- Research Article
- Open Access
- Published:
Existence of Solutions for a Class of Damped Vibration Problems on Time Scales
Advances in Difference Equations volume 2010, Article number: 727486 (2010)
Abstract
We present a recent approach via variational methods and critical point theory to obtain the existence of solutions for a class of damped vibration problems on time scale ,
-a.e.
,
where
denotes the delta (or Hilger) derivative of
at
,
,
is the forward jump operator,
is a positive constant,
,
, and
. 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.
1. Introduction
Consider the damped vibration problem on time-scale

where denotes the delta (or Hilger) derivative of
at
,
,
is the forward jump operator,
is a positive constant,
,
, and
satisfies the following assumption.
(A) is
-measurable in
for every
and continuously differentiable in
for
and there exist
such that

for all and
, where
denotes the gradient of
in
.
Problem (1.1) covers the second-order damped vibration problem (for when )

as well as the second-order discrete damped vibration problem (for when ,
)

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 time-scales 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 , (1.1) is the second-order Hamiltonian system on time-scale

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 , 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.
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 . We start by the definitions of the forward and backward jump operators.
Definition 2.1 (see [3, Definition 1.1]).
Let be a time-scale, for
, the forward jump operator
is defined by

while the backward jump operator is defined by

(supplemented by and
, where
denotes the empty set). A point
is called right scattered, left scattered, if
,
hold, respectively. Points that are right scattered and left scattered at the same time are called isolated. Also, if
and
, then
is called right-dense, and if
and
, then
is called left dense. Points that are right-dense and left dense at the same time are called dense. The set
which is derived from the time-scale
as follows: if
has a left scattered maximum
, the
, otherwise,
. Finally, the graininess function
is defined by

When ,
, we denote the intervals
,
, and
in
by

respectively. Note that if
is left dense and
if
is left scattered. We denote
, therefore
if
is left dense and
if
is left scattered.
Definition 2.2 (see [3, Definition 1.10]).
Assume that is a function and let
. Then we define
to be the number
provided it exists
with the property that given any
, there is a neighborhood
of
(i.e.,
,
for some
such that

We call the delta
or Hilger
derivative of
at
. The function
is delta
or Hilger
differentiable on
provided
exists for all
. The function
is then called the delta derivative of
on
.
Definition 2.3 (see [23, Definition 2.3]).
Assume that is a function,
and let
. Then we define
(provided it exists). We call
the delta
or Hilger
derivative of
at
. The function
is delta
or Hilger
differentiable provided
exists for all
. The function
is then called the delta derivative of
on
.
Definition 2.4 (see [3, Definition 2.7]).
For a function we will talk about the second derivative
provided
is differentiable on
with derivative
.
Definition 2.5 (see [23, Definition 2.5]).
For a function we will talk about the second derivative
provided
is differentiable on
with derivative
.
Definition 2.6 (see [23, Definition 2.6]).
A function 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 is regressive provided

holds. The set of all regressive and rd-continuous functions is denoted by

Definition 2.8 (see [7, Definition 8.2.18]).
If and
, then the unique solution of the initial value problem

is called the exponential function and denoted by .
The exponential function has some important properties.
Lemma 2.9 (see [3, Theorem 2.36]).
If , then

Throughout this paper, we will use the following notations:

The -measure
and
-integration are defined as those in [10].
Definition 2.10 (see [23, Definition 2.7]).
Assume that is a function,
and let
be a
-measurable subset of
.
is integrable on
if and only if
(
) are integrable on
, and

