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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,
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:
−u
ΔΔ
t
f
σ
t
,u
σ
t
,t∈
0,T
T
,t
/
t
j
,j 1, 2, ,p,
1.1
u
t
j
− u
t
−
j
A
j
u
t
−
j
,j 1, 2, ,p, 1.2
u
Δ
t
j
− u
Δ
t
−
j
B
j
u
Δ
t
−
j
I
j
u
t
−
j
,j 1, 2, ,p, 1.3
u
0
0 u
T
, 1.4
where T is a time scale, 0,T
T
:0,T ∩ T,σ00andσTT, f : 0,T
T
× R → R
is a given function, I
j
∈ CR, R, {A
j
}, {B
j
} are real sequences with B
j
1 A
j
−1
− 1and
p
k1
|A
k
| < 1, the impulsive points t
j
∈ 0,T
T
are right-dense and 0 t
0
<t
1
< ··· <t
p
<
t
p1
T, lim
h → 0
u
Δ
t
j
h and lim
h → 0
u
Δ
t
j
− h represent the right and left limits of u
Δ
t
at t t
j
in the sense of the time scale, that is, in terms of h>0 for which t
j
h, t
j
− h ∈ 0,T
T
,
whereas if t
j
is left-scattered, we interpret u
Δ
t
−
j
u
Δ
t
j
and ut
−
j
ut
j
.
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 Samo
˘
ılenko 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,westudy
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, b
T
:a, b ∩ T, a, b
T
: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
σ
t
: inf
{
s ∈ T : s>t
}
,ρ
t
: sup
{
s ∈ T : s<t
}
,μ
t
σ
t
− t for t ∈ T, 2.1
respectively. If σtt, then t is called right-dense otherwise: right-scattered,andifρtt,
then t is called left-dense otherwise: left-scattered. Denote y
σ
tyσt.
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 t hat 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
f
σ
t
− f
s
− f
Δ
t
σ
t
− s
≤ ε
|
σ
t
− 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 right-
dense 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 C
rd
T.
As mentioned in 24, the Lebesgue μ
Δ
-measure can be characterized as follows:
μ
Δ
λ
i∈I
σ
t
i
− t
i
δ
t
i
, 2.3
where λ is the Lebesgue measure on R, and {t
i
}
i∈I
is the at most countable set of all
right-scattered points of T. A f unction f which is measurable with respect to μ
Δ
is called
Δ-measurable, and the Lebesgue integral over a, b
T
is denoted by
b
a
f
t
Δt :
a,b
T
f
t
dμ
Δ
. 2.4
The Lebesgue integral associated with the measure μ
Δ
on T is called the Lebesgue delta
integral.
Lemma 2.5 see 24, Theorem 2.11. If f, g : a, b
T
→ R are absolutely continuous functions
on a, b
T
,thenf · g is absolutely continuous on a, b
T
and the following equality is valid:
a,b
T
f
t
g
Δ
t
Δt
f
t
g
t
b
a
−
a,b
T
f
Δ
t
g
σ
t
Δt. 2.5
For 1 <p<∞, the Banach space L
p
Δ
may be defined in the standard way, namely,
L
p
Δ
a, b
T
:
f :
a, b
T
−→ R | f is Δ-measurable and
b
a
f
t
p
Δt<∞
, 2.6
equipped with the norm
f
L
p
Δ
:
b
a
f
t
p
Δt
1/p
. 2.7
Let H
1
Δ
a, b
T
be the space of the form
H
1
Δ
a, b
T
: W
1,2
Δ
a, b
T
:
f :
0,T
T
−→ R | f is absolutely continuous on
a, b
T
,f
Δ
∈ L
2
Δ
a, b
T
2.8
4 Advances in Difference Equations
its norm is induced by the inner product given by
f, g
H
1
Δ
:
b
a
f
Δ
t
g
Δ
t
Δt
b
a
f
t
g
t
Δt, 2.9
for all f, g ∈ H
1
Δ
a, b
T
.
Let Ca, b
T
denote the linear space of all continuous function f : a, b
T
→ R with
the maximum norm f
C
max
t∈a,b
T
|ft|.
Lemma 2.6 see 24, Corollary 3.8. Let {x
m
}⊂H
1
Δ
a, b
T
, and x ∈ H
1
Δ
a, b
T
. If {x
m
}
converges weakly in H
1
Δ
a, b
T
to x,then{x
m
} converges strongly in Ca, b
T
to x.
