Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2009, Article ID 907368, 16 pages
doi:10.1155/2009/907368
Research Article
Existence of Weak Solutions for
Second-Order Boundary Value Problem of
Impulsive Dynamic Equations on Time Scales
Hongbo Duan and Hui Fang
Department of Applied Mathematics, Kunming University of Science and Technology,
Kunming, Yunnan 650093, China
Correspondence should be addressed to Hui Fang, kmustfanghui@hotmail.com
Received 9 April 2009; Accepted 28 June 2009
Recommended by Victoria Otero-Espinar
We study the existence of weak solutions for second-order boundary value problem of impulsive
dynamic equations on time scales by employing critical point theory.
Copyright q 2009 H. Duan and H. Fang. 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.
1. Introduction
Consider the following boundary value problem:
tj , j 1, 2, . . . , p,
−uΔΔ t fσt, uσ t, t ∈ 0, T T , t /
u tj − u t−j Aj u t−j , j 1, 2, . . . , p,
1.2
uΔ tj − uΔ t−j Bj uΔ t−j
1.3
Ij u t−j ,
u0 0 uT ,
j 1, 2, . . . , p,
1.1
1.4
where T is a time scale, 0, T T : 0, T ∩ T, σ0 0 and σT T, f : 0, T T × R → R
is a given function, Ij ∈ CR, R, {Aj }, {Bj } are real sequences with Bj 1 Aj −1 − 1 and
p
|Ak | < 1, the impulsive points tj ∈ 0, T T are right-dense and 0 t0 < t1 < · · · < tp <
k1
tp 1 T, limh → 0 uΔ tj h and limh → 0 uΔ tj − h represent the right and left limits of uΔ t
at t tj in the sense of the time scale, that is, in terms of h > 0 for which tj h, tj − h ∈ 0, T T ,
whereas if tj is left-scattered, we interpret uΔ t−j uΔ tj and ut−j utj .
2
Advances in Difference Equations
The theory of time scales, which unifies continuous and discrete analysis, was first
introduced by Hilger 1. The study of boundary value problems for dynamic equations on
time scales has recently received a lot of attention, see 2–16. At the same time, there have
been significant developments in impulsive differential equations, see the monographs of
Lakshmikantham et al. 17 and Samoı̆lenko and Perestyuk 18. Recently, Benchohra and
Ntouyas 19 obtained some existence results for second-order boundary value problem of
impulsive differential equations on time scales by using Schaefer’s fixed point theorem and
nonlinear alternative of Leray-Schauder type. However, to the best of our knowledge, few
papers have been published on the existence of solutions for second-order boundary value
problem of impulsive dynamic equations on time scales via critical point theory. Inspired and
motivated by Jiang and Zhou 10, Nieto and O’Regan 20, and Zhang and Li 21, we study
the existence of weak solutions for boundary value problems of impulsive dynamic equations
on time scales 1.1–1.4 via critical point theory.
This paper is organized as follows. In Section 2, we present some preliminary results
concerning the time scales calculus and Sobolev’s spaces on time scales. In Section 3, we
construct a variational framework for 1.1–1.4 and present some basic notation and results.
Finally, Section 4 is devoted to the main results and their proof.
2. Preliminaries about Time Scales
In this section, we briefly present some fundamental definitions and results from the calculus
on time scales and Sobolev’s spaces on time scales so that the paper is self-contained. For
more details, one can see 22–25.
Definition 2.1. A time scale T is an arbitrary nonempty closed subset of R, equipped with the
topology induced from the standard topology on R.
For a, b ∈ T, a < b, a, bT : a, b ∩ T, a, bT : a, b ∩ T.
Definition 2.2. One defines the forward jump operator σ : T → T, the backward jump
operator ρ : T → T, and the graininess μ : T → R 0, ∞ by
σt : inf{s ∈ T : s > t},
ρt : sup{s ∈ T : s < t},
μt σt − t
for t ∈ T,
2.1
respectively. If σt t, then t is called right-dense otherwise: right-scattered, and if ρt t,
then t is called left-dense otherwise: left-scattered. Denote yσ t yσt.
