk), ∀k = 0, . . . , m, and both x(tk+) and x(tk−)

exist such that x(tk−) = x(tk), ∀k = 1, . . . , m},

where xk is the restriction of x to Jk for each k = 0, . . . , m. We introduce

the Banach space X = {x ∈ P C : x(0) = x(σ(T ))} with the norm kxkX =

sup_t∈J|x(t)|.

Definition 1.1 A function x is said to be a solution of (NP) if and only if

x ∈ P CT

C1_{([0, T ]}

We shall apply the well-known Banach’s fixed point theorem and Schaefer’s theorem to establish the existence criteria of solutions for (NP). For readers’ convenience, we provide these two theorems here.

Lemma 1.2 (Banach’s fixed point theorem [20]) A contraction f of a com-plete metric space S has a unique fixed point in S.

Lemma 1.3 (Schaefer’s theorem [20]) Let S ba a normed linear space, and let operator F : S → S be compact. If the set

H(F ) = {x ∈ S : x = µF (x) for some µ ∈ (0, 1)} is bounded, then F has a fixed point in S.

2 Linear problem

In this section we consider the ”linear problem”

x∆+ p(t)xσ = h(t), t ∈ [0, T ]_T, t 6= tk, k = 1, . . . , m,

x(tk+) − x(tk−) = Ik(x(tk−)), k = 1, . . . , m,

x(0) = x(σ(T )).

For convenience, we shall refer to this problem as (LP). Note that (LP) is not really a linear problem since the impulsive functions Ik, k = 1, . . . , m, may or

may not be linear.

The following two basic lemmas will be used later and their proofs can be found in [5].

Lemma 2.1 Suppose that h : J → R is rd-continuous. Then x is a solution of (LP) if and only if x is a solution of

x(t) = Z σ(T ) 0 G(t, s)h(s)∆s + m X k=1 G(t, tk)Ik(x(tk)), t ∈ J, (4) where G(t, s) =          ep(s, t)ep(σ(T ), 0) ep(σ(T ), 0) − 1 , 0 ≤ s ≤ t ≤ σ(T ), ep(s, t) ep(σ(T ), 0) − 1 , 0 ≤ t < s ≤ σ(T ). Lemma 2.2 Let G(t, s) be defined as Lemma 2.1. Then

0 6 G(t, s) 6 ep(σ(T ), 0)

Our existence result for (LP) is as follows.

Theorem 2.3 Suppose that there exist positive constants lk, k = 1, . . . , m,

such that

|Ik(x) − Ik(y)| ≤ lk|x − y| for all x, y ∈ R and k = 1, . . . , m.

If A m X k=1 lk < 1,

then the problem (LP) has a unique solution for any h ∈ P C. Proof. First, we define the operator Ψ : X → X by

Ψx(t) = Z σ(T ) 0 G(t, s)h(s)∆s + m X k=1 G(t, tk)Ik(x(tk)),

so that fixed points of Ψ are solutions of (LP) and vice versa. Next, we claim that Ψ is a contraction mapping. To show this, we consider u, v ∈ X and t ∈ J . It is easy to see that

|(Ψu)(t) − (Ψv)(t)| =      m X k=1 G(t, tk)Ik(u(tk)) − m X k=1 G(t, tk)Ik(v(tk))      ≤ m X k=1 |G(t, tk)||Ik(u(tk)) − Ik(v(tk))| ≤ m X k=1 Alk|u(tk) − v(tk)| ≤ m X k=1 Alkku − vk, and hence kΨu − Ψvk ≤ A m X k=1 lkku − vk.

This means that Ψ is a contraction mapping. Finally, applying Banach’s fixed point theorem, we conclude that Ψ has a unique fixed point x ∈ X so that

(LP) has exactly one solution. 2

3 Nonlinear problem

In this section we study the ”nonlinear problem” (NP). It follows from Lemma 2.1 that x ∈ X is a solution of (NP) if and only if it satisfies

x(t) = Z σ(T ) 0 G(t, s)f (s, x(s))∆s + m X k=1 G(t, tk)Ik(x(tk)), t ∈ J.

Introduce the operator Φ : X → X by the formula Φx(t) = Z σ(T ) 0 G(t, s)f (s, x(s))∆s + m X k=1 G(t, tk)Ik(x(tk)), t ∈ J.

Obviously, fixed points of Φ are solutions of (NP) and conversely.