Lemma 2.7 H
¨
older inequality 25, Theorem 3.1. Let f, g ∈ C
rd
a, b,p>1 and q be the
conjugate number of p. Then
b
a
f
t
g
t
Δt ≤
b
a
f
t
p
Δt
1/p
b
a
g
t
q
Δt
1/q
. 2.10
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 Γ
j
t
j
,t
j1
T
, and
f
Γ
j
t
:
⎧
⎨
⎩
f
t
,t∈
t
j
,t
j1
T
,
f
t
j
,t t
j
,
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 t
j
,f
t
j
exists,
f
Γ
j
is absolutely continuous on Γ
j
, the delta derivative of f
Γ
j
∈ L
2
Δ
t
j
,t
j1
T
,
f satisfies the condition
1.2
for all j 0, 1, ,p, f
0
f
T
0
,
3.2
Advances in Difference Equations 5
its norm is induced by the inner product given by
f, g
H
:
p
j0
t
j1
t
j
f
Δ
Γ
j
t
g
Δ
Γ
j
t
Δt, ∀f, g ∈ H. 3.3
That is
f
H
⎛
⎝
p
j0
t
j1
t
j
f
Δ
Γ
j
t
2
Δt
⎞
⎠
1/2
, 3.4
for any f ∈ H.
First, we give some lemmas which are useful in the proof of theorems.
Lemma 3.1. If
p
k1
|A
k
| < 1, then for any x ∈ H, sup
t∈0,T
T
|xt|≤R
0
x
H
, where R
0
T
1/2
/1 −
p
k1
|A
k
|.
Proof. For any x ∈ H and t ∈ t
j
,t
j1
T
,j 0, 1, ,p,we have
|
x
t
|
x
t
− x
t
j
x
t
j
−···−x
t
1
x
t
1
− x
t
0
x
t
− x
t
j
j−1
k0
x
t
−
k1
− x
t
k
j−1
k0
A
k1
x
t
−
k1
t
t
j
x
Δ
Γ
k
s
Δs
j−1
k0
t
k1
t
k
x
Δ
Γ
k
s
Δs
j−1
k0
A
k1
x
t
−
k1
≤
t
t
j
x
Δ
Γ
j
s
Δs
j−1
k0
t
k1
t
k
x
Δ
Γ
k
s
Δs
j−1
k0
|
A
k1
|
x
t
−
k1
≤
p
k0
t
k1
t
k
x
Δ
Γ
k
s
Δs
j−1
k0
|
A
k1
|
x
t
−
k1
≤ T
1/2
x
H
p
k1
|
A
k
|
sup
t∈0,T
T
|
x
t
|
,
3.5
which implies that
sup
t∈0,T
T
|
x
t
|
≤ R
0
x
H
, ∀x ∈ H. 3.6
Lemma 3.2. H is a Hilbert space.
6 Advances in Difference Equations
Proof. Let {u
k
}
∞
k1
be a Cauchy sequence in H. By Lemma 3.1, we have
f
H
1
Δ
t
j
,t
j1
T
t
j1
t
j
f
Δ
Γ
j
t
2
Δt
t
j1
t
j
f
Γ
j
t
2
Δt
1/2
≤
t
j1
t
j
f
Δ
Γ
j
t
2
Δt R
2
0
t
j1
− t
j
f
2
H
1/2
≤
1 R
2
0
T
1/2
f
H
.
3.7
Set
u
j
k
t
:
u
k
Γ
j
:
⎧
⎨
⎩
u
k
t
,t∈
t
j
,t
j1
T
,
u
k
t
j
,t t
j
,
3.8
for j 0, 1, ,p, k 1, 2, Then {u
j
k
}
∞
k1
be a Cauchy sequence in H
1
Δ
t
j
,t
j1
T
, for
j 0, 1, ,p.Therefore, there exists a u
j
∈ H
1
Δ
t
j
,t
j1
T
, such that {u
j
k
} converges to u
j
in
H
1
Δ
t
j
,t
j1
T
,j 0, 1, ,p.It follows from Lemma 2.6 that {u
j
k
} converges strongly to u
j
in
Ct
j
,t
j1
T
,thatis,u
j
k
− u
j
Ct
j
,t
j1
T
→ 0ask → ∞ for all j 0, 1, ,p.Hence, we have
lim
k → ∞
u
j
k
t
j
u
j
t
j
, lim
k → ∞
u
j−1
k
t
j
u
j−1
t
j
. 3.9
Noting that
lim
k → ∞
u
j
k
t
j
lim
k → ∞
u
k
t
j
lim
k → ∞
1 A
j
u
k
t
−
j
lim
k → ∞
1 A
j
u
k
t
j
1 A
j
lim
k → ∞
u
j−1
k
t
j
1 A
j
u
j−1
t
j
,
3.10
we have
u
j
t
j
1 A
j
u
j−1
t
j
,j 0, 1, ,p. 3.11
Set
u
t
:
⎧
⎨
⎩
u
j
t
,t∈
t
j
,t
j1
T
,j 0, 1, ,p,
u
j−1
t
j
,t t
j
,j 0, 1, ,p.