Definition 2.3. Assume f : T → R is a function and let t ∈ T. Then one defines f Δ t to be the
number provided it exists with the property that given any ε > 0, there is a neighborhood
U of t i.e., U t − δ, t δ ∩ T for some δ > 0 such that
fσt − fs − f Δ tσt − s ≤ ε|σt − s| ∀s ∈ U.
2.2
In this case, f Δ t is called the delta or Hilger derivative of f at t. Moreover, f is said to be
delta or Hilger differentiable on T if f Δ t exists for all t ∈ T.
Advances in Difference Equations
3
Definition 2.4. A function f : T → R is said to be rd-continuous if it is continuous at rightdense points in T and its left-sided limits exist finite at left-dense points in T. The set of
rd-continuous functions f : T → R will be denoted by Crd T.
As mentioned in 24, the Lebesgue μΔ -measure can be characterized as follows:
σti − ti δti ,
μΔ λ
2.3
i∈I
where λ is the Lebesgue measure on R, and {ti }i∈I is the at most countable set of all
right-scattered points of T. A function f which is measurable with respect to μΔ is called
Δ-measurable, and the Lebesgue integral over a, bT is denoted by
b
ftΔt :
a,bT
a
2.4
ftdμΔ .
The Lebesgue integral associated with the measure μΔ on T is called the Lebesgue delta
integral.
Lemma 2.5 see 24, Theorem 2.11. If f, g : a, bT → R are absolutely continuous functions
on a, bT , then f · g is absolutely continuous on a, bT and the following equality is valid:
a,bT
b
ftg Δ tΔt ftgt a −
a,bT
f Δ tg σ tΔt.
2.5
p
For 1 < p < ∞, the Banach space LΔ may be defined in the standard way, namely,
p
LΔ a, bT :
f : a, bT −→ R | f is Δ-measurable and
ftp Δt < ∞ ,
b
2.6
a
equipped with the norm
f
p
LΔ
:
1/p
ftp Δt
b
.
2.7
a
Let HΔ1 a, bT be the space of the form
HΔ1 a, bT : WΔ1,2 a, bT : f : 0, T T −→ R | f is absolutely continuous on a, bT , f Δ ∈ L2Δ a, bT 2.8
4
Advances in Difference Equations
its norm is induced by the inner product given by
f, g
b
HΔ1
:
f Δ tg Δ tΔt
a
b
2.9
ftgtΔt,
a
for all f, g ∈ HΔ1 a, bT .
Let Ca, bT denote the linear space of all continuous function f : a, bT → R with
the maximum norm fC maxt∈a,bT |ft|.
Lemma 2.6 see 24, Corollary 3.8. Let {xm } ⊂ HΔ1 a, bT , and x ∈ HΔ1 a, bT . If {xm }
converges weakly in HΔ1 a, bT to x, then {xm } converges strongly in Ca, bT to x.
Lemma 2.7 Hölder inequality 25, Theorem 3.1. Let f, g ∈ Crd a, b, p > 1 and q be the
conjugate number of p. Then
ftgtΔt ≤
1/p
ftp Δt
b
b
a
a
1/q
gtq Δt
b
.
2.10
a
When p q 2, we obtain the Cauchy-Schwarz inequality.
For more basic properties of Sobolev’s spaces on time scales, one may refer to Agarwal
et al. 24.
3. Variational Framework
In this section, we will establish the corresponding variational framework for problem 1.1–
1.4.
Let Γj tj , tj 1 T , and
⎧
⎨ft,
fΓj t :
⎩f t ,
j
t ∈ t j , tj 1 T ,
t tj ,
3.1
for j 0, 1, . . . , p.
