Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 727486, 27 pages
doi:10.1155/2010/727486
Research Article
Existence of Solutions for a Class of
Damped Vibration Problems on Time Scales
Yongkun Li and Jianwen Zhou
Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
Correspondence should be addressed to Yongkun Li, yklie@ynu.edu.cn
Received 3 June 2010; Revised 20 November 2010; Accepted 24 November 2010
Academic Editor: Kanishka Perera
Copyright q 2010 Y. Li and J. Zhou. 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 present a recent approach via variational methods and critical point theory to obtain the
2
existence of solutions for a class of damped vibration problems on time scale Ì, uΔ t wtuΔ σt ∇Fσt, uσt, Δ-a.e. t ∈ 0, Tκ , u0 − uT 0, uΔ 0 − uΔ T 0, where
2
uΔ t denotes the delta or Hilger derivative of u at t, uΔ t uΔ Δ t, σ is the forward jump
operator, T is a positive constant, w ∈ R 0, T , Ê, ew T, 0 1, and F : 0, T × ÊN → Ê.
By establishing a proper variational setting, three existence results are obtained. Finally, three
examples are presented to illustrate the feasibility and effectiveness of our results.
Ì
Ì
Ì
1. Introduction
Consider the damped vibration problem on time-scale
uΔ t wtuΔ σt ∇Fσt, uσt,
2
u0 − uT 0,
Δ-a.e. t ∈ 0, TκÌ,
uΔ 0 − uΔ T 0,
1.1
where uΔ t denotes the delta or Hilger derivative of u at t, uΔ t uΔ Δ t, σ is the
forward jump operator, T is a positive constant, w ∈ R 0, TÌ, , ew T, 0 1, and F :
0, TÌ × N → satisfies the following assumption.
2
A Ft, x is Δ-measurable in t for every x ∈ N and continuously differentiable in x
for t ∈ 0, TÌ and there exist a ∈ C , , b ∈ L1Δ 0, TÌ, such that
|Ft, x| ≤ a|x|bt,
|∇Ft, x| ≤ a|x|bt
1.2
for all x ∈ N and t ∈ 0, TÌ, where ∇Ft, x denotes the gradient of Ft, x in x.
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Problem 1.1 covers the second-order damped vibration problem for when
üt wtu̇t ∇Ft, ut,
u0 − uT 0,
a.e. t ∈ 0, T,
1.3
u̇0 − u̇T 0,
as well as the second-order discrete damped vibration problem for when
Δ2 t wtΔut 1 ∇Ft 1, ut 1,
u0 − uT 0,
, T ≥ 2
t ∈ 0, T − 1 ∩ ,
Δu0 − ΔuT 0.
1.4
The calculus of time-scales was initiated by Stefan Hilger in his Ph.D. thesis in 1988
in order to create a theory that can unify discrete and continuous analysis. A time-scale is
an arbitrary nonempty closed subset of the real numbers, which has the topology inherited
from the real numbers with the standard topology. The two most popular examples are
and . The time-scales calculus has a tremendous potential for applications in
some mathematical models of real processes and phenomena studied in physics, chemical
technology, population dynamics, biotechnology and economics, neural networks, and social
sciences see 1. For example, it can model insect populations that are continuous while
in season and may follow a difference scheme with variable step-size, die out in winter,
while their eggs are incubating or dormant, and then hatch in a new season, giving rise to a
nonoverlapping population.
In recent years, dynamic equations on time-scales have received much attention. We
refer the reader to the books 2–7 and the papers 8–15. In this century, some authors
have begun to discuss the existence of solutions of boundary value problems on time-scales
see 16–22. There have been many approaches to study the existence and the multiplicity
of solutions for differential equations on time-scales, such as methods of lower and upper
solutions, fixed-point theory, and coincidence degree theory. In 14, the authors have used
the fixed-point theorem of strict-set-contraction to study the existence of positive periodic
solutions for functional differential equations with impulse effects on time-scales. However,
the study of the existence and the multiplicity of solutions for differential equations on timescales using variational method has received considerably less attention see, e.g., 19, 23.
Variational method is, to the best of our knowledge, novel and it may open a new approach
to deal with nonlinear problems on time-scales.
When wt ≡ 0, 1.1 is the second-order Hamiltonian system on time-scale
uΔ t ∇Fσt, uσt,
2
u0 − uT 0,
Δ-a.e. t ∈ 0, TκÌ,
uΔ 0 − uΔ T 0.
1.5
Zhou and Li in 23 studied the existence of solutions for 1.5 by critical point theory on the
Sobolevs spaces on time-scales that they established.
When wt /
≡ 0, to the best of our knowledge, the existence of solutions for problems
1.1 have not been studied yet. Our purpose of this paper is to study the variational structure
of problem 1.1 in an appropriate space of functions and the existence of solutions for
problem 1.1 by some critical point theorems.
Advances in Difference Equations
3
This paper is organized as follows. In Section 2, we present some fundamental
definitions and results from the calculus on time-scales and Sobolev’s spaces on time-scales.
In Section 3, we make a variational structure of 1.1. From this variational structure, we can
reduce the problem of finding solutions of problem 1.1 to one of seeking the critical points of
a corresponding functional. Section 4 is the existence of solutions. Section 5 is the conclusion
of this paper.
2. Preliminaries and Statements
In this section, we present some basic definitions and results from the calculus on time-scales
and Sobolev’s spaces on time-scales that will be required below. We first briefly recall some
basic definitions and results concerning time-scales. Further general details can be found in
3–5, 7, 10, 13.
Through this paper, we assume that 0, T ∈ . We start by the definitions of the forward
and backward jump operators.
Definition 2.1 see 3, Definition 1.1. Let
operator σ : → is defined by
be a time-scale, for t ∈
σt inf{s ∈ , s > t},
while the backward jump operator ρ :
→
, the forward jump
∀t ∈ ,
2.1
is defined by
ρt sup{s ∈ , s < t},
∀t ∈
2.2
supplemented by inf ∅ sup and sup ∅ inf , where ∅ denotes the empty set. A point
t ∈ is called right scattered, left scattered, if σt > t, ρt < t hold, respectively. Points that
are right scattered and left scattered at the same time are called isolated. Also, if t < sup
and σt t, then t is called right-dense, and if t > inf and ρt t, then t is called left
dense. Points that are right-dense and left dense at the same time are called dense. The set κ
which is derived from the time-scale as follows: if has a left scattered maximum m, the
κ
− {m}, otherwise, κ . Finally, the graininess function μ : → 0, ∞ is defined by
μt σt − t.