3.12
Advances in Difference Equations 7
Then we have
u
t
j
u
j
t
j
u
j
t
j
1 A
j
u
j−1
t
j
1 A
j
u
t
j
1 A
j
u
t
−
j
,
u
Γ
j
u
j
,j 0, 1, ,p.
3.13
Thus u ∈ H. Noting that
u
k
− u
H
⎡
⎣
p
j0
t
j1
t
j
u
j
k
Δ
t − u
Δ
Γ
j
t
2
Δt
⎤
⎦
1/2
≤
⎡
⎣
p
j0
u
j
k
− u
j
2
H
1
Δ
t
j
,t
j1
T
⎤
⎦
1/2
,
3.14
we have u
k
converges to u in H as k → ∞. The proof is complete.
Lemma 3.3. If
p
k1
|A
k
| < 1, then for any u ∈ H,
p
j0
t
j1
t
j
u
σ
Γ
j
t
2
Δt ≤ R
2
0
T
u
2
H
, 3.15
where R
0
is given in Lemma 3.1.
Proof. For any u ∈ H, t ∈ t
j
,t
j1
T
, by Lemma 3.1, we have
u
σ
Γ
j
t
≤ R
0
u
H
,j 0, 1, ,p, 3.16
which implies that
p
j0
t
j1
t
j
u
σ
Γ
j
t
2
Δt ≤ R
2
0
T
u
2
H
. 3.17
The proof is complete.
For any u ∈ H satisfying 1.1–1.4, take v ∈ H and multiply 1.1 by v
σ
Γ
j
, then
integrate it between t
j
and t
j1
:
−
t
j1
t
j
u
ΔΔ
t
v
σ
Γ
j
t
Δt
t
j1
t
j
f
σ
t
,u
σ
t
v
σ
Γ
j
t
Δt. 3.18
8 Advances in Difference Equations
The first term is now
t
j1
t
j
u
ΔΔ
t
v
σ
Γ
j
t
Δt u
Δ
t
−
j1
v
Γ
j
t
−
j1
− u
Δ
t
j
v
Γ
j
t
j
−
t
j1
t
j
u
Δ
t
v
Δ
Γ
j
t
Δt. 3.19
Hence, one gets
−
p
j0
t
j1
t
j
u
ΔΔ
t
v
σ
Γ
j
t
Δt
p
j0
u
Δ
t
j
v
Γ
j
t
j
− u
Δ
t
−
j1
v
Γ
j
t
−
j1
p
j0
t
j1
t
j
u
Δ
t
v
Δ
Γ
j
t
Δt
p
j1
u
Δ
t
j
v
t
j
− u
Δ
t
−
j
v
t
−
j
p
j0
t
j1
t
j
u
Δ
t
v
Δ
Γ
j
t
Δt
p
j1
1 A
j
u
Δ
t
j
− u
Δ
t
−
j
v
t
−
j
p
j0
t
j1
t
j
u
Δ
t
v
Δ
Γ
j
t
Δt
p
j1
1 A
j
I
j
u
t
−
j
v
t
−
j
p
j0
t
j1
t
j
u
Δ
t
v
Δ
Γ
j
t
Δt,
3.20
for all u, v ∈ H. Then we have
p
j0
t
j1
t
j
u
Δ
t
v
Δ
Γ
j
t
Δt −
p
j0
t
j1
t
j
f
σ
t
,u
σ
t
v
σ
Γ
j
t
Δt
p
j1
1 A
j
I
j
u
t
−
j
v
t
−
j
0,
3.21
for all u, v ∈ H.