Now we consider the following space:
H : f : 0, T T −→ R | f is continuous from left at each tj , f tj exists,
fΓj is absolutely continuous on Γj , the delta derivative of fΓj ∈ L2Δ tj , tj 1 T ,
f satisfies the condition 1.2 for all j 0, 1, . . . , p, f0 fT 0 ,
3.2
Advances in Difference Equations
5
its norm is induced by the inner product given by
f, g
H
:
p
tj
j0
tj
1
fΓΔj tgΓΔj tΔt,
∀f, g ∈ H.
3.3
That is
⎛
f
H
⎝
p
tj
j0
tj
1
⎞1/2
Δ 2 ⎠
,
fΓj t Δt
3.4
for any f ∈ H.
First, we give some lemmas which are useful in the proof of theorems.
p
Lemma 3.1. If k1 |Ak | < 1, then for any x ∈ H, supt∈0,T T |xt| ≤ R0 xH , where R0 p
T 1/2 /1 − k1 |Ak |.
Proof. For any x ∈ H and t ∈ tj , tj 1 T , j 0, 1, . . . , p, we have
|xt| xt − x tj
xt − x tj
≤
t
tj
t
tj
≤
xΓΔk sΔs
Δ xΓj sΔs
p
tk
k0
tk
1
x t1 − xt0 x tj − · · · − x t1
j−1
− Ak 1 x t k 1 k0
j−1
− x tk 1 − x t k
k0
j−1
tk
k0
tk
j−1
tk
k0
tk
Δ xΓk sΔs
1
xΓΔk sΔs
j−1
Ak 1 x
k0
1
Δ xΓk sΔs
j−1
t−k 1
j−1
|Ak 1 |x t−k 1 3.5
k0
|Ak 1 |x t−k 1 k0
≤ T 1/2 xH
p
|Ak | sup |xt|,
k1
t∈0,T T
which implies that
sup |xt| ≤ R0 xH ,
t∈0,T T
Lemma 3.2. H is a Hilbert space.
∀x ∈ H.
3.6
6
Advances in Difference Equations
Proof. Let {uk }∞
k1 be a Cauchy sequence in H. By Lemma 3.1, we have
f
HΔ1 tj ,tj 1 T tj
1
tj
tj
≤
tj
≤ 1
1
Δ 2
fΓj t Δt
tj
1/2
f
1/2
2
fΓj t Δt
tj
Δ 2
fΓj t Δt
R20 T
1
R20
H
tj
1
− tj
1/2
f
2
H
3.7
.
Set
j
uk t
⎧
⎨uk t,
: uk Γj :
⎩u t ,
k j
t ∈ tj , t j 1 T ,
3.8
t tj ,
1
for j 0, 1, . . . , p, k 1, 2, . . . . Then {uk }∞
k1 be a Cauchy sequence in HΔ tj , tj 1 T , for
j
j
j 0, 1, . . . , p. Therefore, there exists a uj ∈ HΔ1 tj , tj 1 T , such that {uk } converges to uj in
HΔ1 tj , tj 1 T , j 0, 1, . . . , p. It follows from Lemma 2.6 that
Ctj , tj 1 T , that is,
j
uk
− uj Ctj ,tj
1 T → 0 as k →
j lim uk tj uj tj ,
j
{uk }
converges strongly to uj in
∞ for all j 0, 1, . . . , p. Hence, we have
j−1 lim uk
k→ ∞
k→ ∞
tj uj−1 tj .
3.9
Noting that
j lim uk tj lim uk tj lim 1
k→ ∞
k→ ∞
1
k→ ∞
j−1 Aj lim uk tj
k→ ∞
Aj uk t−j lim 1
k→ ∞
1
Aj u
j−1
Aj uk t j
tj ,
3.10
we have
uj tj 1
Aj uj−1 tj ,
j 0, 1, . . . , p.
3.11
Set
⎧
⎨uj t,
t ∈ tj , tj 1 T , j 0, 1, . . . , p,
ut :
⎩uj−1 t , t t , j 0, 1, . . . , p.
j
j
3.12
Advances in Difference Equations
7
Then we have
u tj uj t j uj t j 1
Aj uj−1 tj 1
uΓj u ,
j
Aj u tj 1
Aj u t−j ,
3.13
j 0, 1, . . . , p.