Definition 3.1 Let F be a subset of PC. We say that F is quasiequicontinuous on J if for every  > 0 there exists δ > 0 such that if f ∈ F and k = 0, . . . , m, then

|f (t) − f (˜t)| < , ∀t, ˜t ∈ Jk and |t − ˜t| < δ.

In order to show that Φ is compact, we need the following compactness criteria.

Lemma 3.2 A set F ⊂ P C is relatively compact on J if F is bounded and quasiequicontinuous on J .

Proof. Let {xn} be a sequence in F . From assumption, we know that {xn}

is uniformly bounded and equicontinuous on J0. By Arzela’s theorem, there

is a convergent subsequence {x(1)

n } of {xn} on J0. Since {x(1)n } is uniformly

bounded and equicontinuous on J1, it follows from Arzela’s theorem that there

is a convergent subsequence {x(2)

n } of {x(1)n } on J1. Continuing this process,

we can get a convergent subsequence {x(m+1)

n } of {x(m)n } on Jm. It is clear

that {x(m+1)_n } is a convergent subsequence of {xn} on J. Hence F is relatively

compact. 2

Lemma 3.3 Φ : X → X is compact.

Proof. Let D be a bounded subset of X. The continuity of f and Ik implies

that there exist positive constants M and Mk such that |f (t, x(t))| ≤ M and

|Ik(xtk)| ≤ Mk for all x ∈ D, t ∈ J , and k = 1, . . . , m. Hence we have

|Φx(t)| =      Z σ(T ) 0 G(t, s)f (s, x(s))∆s + m X k=1 G(t, tk)Ik(x(tk))      ≤ Z σ(T ) 0 |G(t, s)||f (s, x(s))|∆s + m X k=1 |G(t, tk)||Ik(x(tk))| ≤ AM σ(T ) + A m X k=1 Mk.

Let x ∈ D and t, ˜t ∈ Jk, where k = 0, . . . , m. We have that |Φx(t) − Φx(˜t)| ≤ MR˜t 0 |G(t, s) − G(˜t, s)|∆s + M Rt ˜ t |G(t, s) − G(˜t, s)|∆s + MRσ(T ) t |G(t, s) − G(˜t, s)|∆s + Pm k=1   G(t, tk) − G(˜t, tk)   Mk = M ARt˜ 0   ep(s, t) − ep(s, ˜t)   ∆s + M ηRt ˜ t   ep(s, t)ep(σ(T ), 0) − ep(s, ˜t)   ∆s + M ηRσ(T ) t   ep(s, t) − ep(s, ˜t)   ∆s + APj−1 k=1   ep(tk, t) − ep(tk, ˜t)   Mk + ηPm k=j   ep(tk, t) − ep(tk, ˜t)   Mk,

where η = 1/(ep(σ(T ), 0) − 1). It follows that |Φx(t) − Φx(˜t)| → 0 uniformly

for x ∈ D as |t − ˜t| → 0. So Φ(D) is quasiequicontinuous on J . By Lemma

3.2, Φ is compact. This completes the proof. 2

Now we are in a position to establish the existence theorems for the problem (NP) by using fixed point theorems.

Theorem 3.4 Suppose that there exist positive constants lk, k = 1, . . . , m,

such that

|Ik(u) − Ik(v)| ≤ lk|u − v| for all u, v ∈ R,

and suppose also that there exists a positive constant l such that |f (t, u) − f (t, v)| ≤ l|u − v| for all t ∈ J and u, v ∈ R. If A σ(T )l + m X k=1 lk ! < 1, then the problem (NP) has a unique solution.

Proof. For any u, v ∈ X and t ∈ J , we can easily get that

|Φu(t) − Φv(t)| ≤ A σ(T )l + m X k=1 lk ! ku − vk, and hence kΦu − Φvk ≤ A σ(T )l + m X k=1 lk ! ku − vk.

This means that Φ is a contraction mapping. By Banach’s fixed point theorem, Φ has a unique fixed point which is the unique solution of (NP). This completes

Theorem 3.5 Suppose that there exist positive constants l and ck, k = 1, ..., m,

such that

|f (t, x)| ≤ l|x| for all t ∈ J and x ∈ R (5)

and

|Ik(x)| ≤ ck for all x ∈ R and k = 1, . . . , m. (6)

If

lAσ(T ) < 1, (7)

then the problem (NP) has at least one solution.