2.3
When a, b ∈ , a < b, we denote the intervals a, bÌ, a, bÌ, and a, bÌ in
a, bÌ a, b ∩ ,
a, bÌ a, b ∩ ,
a, bÌ a, b ∩ ,
by
2.4
respectively. Note that a, bκÌ a, bÌ if b is left dense and a, bκÌ a, bÌ a, ρbÌ if b is
2
2
left scattered. We denote a, bκÌ a, bκÌκ , therefore a, bκÌ a, bÌ if b is left dense and
2
a, bκÌ a, ρbκÌ if b is left scattered.
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Advances in Difference Equations
→ is a function and let t ∈ κ .
Definition 2.2 see 3, Definition 1.10. Assume that f :
Δ
Then we define f t to be the number provided it exists with the property that given any
> 0, there is a neighborhood U of t i.e., U t − δ, t δ ∩ for some δ > 0 such that
fσt − fs − f Δ tσt − s ≤ |σt − s| ∀s ∈ U.
2.5
We call f Δ t the delta or Hilger derivative of f at t. The function f is delta or Hilger
differentiable on κ provided f Δ t exists for all t ∈ κ . The function f Δ : κ → is then
called the delta derivative of f on κ .
Definition 2.3 see 23, Definition 2.3. Assume that f :
→ N is a function, ft Δ
Δ
Δ
1
2
N
κ
Δ
f t, f t, . . . , f t and let t ∈
. Then we define f t f 1 t, f 2 t, . . . , f N t
provided it exists. We call f Δ t the delta or Hilger derivative of f at t. The function
f is delta or Hilger differentiable provided f Δ t exists for all t ∈ κ . The function
f Δ : κ → N is then called the delta derivative of f on κ .
Definition 2.4 see 3, Definition 2.7. For a function f : → we will talk about the second
2
2
2
derivative f Δ provided f Δ is differentiable on κ κ κ with derivative f Δ f Δ Δ :
2
κ
→ .
Definition 2.5 see 23, Definition 2.5. For a function f :
→
2
2
second derivative f Δ provided f Δ is differentiable on κ 2
f Δ Δ : κ → N .
N
we will talk about the
2
with derivative f Δ κ κ
Definition 2.6 see 23, Definition 2.6. A function f :
→ N is called rd-continuous
provided it is continuous at right-dense points in and its left sided limits exist finite at
left dense points in .
Definition 2.7 see 3, Definition 2.25. We we say that a function w :
provided
1 μtwt /
0 ∀t ∈
→
is regressive
2.6
κ
holds. The set of all regressive and rd-continuous functions w :
→
R R R , ,
R , w ∈ R : 1 μtwt > 0 ∀t ∈ .
is denoted by
2.7
Definition 2.8 see 7, Definition 8.2.18. If w ∈ R and t0 ∈ , then the unique solution of the
initial value problem
yΔ wty,
yt0 1
is called the exponential function and denoted by ew ·, t0 .
The exponential function has some important properties.
2.8
Advances in Difference Equations
5
Lemma 2.9 see 3, Theorem 2.36. If w ∈ R, then
e0 t, s ≡ 1,
ew t, t ≡ 1.
2.9
Throughout this paper, we will use the following notations:
Crd Crd
1
1
Crd
Crd
, N
, N
f:
f:
−→ N : f is rd-continuous ,
−→ N : f is differentiable on
κ
and f Δ ∈ Crd 1
1
CT,rd
0, TÌ, N f ∈ Crd
0, TÌ, N : f0 fT .
κ
,
2.10
The Δ-measure mΔ and Δ-integration are defined as those in 10.
Definition 2.10 see 23, Definition 2.7. Assume that f :
→ N is a function, ft 1
2
N
f t, f t, . . . , f t and let A be a Δ-measurable subset of . f is integrable on A if and
only if f i i 1, 2, . . . , N are integrable on A, and
ftΔt f 1 tΔt,
A
f N tΔt .
f 2 tΔt, . . . ,
A
A
2.11
A
Definition 2.11 see 13, Definition 2.3. Let B ⊂ . B is called Δ-null set if mΔ B 0. Say
that a property P holds Δ-almost everywhere Δ-a.e. on B, or for Δ-almost all Δ-a.a. t ∈ B
if there is a Δ-null set E0 ⊂ B such that P holds for all t ∈ B \ E0 .
For p ∈ , p ≥ 1, we set the space
p
LΔ
0, TÌ, N u : 0, TÌ −→ N
:
ftp Δt < ∞
2.12
Ì
0,T with the norm
f p
LΔ
ftp Δt
1/p
Ì
0,T 2.13
.
We have the following lemma.
Lemma 2.12 see 23, Theorem 2.1. Let p ∈ be such that p ≥ 1. Then the space LΔ 0, TÌ, N is a Banach space together with the norm · Lp . Moreover, L2Δ a, bÌ, N is a Hilbert space together
p
Δ
with the inner product given for every f, g ∈ LΔ a, bÌ, N × LΔ a, bÌ, N by
f, g L2 Δ
where ·, · denotes the inner product in N .
p
p
Ì
a,b
ft, gt Δt,
2.14
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Advances in Difference Equations
As we know from general theory of Sobolev spaces, another important class of
functions is just the absolutely continuous functions on time-scales.
Definition 2.13 see 13, Definition 2.9. A function f : a, bÌ → is said to be absolutely
continuous on a, bÌ i.e., f ∈ ACa, bÌ, , if for every > 0, there exists δ > 0 such
that if {ak , bk Ì}nk1 is a finite pairwise disjoint family of subintervals of a, bÌ satisfying
n
n
k1 bk − ak < δ, then
k1 |fbk − fak | < .
Definition 2.14 see 23, Definition 2.11. A function f : a, bÌ → N , ft f 1 t, f 2 t,
. . ., f N t. We say that f is absolutely continuous on a, bÌ i.e., f ∈ ACa, bÌ, N , if for
every > 0, there exists δ > 0 such that if {ak , bk Ì}nk1 is a finite pairwise disjoint family of
subintervals of a, bÌ satisfying nk1 bk − ak < δ, then nk1 |fbk − fak | < .
Absolutely continuous functions have the following properties.
Lemma 2.15 see 23, Theorem 2.2. A function f : a, bÌ →
a, bÌ if and only if f is delta differentiable Δ-a.e. on a, bÌ and
ft fa f Δ sΔs,
Ì
a,t
N
is absolutely continuous on
∀t ∈ a, bÌ.
2.15
Lemma 2.16 see 23, Theorem 2.3. Assume that functions f, g : a, bÌ → N are absolutely
continuous on a, bÌ. Then fg is absolutely continuous on a, bÌ and the following equality is valid:
Ì
a,b
f Δ t, gt f σ t, g Δ t Δt fb, gb − fa, ga
Ì
a,b
ft, g t f Δ t, g σ t Δt.