Thus u ∈ H. Noting that
uk − uH
⎡
p
⎣
tj
j0
tj
1
⎤1/2
2
j Δ
⎦
uk t − uΔ
Γj t Δt
2
HΔ1 tj ,tj 1 T j0
we have uk converges to u in H as k →
Lemma 3.3. If
p
k1
3.14
⎤1/2
⎡
p
j
u k − uj
≤⎣
⎦
,
∞. The proof is complete.
|Ak | < 1, then for any u ∈ H,
p
tj
j0
tj
1
σ 2
uΓj t Δt ≤ R20 T u2H ,
3.15
where R0 is given in Lemma 3.1.
Proof. For any u ∈ H, t ∈ tj , tj 1 T , by Lemma 3.1, we have
σ uΓj t ≤ R0 uH ,
j 0, 1, . . . , p,
3.16
which implies that
p
tj
j0
tj
1
σ 2
uΓj t Δt ≤ R20 T u2H .
3.17
The proof is complete.
For any u ∈ H satisfying 1.1–1.4, take v ∈ H and multiply 1.1 by vΓσj , then
integrate it between tj and tj 1 :
−
tj
tj
1
uΔΔ tvΓσj tΔt tj
tj
1
fσt, uσ tvΓσj tΔt.
3.18
8
Advances in Difference Equations
The first term is now
tj
tj
1
uΔΔ tvΓσj tΔt uΔ t−j 1 vΓj t−j 1 − uΔ tj vΓj tj −
tj
1
tj
uΔ tvΓΔj tΔt.
3.19
Hence, one gets
−
p
tj
j0
tj
1
uΔΔ tvΓσj tΔt
p !
"
uΔ tj vΓj tj − uΔ t−j 1 vΓj t−j 1
j0
p !
"
uΔ tj v tj − uΔ t−j v t−j
p
tj
j0
tj
j1
p !
1
" Aj uΔ tj − uΔ t−j v t−j
j1
Aj Ij u t−j v t−j
p
1
j1
p
tj
j0
tj
1
p
tj
j0
tj
1
1
uΔ tvΓΔj tΔt
uΔ tvΓΔj tΔt
p
tj
j0
tj
1
3.20
uΔ tvΓΔj tΔt
uΔ tvΓΔj t Δt,
for all u, v ∈ H. Then we have
p
tj
j0
tj
1
u
Δ
tvΓΔj tΔt
−
p
tj
j0
tj
1
σ
fσt, u
tvΓσj tΔt
p
1
Aj Ij u t−j v t−j 0,
j1
3.21
for all u, v ∈ H.
This suggests that one defines ϕ : H → R, by
1
2 j0
p
ϕu where Ft, x #x
0
tj
tj
1
p
Δ 2
uΓj t Δt −
j0
ft, sds, and I j 1
tj
tj
1
F σt, uσΓj t Δt
Aj Ij , ∀j 1, 2, . . . , p.
p
utj j1
0
I j sds,
3.22
Advances in Difference Equations
9
By a standard argument, one can prove that the functional ϕ is continuously
differentiable at any u ∈ H and
p
ϕ u, v tj
j0
tj
1
Δ
uΔ
Γj tvΓj tΔt −
p
tj
j0
tj
1
f σt, uσΓj t vΓσj tΔt
p
I j u tj v t j ,
j1
3.23
for all u, v ∈ H.
We call such critical points weak solutions of problem 1.1–1.4.
Let E be a Banach space, ϕ ∈ C1 E, R, which means that ϕ is a continuously Fréchetdifferentiable functional on E. ϕ is said to satisfy the Palais-Smale condition P-S condition
if any sequence {xn } ⊂ E such that {ϕxn } is bounded and ϕ xn → 0 as n → ∞, has a
convergent subsequence in E.