Proof. Let x ∈ X and t ∈ J . Suppose that x is a solution of x = µΦx for some µ ∈ (0, 1). Using (5) and (6), we have

|x(t)| =      µ Z σ(T ) 0 G(t, s)f (s, x(s))∆s + µ m X k=1 G(t, tk)Ik(x(tk))      ≤ µ Z σ(T ) 0 |G(t, s)||f (s, x(s))|∆s + µ m X k=1 |G(t, tk)||Ik(x(tk))| ≤ µAlkxkσ(T ) + µA m X k=1 ck and hence kxk ≤ µAlkxkσ(T ) + µA m X k=1 ck ≤ Alkxkσ(T ) + A m X k=1 ck.

Together with (7), we obtain

kxk ≤ A

Pm

k=1ck

1 − Alσ(T ).

This implies that all solutions of x = µΦx are uniformly bounded independent of µ ∈ (0, 1). From Lemma 1.3, Φ has a fixed point. This completes the

proof. 2

Theorem 3.6 Suppose that there exist positive constants c and ck, k = 1, ..., m,

such that

|f (t, x)| ≤ c for all t ∈ J and x ∈ R (8)

and

|Ik(x)| ≤ lk|x| for all x ∈ R and k = 1, ..., m. (9)

If A m X k=1 lk < 1, (10)

Proof. Let x ∈ X and t ∈ J . Suppose that x is a solution of x = µΦx for some µ ∈ (0, 1). Using (8) and (9), we have

|x(t)| =      µ Z σ(T ) 0 G(t, s)f (s, x(s))∆s + µ m X k=1 G(t, tk)Ik(x(tk))      ≤ µ Z σ(T ) 0 |G(t, s)||f (s, x(s))|∆s + µ m X k=1 |G(t, tk)||Ik(x(tk))| ≤ µAcσ(T ) + µA m X k=1 lkkxk, and hence kxk ≤ µAcσ(T ) + µA m X k=1 lkkxk ≤ Acσ(T ) + A m X k=1 lkkxk.

Together with (10), we obtain that

kxk ≤ Acσ(T )

1 − APm

k=1lk

.

This implies that all solutions of x = µΦx are uniformly bounded independent of µ ∈ (0, 1). Hence it follows from Lemma 1.3 that Φ has a fixed point. So

the proof is complete. 2

When all impulsive functions are linear, we have the following existence result.

Theorem 3.7 For each k = 1, . . . , m, let Ik(x) = lkx, where lk is a constant.

Suppose that the following conditions hold:

(a) |f (t, x)| ≤ c for all (t, x) ∈ J × R, for some positive constant c, (b)

m

Y

k=1

bk 6= ep(σ(T ), 0), where bk = lk+ 1.

Then the problem (NP) has at least one solution.

Proof. In this case, the problem (NP) can be rewritten as

x4+ p(t)xσ = f (t, x), t ∈ [0, T ]_T, t 6= tk, k = 1, . . . , m,

x(tk+) = bkx(tk), k = 1, . . . , m, (11)

We first consider the special case: bk0 = 0 for some 1 ≤ k0 ≤ m. Let y(t) = ep(t, 0)x(t). Then y∆(t) = ep(t, 0)f (t, ep(0, t)y(t)), t ∈ [0, T ]T, t 6= tk, k = 1, . . . , m, y(tk+) = bky(tk), k 6= k0, (12) y(tk0+) = 0, y(0) = y(σ(T )).

We claim that the initial value problem

y∆(t) = ep(t, 0)f (t, ep(0, t)y(t)), t ∈ Jk0,

y(tk0+) = 0, (13)

has at least one solution. To show this, we define an operator Lk0 : C(Jk0) →

C(Jk0) by

(Lk0y)(t) =

Z t

t_k0

ep(s, 0)f (s, ep(0, s)y(s))∆s

so that the fixed points of Lk0 are solutions to (13). Then Lk0 is compact. To

see this, let D ⊆ C(Jk0) be a bounded set. For any y ∈ D and t ∈ Jk0, we

have |(Lk0y)(t)| =      Z t t_k0 ep(s, 0)f (s, ep(0, s)y(s))∆s      ≤ Z t t_k0 |ep(s, 0)||f (s, ep(0, s)y(s))|∆s ≤ cep(σ(T ), 0)(tk0+1− tk0).

This implies that Lk0(D) is uniformly bounded. Also, if t, ˜t ∈ Jk0 and y ∈ D,

then

|(Lk0y)(t) − (Lk0y)(˜t)| ≤ cep(σ(T ), 0)|t − ˜t| → 0,

uniformly for y ∈ D as |t − ˜t| → 0. This implies that Lk0(D) is equicontinuous

on Jk0. Hence Lk0 is compact.