This suggests that one defines ϕ : H → R, by
ϕ
u
1
2
p
j0
t
j1
t
j
u
Δ
Γ
j
t
2
Δt −
p
j0
t
j1
t
j
F
σ
t
,u
σ
Γ
j
t
Δt
p
j1
ut
j
0
I
j
s
ds, 3.22
where Ft, x
x
0
ft, sds, and I
j
1 A
j
I
j
, ∀j 1, 2, ,p.
Advances in Difference Equations 9
By a standard argument, one can prove that the functional ϕ is continuously
differentiable at any u ∈ H and
ϕ
u
,v
p
j0
t
j1
t
j
u
Δ
Γ
j
t
v
Δ
Γ
j
t
Δt −
p
j0
t
j1
t
j
f
σ
t
,u
σ
Γ
j
t
v
σ
Γ
j
t
Δt
p
j1
I
j
u
t
j
v
t
j
,
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 ∈ C
1
E, R, which means that ϕ is a continuously Fr
´
echet-
differentiable functional on E. ϕ is said to satisfy the Palais-Smale condition P-S condition
if any sequence {x
n
}⊂E such that {ϕx
n
} is bounded and ϕ
x
n
→ 0asn →∞, 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 ∈ C
1
E, R, satisfies the P-S condition and the following assumptions:
l
1
there exist constants ρ>0 and a>0 such that ϕx ≥ a for all x ∈ ∂B
ρ
, where B
ρ
{x ∈
E |x
E
<ρ} which will be the open ball in E with radius ρ and centered at 0;
l
2
ϕ0 ≤ 0 and there exists x
0
/
∈ B
ρ
such that ϕx
0
≤ 0.
Then ϕ possesses a critical value c ≥ a. Moreover, c can be characterized as
c inf
h∈Γ
max
s∈0,1
ϕ
h
s
, 3.24
where
Γ
{
h ∈ C
0, 1
; E
| h
0
0,h
1
x
0
}
. 3.25
4. Main Results
Now we introduce some assumptions, which are used hereafter:
H1 the function f : 0,T
T
× R → R is continuous;
H2 lim
x → 0
ft, x/x0 holds uniformly for t ∈ 0,T
T
;
H3 there exist constants μ>2andL>0 such that
0 <μF
t, x
≤ xf
t, x
, ∀
|
x
|
≥ L; 4.1
H4 there exist constants
M
j
, with 0 < M<min{1/2R
2
0
, μ − 2/R
2
0
μ 2} such that
1 A
j
I
j
x
≤
M
j
|
x
|
, ∀x ∈ R,j 1, 2, ,p, 4.2
where
M
p
j1
M
j
, and R
0
T
1/2
/1 −
p
k1
|A
k
|.
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 {u
k
} be the sequence in H satisfying that {ϕu
k
} is bounded and ϕ
u
k
→ 0as
k →∞. Then there exists a constant β>0 such that
ϕ
u
k
≤ β, 4.3
for every k ∈ N. By H3, we know that there exist constants c
1
> 0,c
2
> 0 such that
F
t, x
≥ c
1
|
x
|
μ
− c
2
, 4.4
for all x ∈ R.ByH4 and Lemma 3.1, we have
p
j1
ut
j
0
I
j
s
ds ≥−
p
j1
max{0,ut
j
}
min{0,ut
j
}
I
j
s
ds
≥−
1
2
p
j1
M
j
u
t
j
2
≥−
1
2
MR
2
0
u
2
H
,
4.5
p
j1
I
j
u
t
j
u
t
j
≤
p
j1
I
j
u
t
j
u
t
j
≤
p
j1
M
j
u
t
j
2
≤ MR
2
0
u
2
H
,
4.6
for all u ∈ H.