Lemma 3.4 Mountain pass theorem 26, Theorem 2.2, 27. Let E be a real Hilbert space.
Suppose ϕ ∈ C1 E, R, satisfies the P-S condition and the following assumptions:
l1 there exist constants ρ > 0 and a > 0 such that ϕx ≥ a for all x ∈ ∂Bρ , where Bρ {x ∈
E | xE < ρ} which will be the open ball in E with radius ρ and centered at 0;
∈ Bρ such that ϕx0 ≤ 0.
l2 ϕ0 ≤ 0 and there exists x0 /
Then ϕ possesses a critical value c ≥ a. Moreover, c can be characterized as
c inf max ϕhs,
3.24
Γ {h ∈ C0, 1; E | h0 0, h1 x0 }.
3.25
h∈Γ s∈0,1
where
4. Main Results
Now we introduce some assumptions, which are used hereafter:
H1 the function f : 0, T T × R → R is continuous;
H2 limx → 0 ft, x/x 0 holds uniformly for t ∈ 0, T T ;
H3 there exist constants μ > 2 and L > 0 such that
0 < μFt, x ≤ xft, x,
∀|x| ≥ L;
H4 there exist constants Mj , with 0 < M < min{1/2R20 , μ − 2/R20 μ
1
where M p
j1
Aj Ij x ≤ Mj |x|,
Mj , and R0 T 1/2 /1 −
p
k1
∀x ∈ R, j 1, 2, . . . , p,
|Ak |.
4.1
2} such that
4.2
10
Advances in Difference Equations
Remark 4.1. H3 is the well-known Ambrosetti-Rabinowitz condition from the paper 27.
Lemma 4.2. Suppose that the conditions (H1)–(H4) are satisfied, then ϕ satisfies the Palais-Smale
condition.
Proof. Let {uk } be the sequence in H satisfying that {ϕuk } is bounded and ϕ uk → 0 as
k → ∞. Then there exists a constant β > 0 such that
ϕuk ≤ β,
4.3
for every k ∈ N. By H3, we know that there exist constants c1 > 0, c2 > 0 such that
Ft, x ≥ c1 |x|μ − c2 ,
4.4
for all x ∈ R. By H4 and Lemma 3.1, we have
p
utj j1
0
I j sds ≥ −
I j sds
p
max{0,utj } j1
min{0,utj }
1 2
M j u tj
2 j1
p
≥−
4.5
1
≥ − MR20 u2H ,
2
p
p
I j u tj u tj ≤
I j u tj u tj j1
j1
≤
p
2
Mj u tj 4.6
j1
≤ MR20 u2H ,
for all u ∈ H.
Set
j
j
Ωk t ∈ tj , tj 1 T | uk σt ≥ L ,
⎧
⎨uk t,
j
uk t : uk Γj :
⎩u t ,
k j
for j 0, 1, . . . , p, k 1, 2, . . . .
t ∈ tj , t j 1 T ,
t tj ,
4.7
Advances in Difference Equations
11
It follows from 4.3–4.5, and H3 that
β
γuk H ≥ ϕuk −
$
−
$
−
%
1 1
−
uk 2H
2 μ
p
tj
j0
tj
p
j
Ωk
p
uk tj j1
0
1 I j uk t j u k t j
μ j1
p
I j sds −
j
"
! j
j
F σt, uk σt − f σt, uk σt uk σt Δt
p
uk tj j1
0
1 I j uk t j u k t j
μ j1
p
I j s ds −
4.8
j
"
! j
j
F σt, uk σt − f σt, uk σt uk σt Δt
p
j
tj ,tj 1 T \Ωk
j0
≥
1
%
1 1
−
uk 2H
2 μ
j0
−
1
ϕ uk , uk
μ
j
"
! j
j
F σt, uk σt − f σt, uk σt uk σt Δt
1 1 MR20 MR20
− −
−
2 μ
2
μ
u2H − c3 ,
for some constants γ > 0, c3 > 0, which implies that {uk H } is bounded by the fact that
μ > 2, M < μ − 2/R20 μ 2.