Let µ ∈ (0, 1). We consider the equation

Suppose that y ∈ C(Jk0) is a solution of (14). Then |y(t)| =      µ Z t t_k0 ep(s, 0)f (s, ep(0, s)y(s))∆s      ≤ Z t t_k0 |ep(s, 0)||f (s, ep(0, s)y(s))|∆s ≤ cep(σ(T ), 0)σ(T ),

and hence kyk ≤ cep(σ(T ), 0)σ(T ). It follows that all solutions of y = µLk0y

are bounded independent of µ ∈ (0, 1). From Lemma 1.3, Lk0 has a fixed point.

Hence (13) has at least one solution, saying yk0, on Jk0. This determines the

value of yk0(tk0+1) that we use as the initial value for the following problem

y∆(t) = ep(t, 0)f (t, ep(0, t)y(t)), t ∈ Jk0+1,

y(tk0+1+) = bk0+1yk0(tk0+1). (15)

Similarly, we can get a solution yk0+1on Jk0+1for (15). Continuing this process,

we know that the initial value problem

y∆(t) = ep(t, 0)f (t, ep(0, t)y(t)), t ∈ Jj,

y(tj+) = bjyj−1(tj).

has a solution yj on Jj for each j = k0 + 2, . . . , m. Also, the initial value

problem

y∆(t) = ep(t, 0)f (t, ep(0, t)y(t)), t ∈ J0,

y(0) = ym(σ(T )).

has a solution y0 on J0. As before, we know that the initial value problem

y∆(t) = ep(t, 0)f (t, ep(0, t)y(t)), t ∈ Jj,

y(tj+) = bjyj−1(tj).

has a solution yj on Jj for each j = 1, . . . , k0− 1.

Let y =                    y0, on J0, y1, on J1, .. . ym, on Jm.

Now we consider the only other case: bk 6= 0 for all k = 1, . . . , m. Let x(t)

be any solution of (11). Set

y(t) = x(t) Y

0≤tk<t

b−1_k .

For all k = 1, . . . , m, we have y(tk+) = bkx(tk) Y 0≤ti≤tk b−1_i = x(tk) Y 0≤ti<tk b−1_i = y(tk), y(tk−) = x(tk) Y 0≤ti<tk b−1_k = y(tk).

This shows that y(t) is continuous on J . Furthmore, y(t) satisfies y∆(t) + p(t)y(σ(t)) = F (t, y(t)), t ∈ [0, T ]_T, y(0) = y(σ(T )) m Y k=1 bk, (16) where F (t, y(t)) = f (t, y(t) Y 0≤tk<t bk) Y 0≤tk<t b−1_k .

It follows that (16) has a solution if and only if the integral equation y(t) = Z σ(T ) 0 ˜ G(t, s)F (s, y(s))∆s is solvable. Here, ˜ G(t, s) =        ηep(σ(T ), 0)ep(s, t), 0 ≤ s ≤ t ≤ σ(T ), η m Y k=1 bkep(s, t), 0 ≤ t < s ≤ σ(T ), where η = 1 ep(σ(T ), 0) − m Y k=1 bk .

Define the operator B : C(J ) → C(J ) by By =

Z σ(T )

0

˜

G(t, s)F (s, y(s))∆s.

It is easy to show that B is compact. Let µ ∈ (0, 1) and y ∈ C(J ). Suppose that y is a solution of

on J . Then |y(t)| ≤ Z σ(T ) 0 | ˜G(t, s)||F (s, y(s))|∆s ≤ c1c2σ(T ), where c1 = sup    c Y 0≤tk<t |bk|−1 : t ∈ J    , c2 = ηep(σ(T ), 0) sup ( _m Y k=1 bk, 1 ) .

Hence kyk ≤ c1c2. This implies that all the solutions of (17) are bounded

independent of µ ∈ (0, 1). It follows from Lemma 1.3 that B has a fixed point. Therefore (11) has at least one solution. 2

Theorem 3.8 Suppose that the following conditions hold: (a) lim |x|→∞ f (t, x) x = 0 uniformly for t ∈ J , (b) lim |x|→∞ Ik(x) x = 0 for all k = 1, . . . , m.

Then the problem (NP) has at least one solution.