2.16
Δ
Now, we recall the definition and properties of the Sobolev space on 0, TÌ in 23.
For the sake of convenience, in the sequel, we will let uσ t uσt.
Definition 2.17 see 23, Definition 2.12. Let p ∈ be such that p ≥ 1 and u : 0, TÌ →
1,p
p
N . We say that u ∈ WΔ,T
0, TÌ, N if and only if u ∈ LΔ 0, TÌ, N and there exists
p
g : 0, TκÌ → N such that g ∈ LΔ 0, TÌ, N and
Ì
0,T ut, φΔ t Δt −
gt, φσ t Δt,
Ì
0,T 1
∀φ ∈ CT,rd
0, TÌ, N .
2.17
For p ∈ , p ≥ 1, we denote
1,p
p
VΔ,T 0, TÌ, N x ∈ AC 0, TÌ, N : xΔ ∈ LΔ 0, TÌ, N , x0 xT .
2.18
Remark 2.18 see 23, Remark 2.2. VΔ,T 0, TÌ, N ⊂ WΔ,T 0, TÌ, N is true for every
p ∈ with p ≥ 1.
1,p
1,p
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Lemma 2.19 see 23, Theorem 2.5. Suppose that u ∈ WΔ,T 0, TÌ, N for some p ∈ with
p
p ≥ 1, and that 2.17 holds for g ∈ LΔ 0, TÌ, N . Then, there exists a unique function x ∈
1,p
VΔ,T 0, TÌ, N such that the equalities
1,p
xΔ g
x u,
Δ-a.e.
2.19
on 0, TÌ
are satisfied and
gtΔt 0.
2.20
Ì
0,T Lemma 2.20 see 3, Theorem 1.16. Assume that f :
one has the following.
→
is a function and let t ∈
κ
. Then,
i If f is differentiable at t, then f is continuous at t.
ii If f is differentiable at t, then
fσt ft μtf Δ t.
2.21
By identifying u ∈ WΔ,T 0, TÌ, N with its absolutely continuous representative
1,p
x ∈ VΔ,T 0, TÌ, N for which 2.19 holds, the set WΔ,T 0, TÌ, N can be endowed with
the structure of Banach space.
1,p
1,p
Theorem 2.21. Assume p ∈ and p ≥ 1. The set WΔ,T 0, TÌ, N is a Banach space together with
the norm defined as
1,p
uW 1,p Δ,T
Ì
0,T p
|u t| Δt σ
Δ p
u t Δt
1/p
Ì
0,T 1,p
∀u ∈ WΔ,T 0, TÌ, N .
2.22
1,2
1
Moreover, the set HΔ,T
WΔ,T
0, TÌ, N is a Hilbert space together with the inner product
1
u, vHΔ,T
uσ t, vσ tΔt Ì
0,T Ì
0,T uΔ t, vΔ t Δt
1
∀u, v ∈ HΔ,T
.
2.23
Proof. Let {un }n∈Æ be a Cauchy sequence in WΔ,T 0, TÌ, N . That is, {un }n∈Æ ⊂
p
p
LΔ 0, TÌ, N and there exist gn : 0, Tκ → N such that {gn }n∈Æ ⊂ LΔ 0, TÌ, N and
1,p
Ì
0,T un t, φΔ t Δt −
Ì
0,T gn t, φσ t Δt,
1
∀φ ∈ CT,rd
0, T , N .
Ì
2.24
Thus, by Lemma 2.19, there exists {xn }n∈Æ ⊂ VΔ,T 0, TÌ, N such that
1,p
xn un ,
xnΔ gn
Δ-a.e.
on 0, TÌ.
2.25
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Advances in Difference Equations
Combining 2.24 and 2.25, we obtain
Δ
xn t, φ t Δt −
Ì
0,T Ì
0,T xnΔ t, φσ t Δt,
1
∀φ ∈ CT,rd
0, T , N .
Ì
2.26
Since {un }n∈Æ is a Cauchy sequence in WΔ,T 0, TÌ, N , by 2.22, one has
1,p
|uσn t − uσm t|2 Δt −→ 0 m, n −→ ∞,
2.27
2
Δ
un t − uΔ
m t Δt −→ 0
2.28
Ì
0,T Ì
0,T m, n −→ ∞.
It follows from Lemma 2.20, 2.27, and 2.28 that
|un t − um t|2 Δt Ì
0,T Ì
0,T ≤2
2
σ
Δ
−
u
t
t
Δt
un t − uσm t − μt uΔ
n
m
|uσn t − uσm t|2 Δt 2σT2
Ì
0,T 2
Δ
un t − uΔ
m t Δt
Ì
0,T −→ 0 m, n∞.
2.29
By Lemma 2.12, 2.28 and 2.29, there exist u, g ∈ LΔ 0, TÌ, N such that
p
un − uLp −→ 0 n −→ ∞,
Δ
Δ
un − g p
LΔ
−→ 0
n −→ ∞.
2.30
From 2.26 and 2.30, one has
Ì
0,T ut, φΔ t Δt −
gt, φσ t Δt,
Ì
0,T 1
∀φ ∈ CT,rd
0, TÌ, N .
2.31
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From 2.31, we conclude that u ∈ WΔ,T 0, TÌ, N . Moreover, by Lemma 2.20 and 2.30,
one has
1,p
|uσn t
Ì
0,T 2
− u t| Δt σ
Ì
0,T 2
−
gt
t
Δt
un t − ut μt uΔ
n
Ì
0,T ≤2
2
Δ
un t − ut μt uΔ
n t − u t Δt
|un t − ut|2 Δt 2σT2
Ì
0,T 2
Δ
un t − gt Δt
Ì
0,T −→ 0 n −→ ∞.
2.32
x∈
Thereby, it follows from Remark 2.18, 2.30, 2.32, and Lemma 2.19 that there exists
1,p
N
N
Ì, ⊂ WΔ,T 0, TÌ, such that
1,p
VΔ,T 0, T
un − xW 1,p −→ 0 n −→ ∞.
2.33
Δ,T
1
Obviously, the set HΔ,T
is a Hilbert space together with the inner product
1
u, vHΔ,T
u t, v tΔt Ì
0,T σ
σ
Ì
0,T uΔ t, vΔ t Δt
1
∀u, v ∈ HΔ,T
.
2.34
We will derive some properties of the Banach space WΔ,T 0, TÌ, N .