j
Then {uk HΔ1 tj ,tj 1 T } is bounded in HΔ1 tj , tj 1 T for j 0, 1, . . . , p. Therefore,
j
there exists a subsequence {uk } for simplicity denoted again by {uk } such that {uk }
j
converges weakly to uj in HΔ1 tj , tj 1 T , and by Lemma 2.6, {uk } converges strongly to uj in
j
Ctj , tj 1 T , that is, uk − uj Ctj ,tj
Set
1 T → 0 as k →
∞ for all j 0, 1, . . . , p.
⎧
⎨uj t,
t ∈ tj , tj 1 T , j 0, 1, . . . , p,
ut :
⎩ j−1 u
tj , t tj , j 0, 1, . . . , p.
In a similar way to Lemma 3.2, one can prove that u ∈ H, uΓj uj , j 0, 1, . . . , p.
4.9
12
Advances in Difference Equations
For any v ∈ H, we have
v, uk H p
tj
j0
tj
j Δ
vΓΔj t uk tΔt
1
p j0
−→
j
vΓj , uk
Htj ,tj 1 T p vΓj , uj
j0
p
tj
j0
tj
1
tj
−
Htj ,tj 1 T 1
tj
−
tj
j
vΓj tuk tΔt
1
4.10
vΓj tuj tΔt
tj
Δ
vΓΔj t uj tΔt v, uH ,
which implies that {uk } converges weakly to u in H.
By 3.22 and 3.23, we have
ϕ uk − ϕ u, uk − u
p
tj
j0
tj
−
1
&
j Δ
uk t
p
tj
j0
tj
1
−
uΔ
Γj t
'2
Δt
"! j
"
! j
f σt, uk σt − f σt, uσΓj t uk σt − uσΓj t Δt
p !
" I j uk t j − I j u t j
uk t j − u t j
j1
p
tj
j0
tj
−
1
&
j
uk
p
tj
j0
tj
1
Δ
4.11
'2
t − uΔ
t
Δt
Γj
"! j
"
! j
f σt, uk σt − f σt, uj σt uk σt − uj σt Δt
p ! "! j−1 "
j−1 uk tj − uj−1 tj .
I j uk tj − I j uj−1 tj
j1
By the fact that ϕ uk − ϕ u, uk − u → 0 as k → ∞, and the continuity of f and I j , on
0, T T , j 1, 2, . . . , p, we conclude
p
tj
j0
tj
1
&
j Δ
uk t
−
uΔ
Γj t
'2
Δt −→ 0 as k −→ ∞,
4.12
Advances in Difference Equations
13
that is,
uk − uH −→ 0
as k −→ ∞.
4.13
Thus, {uk } possesses a convergent subsequence in H. Then, the P-S condition is now satisfied.
Theorem 4.3. Suppose that (H1)–(H4) hold. Then problem 1.1–1.4 has at least one nontrivial
weak solution on H.
Proof. In order to show that ϕ has at least one nonzero critical point, it suffices to check the
conditions l1 and l2 . It follows from H2 that there is a constant δ > 0 such that
ft, s ≤
1
|s|,
2R20 T
4.14
for all 0 < |s| ≤ δ and t ∈ 0, T T . Hence, we have
Fσt, uσ t uσ t
fσt, sds
0
≤
max{0,uσ t} min{0,uσ t}
1
≤
2R20 T
≤
fσt, sds
max{0,uσ t}
min{0,uσ t}
4.15
|s|ds
1
|uσ t|2 ,
4R20 T
for all supt∈0,T T |ut| ≤ δ and t ∈ 0, T T . By Lemma 3.3, we obtain
p
tj
j0
tj
1
F σt, uσΓj t Δt ≤
1 4R20 T j0
p
tj
tj
1
σ 2
uΓj t Δt
4.16
1
≤ u2H ,
4
for all supt∈0,T T |ut| ≤ δ and t ∈ 0, T T . It follows from 4.5 and 4.16 that
1
1
1
u2H − u2H − MR20 u2H
2
4
2
$
%
1 1
− MR20 u2H ,
2 2
ϕu ≥
4.17
14
Advances in Difference Equations
for all supt∈0,T T |ut| ≤ δ and t ∈ 0, T T . Therefore, by 3.6, one gets
ϕu ≥ α,
4.18
for all u ∈ ∂Bρ , where Bρ {u ∈ H | uH < ρ δ/R0 } and α δ2 /2R20 1/2 − MR20 > 0.