Proof. Let HΦ = {x ∈ X : x = µΦx for some µ ∈ (0, 1)}. Then HΦ is bounded. Indeed, if HΦ is unbounded, then there exist sequences {xn}∞n=1 in

X and {µn}∞n=1 in (0, 1) such that kxnk ≥ n and

x4_n(t) + p(t)xn(σ(t)) = µnf (t, xn(t)), t ∈ [0, T ]T, t 6= tk, k = 1, . . . , m,

xn(tk+) − xn(tk−) = µnIk(xn(tk)), k = 1, . . . , m,

xn(0) = xn(σ(T )).

Now we let vn= xn/kxnk. Then kvnk = 1 and vn satisfies

v4_n(t) + p(t)vn(σ(t)) = gn(t), , t ∈ [0, T ]T, t 6= tk, k = 1, . . . , m, vn(tk+) − vn(tk−) = θn,k, k = 1, . . . , m, vn(0) = vn(σ(T )), where gn(t) = µnf (t, xn(t)) kxnk and θn,k = µnIk(xn(tk)) kxnk . By Lemma 2.1, we get vn(t) = Z σ(T ) 0 G(t, s)gn(s)∆s + m X k=1 G(t, tk)θn,k, t ∈ J.

From assumptions (a) and (b), we have |gn(t)| ≤

|f (t, xn(t))|

kxnk

→ 0, uniformly for t ∈ J and

|θn,k| ≤ |Ik(xn(tk))| kxnk → 0 , k = 1, . . . , m, as n → ∞, so that |vn(t)| ≤ A ( Z σ(T ) 0 |gn(s)|∆s + m X k=1 |θn,k| ) → 0,

uniformly for t ∈ J as n → ∞. Hence kvnk → 0 as n → ∞, which contradicts

the fact that kvnk=1. From Lemma 1.3, the problem (NP) has at least one

solution. Therefore the proof is complete. 2

The following corollaries can be immediately obtained from Theorem 3.8. Corollary 3.9 (Bounded case) Assume that the nonlinearity f is bounded and that the impulsive functions Ik, k = 1, . . . , m, are bounded. Then the nonlinear

problem (NP) has at least one solution.

Corollary 3.10 (Sublinear growth) Suppose that there exist a ∈ P C, b ∈ R and α ∈ [0, 1) such that

|f (t, x)| ≤ a(t) + b|x|α

for all t ∈ J and x ∈ R,

and suppose also that there exist positive constants ak, bk∈ R, and αk ∈ [0, 1)

such that

|Ik(x)| ≤ ak+ bk|x|αk for all x ∈ R and k = 1, . . . , m.

Then the problem (NP) has at least one solution.

4 Examples

Example 5.1 Let T = [0, 1] ∪ Z. We consider the following PBVP on T x∆+ (t + 1)xσ(t) = x 10et, t ∈ [0, 4]T, t 6= 1 2, x ₁ 2+  − x ₁ 2−  = 1 4sin  x ₁ 2−  , x(0) = x(σ(4)).

Let

p(t) = t + 1, f (t, x) = x

10et, and I(x) =

1 4sin x. It is easy to see that

|f (t, u) − f (t, v)| ≤ 1

10|u − v|, for all t ∈ [0, σ(4)]T and u, v ∈ R, and

|I(u) − I(v)| ≤ 1

4|u − v| for all u, v ∈ R. Also, by a simple computation, we get A = 360e32/(360e

3 2 − 1) and hence A  σ(4) 1 10 + 1 4  = 3 4A < 1.

Hence by Theorem 3.4 the PBVP has at least one solution.

Example 5.2 Let T = [0,1

2] ∪ 2N

0_{. We consider the following PBVP on T}

x∆+ p(t)xσ(t) = f (t, x), t ∈ [0, 4]_T, t 6= 1 4, x(1 4+) − x( 1 4−) = I(x( 1 4−)), x(0) = x(σ(4)), where p(t) =      t, t ∈ [0,1₂], 1, t ∈ 2N0_, , f (t, x) = 2 sin t x2_{+ 1}, and I(x) = 1 24x.

It is easy to see that

|f (t, x)| ≤ 2 for all t ∈ [0, σ(4)]_{T and x ∈ R,}

and

|I(x)| ≤ 1

24|x| for all x ∈ R. By a simple computation, we get A = 75e18/(75e

1

8− 2) and so A/24 < 1. Then

Example 5.3 Let T = N2

0∪ [6, 8]. We consider the following PBVP on T

x∆+ p(t)x(σ(t)) = f (t, x), t ∈ [0, 8]_T, t 6= 7, x(7+) − x(7−) = I(x(7−)), x(0) = x(σ(8)), where p(t) =      1, t ∈ {0, 1, 4}, t, t ∈ [6, 8], , f (t, x) = x

t + 18, and I(x) = sin x. It is easy to see that

|f (t, x)| ≤ 1

18|x| for all t ∈ [0, σ(8)]T and x ∈ R, and

|I(x)| ≤ 1 for all x ∈ R.