1,p
Lemma 2.22 see 10, Theorem A.2. Let f : a, bÌ → be a continuous function on a, bÌ
which is delta differentiable on a, bÌ. Then there exist ξ, τ ∈ a, bÌ such that
f Δ τ ≤
fb − fa
≤ f Δ ξ.
b−a
2.35
Theorem 2.23. There exists K Kp > 0 such that the inequality
u∞ ≤ KuW 1,p
2.36
Δ,T
holds for all u ∈ WΔ,T 0, TÌ, N , where u∞ maxt∈0,T |ut|.
Moreover, if 0,T utΔt 0, then
Ì
1,p
Ì
u∞ ≤ K uΔ p .
LΔ
2.37
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Proof. Going to the components of u, we can assume that N 1. If u ∈ WΔ,T 0, TÌ, ,
by Lemma 2.19, Ut 0,t usΔs is absolutely continuous on a, bÌ. It follows from
Lemma 2.22 that there exists ζ ∈ a, bÌ such that
1,p
Ì
UT − U0 1
uζ ≤
T
T
2.38
usΔs.
Ì
0,T Hence, for t ∈ a, bÌ, using Lemma 2.15, 2.38, and Hölder’s inequality, one has that
|ut| uζ ζ,t
≤ |uζ| u sΔs
Δ
Ì
Δ u sΔs
2.39
Ì
0,T 1/p
1 Δ p
1/q
usΔs T
,
≤ u s Δs
T 0,T 0,T Ì
Ì
where 1/p 1/q 1. If 0,T utΔt 0, by 2.39, we obtain 2.37. In the general case, for
t ∈ a, bÌ, by Lemma 2.20 and Hölder’s inequality, we get
Ì
1/p
1 Δ p
1/q
usΔs T
|ut| ≤ u s Δs
T 0,T 0,T 1
≤
T
1
T
1
T
≤T
Ì
|us|Δs T
Ì
0,T Ì
1/q
σ
Δ
1/q
u s − μsu sΔs T
1
|u s|Δs σT
T
Ì
0,T −1/p
p
|u s| Δs
Ì
0,T σ
Δ p
u s Δs
Δ p
u s Δs
Δ 1/q
u sΔs T
T
1/p
Ì
0,T Ì
0,T 1/p
T 1/q
σ
1/p
Ì
0,T Ì
0,T Δ p
u s Δs
Ì
0,T −1/p
σT
1/p
Δ p
u s Δs
Δ p
u s Δs
1/p
2.40
1/p
Ì
0,T Ì
0,T ≤ T −1/p T −1/p σT T 1/q uW 1,p .
Δ,T
From 2.40, 2.36 holds.
Remark 2.24. It follows from Theorem 2.23 that WΔ,T 0, TÌ, N is continuously immersed
into C0, TÌ, N with the norm · ∞ .
1,p
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11
Theorem 2.25. If the sequence {uk }k∈Æ
WΔ,T 0, TÌ, N converges weakly to u in
1,p
⊂
WΔ,T 0, TÌ, N , then {uk }k∈Æ converges strongly in C0, TÌ, N to u.
1,p
Proof. Since uk u in WΔ,T 0, TÌ, N , {uk }k∈Æ is bounded in WΔ,T 0, TÌ, N and,
hence, in C0, TÌ, N . It follows from Remark 2.24 that uk u in C0, TÌ, N . For
t1 , t2 ∈ 0, TÌ, t1 ≤ t2 , there exists C1 > 0 such that
1,p
1,p
|uk t2 − uk t1 | ≤
Δ uk sΔs
Ì
t1 ,t2 1/q
≤ t2 − t1 Δ p
uk s Δs
1/p
Ì
t1 ,t2 2.41
≤ t2 − t1 1/q uk W 1,p
Δ,T
≤ C1 t2 − t1 1/q .
Hence, the sequence {uk }k∈Æ is equicontinuous. By Ascoli-Arzela theorem, {uk }k∈Æ is
relatively compact in C0, TÌ, N . By the uniqueness of the weak limit in C0, TÌ, N ,
every uniformly convergent subsequence of {uk }k∈Æ converges to u. Thus, {uk }k∈Æ converges
strongly in C0, TÌ, N to u.
Remark 2.26. By Theorem 2.25, the immersion WΔ,T 0, TÌ, N →
compact.
1,p
C0, TÌ, N is
Theorem 2.27. Let L : 0, TÌ × N × N → , t, x, y → Lt, x, y be Lebesgue Δ-measurable
in t for each x, y ∈ N × N and continuously differentiable in x, y for every t ∈ 0, TÌ. If there
q
exist a ∈ C , , b, c ∈ 0, TÌ → , bσ ∈ L1Δ 0, TÌ, and cσ ∈ LΔ 0, TÌ, 1 < q <
∞ such that for Δ-almost t ∈ 0, TÌ and every x, y ∈ N × N , one has
L t, x, y ≤ a|x| bt yp ,
Lx t, x, y ≤ a|x| bt yp ,
Ly t, x, y ≤ a|x| ct yp−1 ,
where 1/p 1/q 1, then the functional Φ : WΔ,T 0, TÌ, N →
1,p
Φu defined as
L σt, uσ t, uΔ t Δt
Ì
0,T 2.42
2.43
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is continuously differentiable on WΔ,T 0, TÌ, N and
1,p
Φ u, v Ì
0,T Lx σt, uσ t, uΔ t , vσ t Ly σt, uσ t, uΔ t , vΔ t Δt.
2.44
Proof. It suffices to prove that Φ has at every point u a directional derivative Φ u ∈
1,p
WΔ,T 0, TÌ, N ∗ given by 2.44 and that the mapping
1,p ∗
1,p
Φ : WΔ,T 0, TÌ, N −→ WΔ,T 0, TÌ, N
2.45
is continuous.
1,p
Firstly, it follows from 2.42 that Φ is everywhere finite on WΔ,T 0, TÌ, N . We
define, for u and v fixed in WΔ,T 0, TÌ, N , t ∈ 0, TÌ, λ ∈ −1, 1,
1,p
Gλ, t L σt, uσ t λvσ t, uΔ t λvΔ t ,
Ψλ Gλ, tΔt Φu λv.
2.46
Ì
0,T From 2.42, one has
|Dλ Gλ, t| ≤ Dx L σt, uσ t λvσ t, uΔ t λvΔ t , vσ t Dy L σt, uσ t λvσ t, uΔ t λvΔ t , vΔ t p ≤ a|uσ t λvσ t| bσ t uΔ t λvΔ t |vσ t|
p−1 Δ a|uσ t λvσ t| cσ t uΔ t λvΔ t
v t
2.47
p ≤ a bσ t uΔ t vΔ t
|vσ t|
p−1 Δ a cσ t uΔ t vΔ t
v t
dt,
where
a
max
a|ut λvt|,
Ì
λ,t∈−1,1×0,T 2.48
Advances in Difference Equations
13
thus, d ∈ L1Δ 0, TÌ, . Since bσ ∈ L1Δ 0, TÌ, , |uΔ | |vΔ |p ∈ L1Δ 0, TÌ, , cσ ∈
q
LΔ 0, TÌ, , one has
|Dλ Gλ, t| ≤ dt,
Ψ 0 Dλ G0, tΔt
Ì
0,T Ì
0,T 2.49
Dx L σt, uσ t, uΔ t , vσ t Dy L σt, uσ t, uΔ t , vΔ t Δt.