Then l1 is verified. Next we verify l2 . By 4.4, one has
p
tj
j0
tj
1
p
F σt, uσΓj t Δt ≥ c1
tj
tj
j0
1
σ μ
uΓj t Δt − c2 T,
4.19
for all u ∈ H, and by H4 and Lemma 3.1, we have
p
utj j1
0
I j sds ≤
I j sds
p
max 0,utj j1
min 0,utj 1 2
M j u tj
2 j1
p
≤
4.20
≤ c4 u2H ,
for all u ∈ H and some positive constant c4 .
Let v0 ∈ H and v0 H 1. For any τ > 0, by 4.19 and 4.20, one obtains
τ2
v0 2H −
2
j0
p
ϕτv0 ≤
1
tj
p
τ2
v0 2H − c1 τ μ
2
j0
$
≤
tj
1
2
F σt, τv0 σΓj t Δt
tj
tj
1
μ
v0 σΓj t Δt
%
p
c4 τ 2 v0 2H − c1 τ μ
j0
tj
tj
1
c2 T
p
τv0 tj j1
0
I j sds
c4 τ 2 v0 2H
μ
v0 σΓj t Δt
4.21
c2 T,
which implies that
ϕτv0 −→ −∞,
4.22
as τ → ∞ for μ > 2. Hence, we can choose sufficiently large τ0 > ρ such that u0 τ0 v0 ∈ H,
∈ Bρ , and ϕu0 < 0. Assumption l2 is verified. Theorem 4.3 is now proved.
u0 /
Advances in Difference Equations
15
Example 4.4. Let 0, T T 0, 0.01 ∪ 0.02, 0.05, t1 0.02, T 0.05. Then the system
−uΔΔ t 4uσt3 exp uσt4 ,
t ∈ 0, T T , t /
t1 ,
1 u t1 − u t−1 u t−1 ,
2
1 u t1 − uΔ t−1 u t−1 − uΔ t−1 ,
3
4.23
Δ
u0 uT 0
is solvable.
Acknowledgment
This research is supported by the National Natural Science Foundation of China no.
10561004.
References
1 S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,”
Results in Mathematics, vol. 18, no. 1-2, pp. 18–56, 1990.
2 R. P. Agarwal, M. Bohner, and P. J. Y. Wong, “Sturm-Liouville eigenvalue problems on time scales,”
Applied Mathematics and Computation, vol. 99, no. 2-3, pp. 153–166, 1999.
3 R. Agarwal, M. Bohner, D. O’Regan, and A. Peterson, “Dynamic equations on time scales: a survey,”
Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 1–26, 2002.
4 P. Amster and P. D. Nápoli, “Variational methods for two resonant problems on time scales,”
International Journal of Difference Equations, vol. 2, no. 1, pp. 1–12, 2007.
5 D. R. Anderson, “Eigenvalue intervals for a two-point boundary value problem on a measure chain,”
Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 57–64, 2002.
6 D. R. Anderson and R. I. Avery, “An even-order three-point boundary value problem on time scales,”
Journal of Mathematical Analysis and Applications, vol. 291, no. 2, pp. 514–525, 2004.
7 F. M. Atici and G. Sh. Guseinov, “On Green’s functions and positive solutions for boundary value
problems on time scales,” Journal of Computational and Applied Mathematics, vol. 141, no. 1-2, pp. 75–99,
2002.