Definition 2.11 (see [13, Definition 2.3]).
Let .
is called
-null set if
. Say that a property
holds
-almost everywhere (
-a.e.) on
, or for
-almost all (
-a.a.)
if there is a
-null set
such that
holds for all
.
For ,
, we set the space

with the norm

We have the following lemma.
Lemma 2.12 (see [23, Theorem 2.1]).
Let be such that
. Then the space
is a Banach space together with the norm
. Moreover,
is a Hilbert space together with the inner product given for every
by

where denotes the inner product in
.
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 is said to be absolutely continuous on
(i.e.,
), if for every
, there exists
such that if
is a finite pairwise disjoint family of subintervals of
satisfying
, then
.
Definition 2.14 (see [23, Definition 2.11]).
A function ,
,
,
. We say that
is absolutely continuous on
(i.e.,
), if for every
, there exists
such that if
is a finite pairwise disjoint family of subintervals of
satisfying
, then
.
Absolutely continuous functions have the following properties.
Lemma 2.15 (see [23, Theorem 2.2]).
A function is absolutely continuous on
if and only if
is delta differentiable
-a.e. on
and

Lemma 2.16 (see [23, Theorem 2.3]).
Assume that functions are absolutely continuous on
. Then
is absolutely continuous on
and the following equality is valid:

Now, we recall the definition and properties of the Sobolev space on in [23]. For the sake of convenience, in the sequel, we will let
.
Definition 2.17 (see [23, Definition 2.12]).
Let be such that
and
. We say that
if and only if
and there exists
such that
and

For , we denote

Remark 2.18 (see [23, Remark 2.2]).
is true for every
with
.
Lemma 2.19 (see [23, Theorem 2.5]).
Suppose that for some
with
, and that (2.17) holds for
. Then, there exists a unique function
such that the equalities

are satisfied and

Lemma 2.20 (see [3, Theorem 1.16]).
Assume that is a function and let
. Then, one has the following.
(i)If is differentiable at
, then
is continuous at
.
(ii)If is differentiable at
, then

By identifying with its absolutely continuous representative
for which (2.19) holds, the set
can be endowed with the structure of Banach space.
Theorem 2.21.
Assume and
. The set
is a Banach space together with the norm defined as

Moreover, the set is a Hilbert space together with the inner product

Proof.
Let be a Cauchy sequence in
. That is,
and there exist
such that
and

Thus, by Lemma 2.19, there exists such that

Combining (2.24) and (2.25), we obtain

Since is a Cauchy sequence in
, by (2.22), one has


It follows from Lemma 2.20, (2.27), and (2.28) that

By Lemma 2.12, (2.28) and (2.29), there exist such that

From (2.26) and (2.30), one has

From (2.31), we conclude that . Moreover, by Lemma 2.20 and (2.30), one has

Thereby, it follows from Remark 2.18, (2.30), (2.32), and Lemma 2.19 that there exists such that

Obviously, the set is a Hilbert space together with the inner product

We will derive some properties of the Banach space .
Lemma 2.22 (see [10, Theorem A.2]).
Let be a continuous function on
which is delta differentiable on
. Then there exist
such that

Theorem 2.23.
There exists such that the inequality

holds for all , where
.
Moreover, if , then

Proof.
Going to the components of , we can assume that
. If
, by Lemma 2.19,
is absolutely continuous on
. It follows from Lemma 2.22 that there exists
such that

Hence, for , using Lemma 2.15, (2.38), and Hölder's inequality, one has that

where . If
, by (2.39), we obtain (2.37). In the general case, for
, by Lemma 2.20 and Hölder's inequality, we get

From (2.40), (2.36) holds.
Remark 2.24.
It follows from Theorem 2.23 that is continuously immersed into
with the norm
.
Theorem 2.25.
If the sequence converges weakly to
in
,
, then
converges strongly in
to
.
Proof.
Since in
,
is bounded in
and, hence, in
. It follows from Remark 2.24 that
in
. For
,
, there exists
such that

Hence, the sequence is equicontinuous. By Ascoli-Arzela theorem,
is relatively compact in
. By the uniqueness of the weak limit in
, every uniformly convergent subsequence of
converges to
. Thus,
converges strongly in
to
.
Remark 2.26.
By Theorem 2.25, the immersion is compact.
Theorem 2.27.
Let be Lebesgue
-measurable in
for each
and continuously differentiable in
for every
. If there exist
,
,
and
such that for
-almost
and every
, one has

where , then the functional
defined as

is continuously differentiable on and

Proof.
It suffices to prove that has at every point
a directional derivative
given by (2.44) and that the mapping

is continuous.
Firstly, it follows from (2.42) that is everywhere finite on
. We define, for
and
fixed in
,
,
,