2.50
On the other hand, it follows from 2.42 that
p Dx L σt, uσ t, uΔ t ≤ a|uσ t| bσ t uΔ t ψ1 t,
p−1 ψ2t,
Dy L σt, uσ t, uΔ t ≤ a|uσ t| cσ t uΔ t
2.51
thus ψ1 ∈ L1Δ 0, TÌ, , ψ2 ∈ LΔ 0, TÌ, . Thereby, by Theorem 2.23, 2.50, and 2.51,
there exist positive constants C2 , C3 , C4 such that
q
Ì
0,T Dx L σt, uσ t, uΔ t , vσ t Dy L σt, uσ t, uΔ t , vΔ t Δt
≤ C2 v∞ C3 vΔ 2.52
p
LΔ
≤ C4 vW 1,p
Δ,T
and Φ has a directional derivative at u and Φ u ∈ WΔ,T 0, TÌ, N ∗ given by 2.44.
1,p
Moreover, 2.42 implies that the mapping from WΔ,T 0, TÌ, N into L1Δ 0, TÌ,
N
× LqΔ 0, TÌ, N defined by
1,p
u −→ Dx L ·, uσ , uΔ , Dy L ·, uσ , uΔ
2.53
is continuous, so that Φ is continuous from WΔ,T 0, TÌ, N into WΔ,T 0, TÌ, N ∗ .
1,p
1,p
3. Variational Setting
In this section, in order to apply the critical point theory, we make a variational structure.
From this variational structure, we can reduce the problem of finding solutions of problem
1.1 to one of seeking the critical points of a corresponding functional.
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1,2
1
WΔ,T
0, TÌ, N with the inner product
By Theorem 2.21, the space HΔ,T
1
u, vHΔ,T
u t, v tΔt σ
Ì
0,T σ
Ì
0,T uΔ t, vΔ t Δt
3.1
and the induced norm
1
uHΔ,T
|u t| Δt Ì
0,T 2
σ
Δ 2
u t Δt
1/2
3.2
Ì
0,T is Hilbert space.
Since w ∈ R 0, TÌ, , by Theorem 2.44 in 3, one has that
ew t, 0 > 0
3.3
∀t ∈ 0, TÌ,
1
in HΔ,T
, we also consider the inner product
u, v ew t, 0uσ t, vσ tΔt Ì
0,T ew t, 0 uΔ t, vΔ t Δt
Ì
0,T 3.4
and the induced norm
u Ì
0,T 2
ew t, 0|u t| Δt σ
2
ew t, 0uΔ t Δt
Ì
0,T 1/2
,
3.5
we prove the following theorem.
1
are equivalent.
Theorem 3.1. The norm · and · HΔ,T
Proof. Since ew ·, 0 is continuous on 0, TÌ and
ew t, 0 > 0
3.6
∀t ∈ 0, TÌ,
there exist two positive constants M1 and M2 such that
M1 min ew t, 0,
Ì
t∈0,T M2 max ew t, 0.
Ì
3.7
1
∀u ∈ HΔ,T
.
3.8
t∈0,T Hence, one has
M1 uH 1 ≤ u ≤ M2 uH 1 ,
Δ,T
Δ,T
1
Consequently, the norm · and · HΔ,T
are equivalent.
Advances in Difference Equations
15
defined by
1
→
Consider the functional ϕ : HΔ,T
1
ϕu 2
Δ 2
ew t, 0u t Δt Ì
0,T ew t, 0Fσt, uσ tΔt.
Ì
0,T 3.9
We have the following facts.
1
Theorem 3.2. The functional ϕ is continuously differentiable on HΔ,T
and
ϕ u, v Δ
Δ
ew t, 0 u t, v t Δt Ì
0,T ew t, 0∇Fσt, uσ t, vσ tΔt 3.10
Ì
0,T 1
.
for all v ∈ HΔ,T
Proof. Let Lt, x, y ew ρt, 0 1/2|y|2 Ft, x for all x, y ∈ N and t ∈ 0, TÌ. Then,
by condition A, Lt, x, y satisfies all assumptions of Theorem 2.27. Hence, by Theorem 2.27,
1
we know that the functional ϕ is continuously differentiable on HΔ,T
and
ϕ u, v ew t, 0 uΔ t, vΔ t Δt Ì
0,T ew t, 0∇Fσt, uσ t, vσ tΔt 3.11
Ì
0,T 1
.
for all v ∈ HΔ,T
1
1
Theorem 3.3. If u ∈ HΔ,T
is a critical point of ϕ in HΔ,T
, that is, ϕ u 0, then u is a solution of
problem 1.1.
Proof. Since ϕ u 0, it follows from Theorem 3.2 that
ew t, 0 uΔ t, vΔ t Δt Ì
0,T ew t, 0∇Fσt, uσ t, vσ tΔt 0
Ì
0,T 3.12
1
, that is,
for all v ∈ HΔ,T
Δ
Δ
ew t, 0 u t, v t Δt −
Ì
0,T ew t, 0∇Fσt, uσ t, vσ tΔt
Ì
0,T 3.13
1
1
. From condition A and Definition 2.17, one has that ew ·, 0uΔ ∈ HΔ,T
. By
for all v ∈ HΔ,T
1,2
0, TÌ, N such that
Lemma 2.19 and 2.20, there exists a unique function x ∈ VΔ,T
x u,
Δ
ew t, 0xΔ t ew t, 0∇Fσt, uσ t Δ-a.e.
ew t, 0∇Fσt, uσ tΔt 0.
Ì
0,T Ì
on 0, Tκ ,
3.14
3.15
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Advances in Difference Equations
By 3.14, one has
Ì
ew t, 0xΔ t wtew t, 0xΔ σt ew t, 0∇Fσt, uσ t Δ-a.e. on 0, Tκ .
2
3.16
Combining 3.14, 3.15, 3.16, and Lemma 2.19, we obtain
xΔ t wtxΔ σt ∇Fσt, uσ t Δ-a.e.
2
Δ
x0 − xT 0,
Ì
on 0, Tκ ,
Δ
x 0 − x T 0.