8 A. Cabada, “Extremal solutions and Green’s functions of higher order periodic boundary value
problems in time scales,” Journal of Mathematical Analysis and Applications, vol. 290, no. 1, pp. 35–54,
2004.
9 Z. He, “Existence of two solutions of m-point boundary value problem for second order dynamic
equations on time scales,” Journal of Mathematical Analysis and Applications, vol. 296, no. 1, pp. 97–109,
2004.
10 L. Jiang and Z. Zhou, “Existence of weak solutions of two-point boundary value problems for secondorder dynamic equations on time scales,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69,
no. 4, pp. 1376–1388, 2008.
11 S. G. Topal, “Second-order periodic boundary value problems on time scales,” Computers &
Mathematics with Applications, vol. 48, no. 3-4, pp. 637–648, 2004.
12 A. Cabada, “Existence results for ϕ-Laplacian boundary value problems on time scales,” Advances in
Difference Equations, vol. 2006, Article ID 21819, 11 pages, 2006.
13 F. A. Davidson and B. P. Rynne, “The formulation of second-order boundary value problems on time
scales,” Advances in Difference Equations, vol. 2006, Article ID 31430, 10 pages, 2006.
14 B. Karna and B. A. Lawrence, “An existence result for a multipoint boundary value problem on a time
scale,” Advances in Difference Equations, vol. 2006, Article ID 63208, 8 pages, 2006.
16
Advances in Difference Equations
15 R. P. Agarwal, V. Otero-Espinar, K. Perera, and D. R. Vivero, “Multiple positive solutions in the sense
of distributions of singular BVPs on time scales and an application to Emden-Fowler equations,”
Advances in Difference Equations, vol. 2008, Article ID 796851, 13 pages, 2008.
16 M. Benchohra, J. Henderson, and S. K. Ntouyas, “Eigenvalue problems for systems of nonlinear
boundary value problems on time scales,” Advances in Difference Equations, vol. 2007, Article ID
031640, 10 pages, 2007.
17 V. Lakshmikantham, D. D. Baı̆nov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6
of Series in Modern Applied Mathematics, World Scientific, Teaneck, NJ, USA, 1989.
18 A. M. Samoı̆lenko and N. A. Perestyuk, Impulsive Differential Equations, vol. 14 of World Scientific Series
on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, River Edge, NJ, USA, 1995.
19 M. Benchohra, S. K. Ntouyas, and A. Ouahab, “Existence results for second order boundary value
problem of impulsive dynamic equations on time scales,” Journal of Mathematical Analysis and
Applications, vol. 296, no. 1, pp. 65–73, 2004.
20 J. J. Nieto and D. O’Regan, “Variational approach to impulsive differential equations,” Nonlinear
Analysis: Real World Applications, vol. 10, no. 2, pp. 680–690, 2009.
21 H. Zhang and Z. X. Li, “Periodic solutions of second-order nonautonomous impulsive differential
equations,” International Journal of Qualitative Theory of Differential Equations and Applications, vol. 2,
no. 1, pp. 112–124, 2008.
22 M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introdution with Applications,
Birkhäuser, Boston, Mass, USA, 2001.
23 M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2001.
24 R. P. Agarwal, V. Otero-Espinar, K. Perera, and D. R. Vivero, “Basic properties of Sobolev’s spaces on
time scales,” Advances in Difference Equations, vol. 2006, Article ID 38121, 14 pages, 2006.
25 R. Agarwal, M. Bohner, and A. Peterson, “Inequalities on time scales: a survey,” Mathematical
Inequalities & Applications, vol. 4, no. 4, pp. 535–557, 2001.
26 P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,
vol. 65 of CBMS Regional Conference Series in Mathematics, CBMS AMS, Providence, RI, USA, 1986.
27 A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and
applications,” Journal of Functional Analysis, vol. 14, no. 4, pp. 349–381, 1973.