Set
Ω
j
k
t ∈
t
j
,t
j1
T
|
u
j
k
σ
t
≥ L
,
u
j
k
t
:
u
k
Γ
j
:
⎧
⎨
⎩
u
k
t
,t∈
t
j
,t
j1
T
,
u
k
t
j
,t t
j
,
4.7
for j 0, 1, ,p, k 1, 2,
Advances in Difference Equations 11
It follows from 4.3–4.5,andH3 that
β γ
u
k
H
≥ ϕ
u
k
−
1
μ
ϕ
u
k
,u
k
1
2
−
1
μ
u
k
2
H
p
j1
u
k
t
j
0
I
j
s
ds −
1
μ
p
j1
I
j
u
k
t
j
u
k
t
j
−
p
j0
t
j1
t
j
F
σ
t
,u
j
k
σ
t
− f
σ
t
,u
j
k
σ
t
u
j
k
σ
t
Δt
1
2
−
1
μ
u
k
2
H
p
j1
u
k
t
j
0
I
j
s
ds −
1
μ
p
j1
I
j
u
k
t
j
u
k
t
j
−
p
j0
Ω
j
k
F
σ
t
,u
j
k
σ
t
− f
σ
t
,u
j
k
σ
t
u
j
k
σ
t
Δt
−
p
j0
t
j
,t
j1
T
\Ω
j
k
F
σ
t
,u
j
k
σ
t
− f
σ
t
,u
j
k
σ
t
u
j
k
σ
t
Δt
≥
1
2
−
1
μ
−
MR
2
0
2
−
MR
2
0
μ
u
2
H
− c
3
,
4.8
for some constants γ>0,c
3
> 0, which implies that {u
k
H
} is bounded by the fact that
μ>2,
M<μ − 2/R
2
0
μ 2.
Then {u
j
k
H
1
Δ
t
j
,t
j1
T
} is bounded in H
1
Δ
t
j
,t
j1
T
for j 0, 1, ,p. Therefore,
there exists a subsequence {u
k
} for simplicity denoted again by {u
k
} such that {u
j
k
}
converges weakly to u
j
in H
1
Δ
t
j
,t
j1
T
, and by Lemma 2.6, {u
j
k
} converges strongly to u
j
in
Ct
j
,t
j1
T
,thatis,u
j
k
− u
j
Ct
j
,t
j1
T
→ 0ask → ∞ for all j 0, 1, ,p.
Set
u
t
:
⎧
⎨
⎩
u
j
t
,t∈
t
j
,t
j1
T
,j 0, 1, ,p,
u
j−1
t
j
,t t
j
,j 0, 1, ,p.
4.9
In a similar way to Lemma 3.2, one can prove that u ∈ H, u
Γ
j
u
j
,j 0, 1, ,p.
12 Advances in Difference Equations
For any v ∈ H, we have
v, u
k
H
p
j0
t
j1
t
j
v
Δ
Γ
j
t
u
j
k
Δ
t
Δt
p
j0
v
Γ
j
,u
j
k
Ht
j
,t
j1
T
−
t
j1
t
j
v
Γ
j
t
u
j
k
t
Δt
−→
p
j0
v
Γ
j
,u
j
Ht
j
,t
j1
T
−
t
j1
t
j
v
Γ
j
t
u
j
t
Δt
p
j0
t
j1
t
j
v
Δ
Γ
j
t
u
j
Δ
t
Δt
v, u
H
,
4.10
which implies that {u
k
} converges weakly to u in H.
By 3.22 and 3.23, we have
ϕ
u
k
− ϕ
u
,u
k
− u
p
j0
t
j1
t
j
u
j
k
Δ
t − u
Δ
Γ
j
t
2
Δt
−
p
j0
t
j1
t
j
f
σ
t
,u
j
k
σ
t
− f
σ
t
,u
σ
Γ
j
t
u
j
k
σ
t
− u
σ
Γ
j
t
Δt
p
j1
I
j
u
k
t
j
−
I
j
u
t
j
u
k
t
j
− u
t
j
p
j0
t
j1
t
j
u
j
k
Δ
t − u
Δ
Γ
j
t
2
Δt
−
p
j0
t
j1
t
j
f
σ
t
,u
j
k
σ
t
− f
σ
t
,u
j
σ
t
u
j
k
σ
t
− u
j
σ
t
Δt
p
j1
I
j
u
j−1
k
t
j
−
I
j
u
j−1
t
j
u
j−1
k
t
j
− u
j−1
t
j
.