3.17
1,2
1
with its absolutely continuous representative x ∈ VΔ,T
0, TÌ, N for
We identify u ∈ HΔ,T
which 3.14 holds. Thus u is a solution of problem 1.1.
1
Theorem 3.4. The functional ϕ is weakly lower semicontinuous on HΔ,T
.
Proof. Let
1
ϕ1 u 2
ϕ2 u 2
ew t, 0uΔ t Δt,
Ì
0,T ew t, 0Fσt, u tΔt.
Ì
0,T 3.18
σ
Then, ϕ1 is continuous and convex. Hence, ϕ1 is weakly lower semicontinuous. On the other
1
1
, un u in HΔ,T
. By Theorem 2.25, {uk }k∈Æ converges strongly in
hand, let {un }n∈Æ ⊂ HΔ,T
N
C0, TÌ, to u. By condition A, one has
σ
σ
ϕ2 un − ϕ2 u ew t, 0Fσt, un tΔt −
ew t, 0Fσt, u tΔt
0,T 0,T ≤ M2
|Fσt, uσn t − Fσt, uσ t|Δt
Ì
Ì
Ì
3.19
0,T −→ 0.
Thus, ϕ2 is weakly continuous. Consequently, ϕ ϕ1 ϕ2 is weakly lower semicontinuous.
To prove the existence of solutions for problem 1.1, we need the following
definitions.
Definition 3.5 see 23, page 81. Let X be a real Banach space, I ∈ C1 X, and c ∈ .
I is said to satisfy PSc -condition on X if the existence of a sequence {xn } ⊆ X such that
Ixn → c and I xn → 0 as n → ∞, implies that c is a critical value of I.
Definition 3.6 see 23, page 81. Let X be a real Banach space and I ∈ C1 X, . I is said
to satisfy P.S. condition on X if any sequence {xn } ⊆ X for which Ixn is bounded and
I xn → 0 as n → ∞, possesses a convergent subsequence in X.
Advances in Difference Equations
17
Remark 3.7. It is clear that the P.S. condition implies the PSc -condition for each c ∈ .
We also need the following result to prove our main results of this paper.
Lemma 3.8 see 24, Theorem 1.1. If ϕ is weakly lower semicontinuous on a reflexive Banach
space X and has a bounded minimizing sequence, then ϕ has a minimum on X.
Lemma 3.9 see 24, Theorem 4.7. Let X be a Banach space and let J ∈ C1 X, , R > 0. Assume
that X splits into a direct sum of closed subspace X X − ⊕ X with dim X − < ∞ and supS− J <
R
infX J, where S−R {u ∈ X − : u R}. Let BR− {u ∈ X − : u ≤ R}, M {h ∈ CBR− , X :
hs s if s ∈ S−R } and c infh∈M maxs∈BR− Jhs. Then, if J satisfies the PSc condition, c is a
critical value of J.
Lemma 3.10 see 24, Proposition 1.4. Let G ∈ C1 N , be a convex function. Then, for all
x, y ∈ N one has
Gx ≥ G y ∇G y , x − y .
3.20
4. Existence of Solutions
1
For u ∈ HΔ,T
, let u 1/T 0,T utΔt and u
t ut − u, then 0,T u
tΔt 0. We have
the following existence results.
Ì
Ì
Theorem 4.1. Assume that (A) and the following conditions are satisfied.
i There exist f, g : 0, TÌ →
and α ∈ 0, 1 such that f σ , g σ ∈ L1Δ 0, TÌ, and
|∇Ft, x| ≤ ft|x|α gt
4.1
for all x ∈ N and Δ-a.e. t ∈ 0, TÌ.
ii |x|−2α
Ìew t, 0Fσt, xΔt → ∞ as |x| → ∞.
0,T Then problem 1.1 has at least one solution which minimizes the function ϕ.
Proof. By Theorem 2.23, there exists C5 > 0 such that
u2∞ ≤ C5
Δ 2
u t Δt.
Ì
0,T 4.2
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Advances in Difference Equations
It follows from i, Theorem 2.23 and 4.2 that
σ
ew t, 0Fσt, u t − Fσt, uΔt
0,T 1
≤
ew t, 0
uσ t, u
σ tds Δt
∇Fσt, u s
0,T 0
Ì
Ì
≤ M2
1
Ì
0,T σ
0
uα∞ u ∞
α
≤ 2M2 |u| α
σ
f tΔt M2 u ∞
Ì
0,T 1
f t|u s
u t| |
u t|ds Δt M2
σ
σ
Ì
0,T g t|
u t|ds Δt
σ
0
g σ tΔt
Ì
0,T 2
4M22 C5 2α
M1
2
σ
≤
f tΔt
u ∞ |u|
4C5
M1
0,T α1
σ
2M2 f tdt M2 g σ tΔt
u ∞
u ∞
σ
Ì
Ì
Ì
0,T M1
≤
4
C8
Δ 2
2α
u t Δt C6 |u| C7
Ì
0,T 0,T Δ 2
u t Δt
Δ 2
u t Δt
α1/2
Ì
0,T 1/2
Ì
0,T 4.3
1
for all u ∈ HΔ,T
, where C6 4M22 C5 /M1 0,T f σ tdt2 , C7 2M2 C5 α1/2 0,T f σ tΔt,
C8 M2 C5 1/2 0,T g σ tΔt. Therefore, one has
Ì
Ì
ϕu 1
2
1
2
2
ew t, 0uΔ t Δt Ì
0,T 2
ew t, 0uΔ t Δt Ì
0,T 1
≥ M1
4
− C7
ew t, 0Fσt, uσ t − Fσt, uΔt
Ì
0,T 4.4
Δ 2
2α
|u|−2α
u t Δt |u|
Ì
0,T ew t, 0Fσt, uσ tΔt
Ì
0,T ew t, 0Fσt, uΔt
Ì
0,T Ì
Δ 2
u t Δt
α1/2
− C8
ew t, 0Fσt, uΔt − C6
Ì
0,T Ì
0,T Δ 2
u t Δt
Ì
0,T 1/2
Advances in Difference Equations
1
for all u ∈ HΔ,T
. As u → ∞ if and only if |u|2 that
19
T
0
|uΔ t|2 dt1/2 → ∞, 4.4 and ii imply
ϕu −→ ∞ as u −→ ∞.
4.5
1
By Lemma 3.8 and Theorem 3.4, ϕ has a minimum point on HΔ,T
, which is a critical point of
ϕ. From Theorem 3.3, problem 1.1 has at least one solution.
, T 2π, N 3. Consider the damped vibration problem on time-scale
Example 4.2. Let
üt cos tu̇t ∇Ft, ut,
a.e. t ∈ 0, 2π,
4.6
u0 − u2π u̇0 − u̇2π 0,
where Ft, x 4/3 t|x|3/2 .