Also, by a simple computation, we get A = 216e14_/(216e14 _{− 1) and so}

Aσ(8)/18 = A/2 < 1. Then by Theorem 3.5, the PBVP has at least one solution.

Example 5.4 Let T be a time scale and let 0, T ∈ T. We consider the following PBVP on T x∆+ xσ = ex1 sin t, t ∈ [0, T ] T, t 6= tk, k = 1, . . . , m, x(tk+) − x(tk−) = x(tk−) 1 2, k = 1, . . . , m, x(0) = x(σ(T )),

where tk ∈ (0, T )T are right-dense for all k = 1, . . . , m. Let f (t, x) = e

1 x sin t

and Ik(x) = x

1

2. Then it is easy to see that

lim |x|→∞ f (t, x) x = 0 and |x|→∞lim Ik(x) x = 0.

Hence it follows from Theorem 3.8 that the PBVP has least one solution .

References

[1] R.P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications, Results Math. 35 (1999) 3-22.

[2] M. Bohner and A. Peterson, Dynamic equations on time scales, Birkh¨auser, Boston, 2001.

[3] M. Bohner and A. Peterson, Advances in dynamic equations on time scales, Birkh¨auser, Boston, 2003.

[4] F. Geng, Y. Xu, and D. Zhu Periodic boundary value problems for first-order impulsive dynamic equations on time scales, Nonlinear Anal. 69 (2008) 4074-4087.

[5] D.B. Wang, Positive solutions for nonlinear first-order periodic boundary value problems of impulsive dynamic equations on time scales, Comput. Math. Appl. 56 (2008) 1496-1504.

[6] J.J. Nieto, Basic theory for nonresonance impulsive periodic problems of first order, J. Math. Anal. Appl. 205 (1997) 423-433.

[7] J. Li, J.J. Nieto, and J. Shen, Impulsive periodic boundary value problems of first-order differential equations, J. Math. Anal. Appl. 325 (2007) 226-236. [8] A. Cabada, Extremal solutions and Green’s functions of higher order periodic

boundary value problems in time scales, J. Math. Anal. Appl. 290 (2004) 35-54. [9] Q. Dai and C.C. Tisdell, Existence of solutions to first-order dynamic boundary

valued problems, Int. J. Differ. Equ. 1 (2006) 1-17.

[10] J.P. Sun and W.T. Li, Existence of solutions to nonlinear first-order PBVPs on time scales, Nonlinear Anal. 67 (2007) 883-888.

[11] J.P. Sun and W.T. Li, Existence and multiplicity of positive solutions to nonlinear first-order PBVPs on time scales, Comput. Math. Appl. 54 (2007) 861-871.

[12] D.D. Bainov and P.S. Simeonov, Systems with impulse effect: stability theory and applications, Horwood, Chichester, 1989.

[13] D.D. Bainov and P.S. Simeonov, Impulsive differential equations: periodic solutions and applications, Longman Scientific and Technical, Harlow, 1993. [14] D. Franco and J.J. Nieto, Maximum principles for periodic impulsive first order

problems, J. Comput. Appl. Math. 88 (1998) 144-159.

[15] D. Franco, J.J. Nieto, E. Liz, and Y.V. Rogovchenko, A contribution to the study of functional differential equations with impulses, Math. Nachr. 218 (2000) 49-60.

[16] J.J. Nieto, Impulsive resonance periodic problems of first order, Appl. Math. Lett. 15 (2002) 489-493.

[17] J.J. Nieto, Periodic boundary value problems for first order impulsive ordinary differential equations, Nonlinear Appl. 51 (2002) 1223-1232.

[18] Y. Liu, Positive solutions of periodic boundary value problems for nonlinear first-order impulsive differential equations, Nonlinear Anal. 70 (2009) 2106-2122.

[19] Z. He and X. Zhang, Monotone iterative technique for first order impulsive difference equations with periodic boundary conditions, Applied. Math. Comput. 156 (2004) 605-620.

[20] D.R. Smart, Fixed point theorems, Cambridge University Press, Cambridge, 1980.

98 年度專題研究計畫研究成果彙整表

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