4.11
By the fact that ϕ
u
k
− ϕ
u,u
k
− u → 0ask →∞, and the continuity of f and I
j
, on
0,T
T
,j 1, 2, ,p,we conclude
p
j0
t
j1
t
j
u
j
k
Δ
t − u
Δ
Γ
j
t
2
Δt −→ 0ask −→ ∞ , 4.12
Advances in Difference Equations 13
that is,
u
k
− u
H
−→ 0ask −→ ∞ . 4.13
Thus, {u
k
} 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 l
1
and l
2
. It follows from H2 that there is a constant δ>0 such that
f
t, s
≤
1
2R
2
0
T
|
s
|
, 4.14
for all 0 < |s|≤δ and t ∈ 0,T
T
. Hence, we have
F
σ
t
,u
σ
t
u
σ
t
0
f
σ
t
,s
ds
≤
max{0,u
σ
t}
min{0,u
σ
t}
f
σ
t
,s
ds
≤
1
2R
2
0
T
max{0,u
σ
t}
min{0,u
σ
t}
|
s
|
ds
≤
1
4R
2
0
T
|
u
σ
t
|
2
,
4.15
for all sup
t∈0,T
T
|ut|≤δ and t ∈ 0,T
T
. By Lemma 3.3,weobtain
p
j0
t
j1
t
j
F
σ
t
,u
σ
Γ
j
t
Δt ≤
1
4R
2
0
T
p
j0
t
j1
t
j
u
σ
Γ
j
t
2
Δt
≤
1
4
u
2
H
,
4.16
for all sup
t∈0,T
T
|ut|≤δ and t ∈ 0,T
T
. It follows from 4.5 and 4.16 that
ϕ
u
≥
1
2
u
2
H
−
1
4
u
2
H
−
1
2
MR
2
0
u
2
H
1
2
1
2
−
MR
2
0
u
2
H
,
4.17
14 Advances in Difference Equations
for all sup
t∈0,T
T
|ut|≤δ and t ∈ 0,T
T
. Therefore, by 3.6,onegets
ϕ
u
≥ α, 4.18
for all u ∈ ∂B
ρ
, where B
ρ
{u ∈ H |u
H
<ρ δ/R
0
} and α δ
2
/2R
2
0
1/2 − MR
2
0
> 0.
Then l
1
is verified. Next we verify l
2
. By 4.4, one has
p
j0
t
j1
t
j
F
σ
t
,u
σ
Γ
j
t
Δt ≥ c
1
p
j0
t
j1
t
j
u
σ
Γ
j
t
μ
Δt − c
2
T, 4.19
for all u ∈ H,andbyH4 and Lemma 3.1, we have
p
j1
ut
j
0
I
j
s
ds ≤
p
j1
max 0,ut
j
min 0,ut
j
I
j
s
ds
≤
1
2
p
j1
M
j
u
t
j
2
≤ c
4
u
2
H
,
4.20
for all u ∈ H and some positive constant c
4
.
Let v
0
∈ H and v
0
H
1. For any τ>0, by 4.19 and 4.20,oneobtains
ϕ
τv
0
τ
2
2
v
0
2
H
−
p
j0
t
j1
t
j
F
σ
t
,τ
v
0
σ
Γ
j
t
Δt
p
j1
τv
0
t
j
0
I
j
s
ds
≤
τ
2
2
v
0
2
H
− c
1
τ
μ
p
j0
t
j1
t
j
v
0
σ
Γ
j
t
μ
Δt c
2
T c
4
τ
2
v
0
2
H
≤
1
2
c
4
τ
2
v
0
2
H
− c
1
τ
μ
p
j0
t
j1
t
j
v
0
σ
Γ
j
t
μ
Δt c
2
T,
4.21
which implies that
ϕ
τv
0
−→ − ∞ , 4.22
as τ →∞for μ>2. Hence, we can choose sufficiently large τ
0
>ρsuch that u
0
τ
0
v
0
∈ H,
u
0
/
∈ B
ρ
, and ϕu
0
< 0. Assumption l
2
is verified. Theorem 4.3 is now proved.
Advances in Difference Equations 15
Example 4.4. Let 0,T
T
0, 0.01 ∪ 0.02, 0.05,t
1
0.02,T 0.05. Then the system
−u
ΔΔ
t
4
u
σ
t
3
exp
u
σ
t
4
,t∈
0,T
T
,t
/
t
1
,
u
t
1
− u
t
−
1
1
2
u
t
−
1
,
u
Δ
t
1
− u
Δ
t
−
1
u
t
−
1
−
1
3
u
Δ
t
−
1
,
u
0
u
T
0
4.23
is solvable.
Acknowledgment
This research is supported by the National Natural Science Foundation of China no.
10561004.
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