Since, Ft, x 4/3 t|x|3/2 , wt cos t, ∇Ft, x 3/24/3 tx|x|−1/2 , α 1/2,
3 4
t |x|1/2 ,
2 3
2π
2π 4
4
t |x|3/2 dt ≥ |x|1/2 e−1
t dt
esin t
|x|−2×1/2
3
3
0
0
8
π 2π 2 e−1 |x|1/2 −→ ∞
3
|∇Ft, x| ≤
4.7
as |x| −→ ∞,
all conditions of Theorem 4.1 hold. According to Theorem 4.1, problem 4.6 has at least one
solution. Moreover, 0 is not the solution of problem 4.6. Thus, problem 4.6 has at least one
nontrivial solution.
Theorem 4.3. Suppose that assumption A and the condition i of Theorem 4.1 hold. Assume that
iii |x|−2α
Ìew t, 0Fσt, xΔt → −∞ as |x| → ∞.
0,T Then problem 1.1 has at least one solution.
Firstly, we prove the following lemma.
Lemma 4.4. Suppose that the conditions of Theorem 4.3 hold. Then ϕ satisfies P.S. condition.
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Advances in Difference Equations
1
be a P.S. sequence for ϕ, that is, {ϕun } is bounded and ϕ un → 0 as
Proof. Let {un } ⊆ HΔ,T
n → ∞. It follows from i, Theorem 2.23 and 4.2 that
σ
ew t, 0Fσt, un t − Fσt, un Δt
0,T 1
σ
σ
≤
ew t, 0
un t, u
n tds Δt
∇Fσt, un s
0,T 0
Ì
≤ M2
Ì
1
f t|un σ
Ì
0,T 0
s
uσn t|α |
uσn t|ds
un α∞ un ∞
≤ 2M2 |un |α ≤
Δt M2
f σ tΔt M2 un ∞
Ì
0,T 4M22 C5
M1
f σ tΔt
un 2∞ |un |2α
4C5
M1
0,T σ
2M2 f
un α1
M
u
tdt
2
n ∞
∞
g
Ì
0,T ≤
1
σ
0
uσn t|ds
t|
Δt
g σ tΔt
Ì
0,T 2
Ì
M1
4
Ì
0,T C7
Ì
0,T Δ 2
2α
un t Δt C6 |un |
Ì
0,T Δ 2
un t Δt
α1/2
Ì
0,T C8
g σ tΔt
Δ 2
un t Δt
1/2
Ì
0,T 4.8
for all n. By 4.8 and i, one has
un ≥ ϕ un , u
n
Δ 2
ew t, 0un t Δt Ì
0,T 3M1
≥
4
Δ 2
2α
un t Δt − C6 |un | − C7
Ì
0,T − C8
ew t, 0∇Fσt, uσn t, u
n tΔt
Ì
0,T Δ 2
un t Δt
α1/2
4.9
Ì
0,T 1/2
Δ 2
un t Δt
Ì
0,T for all large n. It follows from 3.5 and 4.2 that
M1
Δ 2
un 2 ≤ M2 1 TC5 un t Δt ≤ Ì
0,T Δ 2
un t dt.
Ì
0,T 4.10
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21
The inequalities 4.9 and 4.10 imply that
α
C9 |un | ≥
Δ 2
un t Δt
1/2
4.11
− C10
Ì
0,T for all large n and some positive constants C9 and C10 . Similar to the proof of Theorem 4.1,
one has
σ
ew t, 0Fσt, un t − Fσt, un Δt
0,T M1
Δ 2
2α
≤
un t dt C6 |un |
4 0,T Ì
Ì
C7
Δ 2
un t Δt
α1/2
C8
Ì
0,T 4.12
Δ 2
un t Δt
1/2
Ì
0,T for all n. By the boundedness of {ϕun }, 4.11 and 4.12, there exists constant C11 such that
C11 ≤ ϕun 1
Δ 2
ew t, 0un t dt ew t, 0Fσt, uσn t − Fσt, un Δt
2 0,T 0,T ew t, 0Fσt, un Δt
Ì
Ì
Ì
0,T 3
≤ M2
4
Ì
0,T C7
≤ |un |
|un |
Δ 2
un t Δt
−2α
α1/2
C8
ew t, 0Fσt, un Δt
Ì
0,T Ì
0,T 2α
Δ 2
2α
un t Δt C6 |un | Δ 2
un t Δt
Ì
0,T 4.13
1/2
ew t, 0Fσt, un Δt C12
Ì
0,T for all large n and some constant C12 . It follows from 4.13 and iii that {|un |} is bounded.
1
by 4.10 and 4.11. Therefore, there exists a subsequence of
Hence {un } is bounded in HΔ,T
{un } for simplicity denoted again by {un } such that
un u
1
in HΔ,T
.
4.14
By Theorem 2.25, one has
un −→ u
in C 0, TÌ, N .
4.15
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On the other hand, one has
ϕ un − ϕ u, un − u
2
Δ
ew t, 0uΔ
n t − u t Δt
Ì
4.16
0,T ew t, 0∇Fσt, uσn t − ∇Fσt, uσ t, uσn t − uσ tΔt.
Ì
0,T 1
From 4.14, 4.15, 4.16, and A, it follows that un → u in HΔ,T
. Thus, ϕ satisfies P.S.
condition.
Now, we prove Theorem 4.3.
1
given by
Proof. Let W be the subspace of HΔ,T
W
u∈
utΔt 0 ,
4.17
as u ∈ W, u −→ ∞.
4.18
1
HΔ,T
:
Ì
0,T 1
then, HΔ,T
N ⊕ W. We show that
ϕu −→ ∞
Indeed, for u ∈ W, then u 0, similar to the proof of Theorem 4.1, one has
σ
ew t, 0Fσt, u t − Fσt, 0Δt
0,T Ì
M1
≤
4
Δ 2
u t Δt C7
Ì
0,T Δ 2
u t Δt
Ì
0,T α1/2
C8
Δ 2
u t Δt
1/2
Ì
0,T .
4.19
It follows from 4.19 that
1
ϕu 2
Ì
0,T ew t, 0Fσt, 0Δt
Δ 2
u t Δt − C7
Ì
0,T Δ 2
u t Δt
Ì
0,T α1/2
− C8
Δ 2
u t Δt
1/2
Ì
0,T ew t, 0Fσt, 0Δt
Ì
0,T ew t, 0Fσt, uσ t − Fσt, 0Δt
Ì
0,T Ì
0,T M1
≥
4
Δ 2
ew t, 0u t Δt 4.20
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23
for all u ∈ W. By Theorem 2.23 and Theorem 3.1, one has
u −→ ∞ ⇐⇒ uΔ L2
−→ ∞
4.21
on W. Hence 4.18 follows from 4.20.
On the other hand, by iii, one has
ϕu ew t, 0Fσt, uΔt
Ì
0,T ≤ |u|2α |u|−2α
4.22
ew t, 0Fσt, uΔt
Ì
0,T −→ −∞
as u ∈ N and |u| → ∞. By Theorem 3.3, Lemmas 3.9 and 4.4, problem 1.1 has at least one
solution.
, T 20, N 5. Consider the damped vibration problem on time-scale
Example 4.5. Let
Δ2 t wtΔut 1 ∇Ft 1, ut 1,
t ∈ 0, 19 ∩ ,
u0 − u20 Δu0 − Δu20 0,
4.23
where Ft, x −|x|5/3 1, 1, 2, 1, 0, x and
⎧
1
⎪
⎪
⎨− ,
2
wt ⎪
⎪
⎩ 18
2 − 1,
t ∈ 0, 18 ∩ ,
4.24
t 19.
Since, Ft, x −|x|5/3 1, 1, 2, 1, 0, x, ∇Ft, x −5/3x|x|−1/3 1, 1, 2, 1, 0, α 2/3,
!
ew t, 0 t−1
s0 1 ws, ew 20, 0 1,
|∇Ft, x| ≤
|x|
−2×2/3
5 2/3 √
|x| 7,
3
ew t, 0Fσt, xΔt
Ì
0,T |x|−4/3 −|x|5/3 1, 1, 2, 1, 0, x
≤ −|x|1/3
ew t, 0Δt Ì
0,T √
7|x|−1/3
4.25
ew t, 0Δt
Ì
0,T ew t, 0Δt −→ −∞ as |x| −→ ∞,
Ì
0,T all conditions of Theorem 4.3 hold. According to Theorem 4.3, problem 4.23 has at least one
solution. Moreover, 0 is not the solution of problem 4.23. Thus, problem 4.23 has at least
one nontrivial solution.
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Theorem 4.6. Suppose that assumption A and the following condition are satisfied.
iv Ft, · is convex for Δ-a.e. t ∈ 0, TÌ and that
ew t, 0Fσt, xΔt −→ ∞ as |x| −→ ∞.
Ì
0,T 4.26
Then problem 1.1 has at least one solution which minimizes the function ϕ.
Proof. By assumption, the function G : N →
defined by
Gx ew t, 0Fσt, xΔt
Ì
0,T 4.27
has a minimum at some point x for which
ew t, 0∇Fσt, xΔt 0.
Ì
0,T 4.28
Let {uk } be a minimizing sequence for ϕ. From Lemma 3.10 and 4.28, one has
1
ϕuk 2
Ì
0,T 1
≥
2
ew t, 0Fσt, xΔt
Ì
0,T ew t, 0 ∇Fσt, x, uσk t − x Δt
Ì
0,T Δ 2
ew t, 0uk t Δt Ì
0,T Δ 2
ew t, 0uk t Δt Ì
1
2
Ì
0,T ew t, 0Fσt, xΔt
0,T ew t, 0 F σt, uσk t − Fσt, x Δt
Ì
0,T Δ 2
ew t, 0uk t Δt ew t, 0 ∇Fσt, x, u
σk t Δt,
Ì
0,T ew t, 0Fσt, xΔt
Ì
0,T 4.29
Advances in Difference Equations
25
where u
k t uk t−uk , uk 1/T
1
ϕuk ≥
2
Ìuk tΔt. By 4.29, A and Theorem 2.23, we obtain
0,T Δ 2
ew t, 0uk t Δt Ì
0,T − M2
1
≥ M1
2
ew t, 0Fσt, xΔt
Ì
0,T uk ∞
|∇Fσt, x|Δt 4.30
Ì
0,T Δ 2
ew t, 0uk t Δt − C13 − C14
Ì
0,T Δ 2
uk t Δt
1/2
Ì
0,T for some positive constants C13 and C14 . Thus, by 4.30, there exists C15 > 0 such that
Δ 2
uk t Δt ≤ C15 .
4.31
Ì
0,T Theorem 2.23 and 4.31 imply that there exists C16 > 0 such that
uk ∞ ≤ C16 .
4.32
By iv, one has
uk
F σt,
2
F σt,
uσk t − u
σk t
2
1 1 uσk t
≤ F σt, uσk t F σt, −
2
2
4.33
for Δ-a.e. t ∈ 0, TÌ and all k ∈ . It follows from 3.9 and 4.33 that
ϕuk ≥
1
2
−
2
ew t, 0uΔ
Δt
2
t
k
Ì
0,T uk
Δt
ew t, 0F σt,
2
Ì
0,T ew t, 0F σt, −
uσk t Δt.
4.34
Ì
0,T Combining 4.32 and 4.34, there exists C17 > 0 such that
ϕuk ≥ 2
uk
Δt − C17 .
ew t, 0F σt,
2
Ì
0,T 4.35
1
Therefore, by 4.35 and iv, {uk } is bounded. Hence {uk } is bounded in HΔ,T
by
1
Theorem 2.23 and 4.31. By Lemma 3.8 and Theorem 3.4, ϕ has a minimum point on HΔ,T ,
which is a critical point of ϕ. Hence, problem 1.1 has at least one solution which minimizes
the function ϕ.
26
Advances in Difference Equations
Example 4.7. Let 0 ∪{0},
problem on time-scale
{2k : k ∈ 0 }, T 64, N 1. Consider the damped vibration
uΔ t wtuΔ 2t ∇F2t, u2t,
2
t ∈ {1, 2, 4, 8, 16, 32},
u0 − u64 uΔ 0 − uΔ 64 0,
4.36
where Ft, x x2 2x and
⎧
1
⎪
⎪
⎪
⎨− 2t ,
wt ⎪
⎪
⎪
⎩ 31 ,
32
t ∈ {1, 2, 4, 8, 16},
4.37
t 32.
!
Since, Ft, x x2 2x, ew t, 0 s∈Ì∩0,t1 sws, ew 64, 0 1, all conditions
of Theorem 4.6 hold. According to Theorem 4.6, problem 4.36 has at least one solution.
Moreover, 0 is not the solution of problem 4.36. Thus, problem 4.36 has at least one
nontrivial solution.
5. Conclusion
In this paper, we present a new approach via variational methods and critical point theory
to obtain the existence of solutions for a class of damped vibration problems on time-scales.
Three existence results are obtained. Three examples are presented to illustrate the feasibility
and effectiveness of our results.
Acknowledgment
This work is supported by the National Natural Sciences Foundation of People’s Republic of
China under Grant 10971183.
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