From (2.42), one has

where

thus, . Since
,
,
, one has


On the other hand, it follows from (2.42) that

thus ,
. Thereby, by Theorem 2.23, (2.50), and (2.51), there exist positive constants
such that

and has a directional derivative at
and
given by (2.44).
Moreover, (2.42) implies that the mapping from into
defined by

is continuous, so that is continuous from
into
.
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.
By Theorem 2.21, the space with the inner product

and the induced norm

is Hilbert space.
Since , by Theorem 2.44 in [3], one has that

in , we also consider the inner product

and the induced norm

we prove the following theorem.
Theorem 3.1.
The norm and
are equivalent.
Proof.
Since is continuous on
and

there exist two positive constants and
such that

Hence, one has

Consequently, the norm and
are equivalent.
Consider the functional defined by

We have the following facts.
Theorem 3.2.
The functional is continuously differentiable on
and

for all .
Proof.
Let for all
and
. Then, by condition
,
satisfies all assumptions of Theorem 2.27. Hence, by Theorem 2.27, we know that the functional
is continuously differentiable on
and

for all .
Theorem 3.3.
If is a critical point of
in
, that is,
, then
is a solution of problem (1.1).
Proof.
Since , it follows from Theorem 3.2 that

for all , that is,

for all . From condition (A) and Definition 2.17, one has that
. By Lemma 2.19 and (2.20), there exists a unique function
such that


By (3.14), one has

Combining (3.14), (3.15), (3.16), and Lemma 2.19, we obtain

We identify with its absolutely continuous representative
for which (3.14) holds. Thus
is a solution of problem (1.1).
Theorem 3.4.
The functional is weakly lower semicontinuous on
.
Proof.
Let

Then, is continuous and convex. Hence,
is weakly lower semicontinuous. On the other hand, let
in
. By Theorem 2.25,
converges strongly in
to
. By condition
, one has

Thus, is weakly continuous. Consequently,
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 be a real Banach space,
and
.
is said to satisfy
-condition on
if the existence of a sequence
such that
and
as
, implies that
is a critical value of
.
Definition 3.6 (see [23, page 81]).
Let be a real Banach space and
.
is said to satisfy P.S. condition on
if any sequence
for which
is bounded and
as
, possesses a convergent subsequence in
.
Remark 3.7.
It is clear that the P.S. condition implies the -condition for each
.
We also need the following result to prove our main results of this paper.
Lemma 3.8 (see [24, Theorem 1.1]).
If is weakly lower semicontinuous on a reflexive Banach space
and has a bounded minimizing sequence, then
has a minimum on
.
Lemma 3.9 (see [24, Theorem 4.7]).
Let be a Banach space and let
. Assume that
splits into a direct sum of closed subspace
with
and
, where
. Let
,
and
. Then, if
satisfies the
condition,
is a critical value of
.
Lemma 3.10 (see [24, Proposition 1.4]).
Let be a convex function. Then, for all
one has

4. Existence of Solutions
For , let
and
, then
. We have the following existence results.
Theorem 4.1.
Assume that (A) and the following conditions are satisfied.
(i)There exist and
such that
and

for all and
-a.e.
.
(ii) as
.
Then problem (1.1) has at least one solution which minimizes the function .
Proof.
By Theorem 2.23, there exists such that

It follows from (i), Theorem 2.23 and (4.2) that

for all , where
,
,
. Therefore, one has

for all . As
if and only if
, (4.4) and
imply that

By Lemma 3.8 and Theorem 3.4, has a minimum point on
, which is a critical point of
. From Theorem 3.3, problem (1.1) has at least one solution.
Example 4.2.
Let . Consider the damped vibration problem on time-scale

where .
Since, ,
,
,
,

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) as
.
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 satisfies P.S. condition.
Proof.
Let be a P.S. sequence for
, that is,
is bounded and
as
. It follows from (i), Theorem 2.23 and (4.2) that

for all . By (4.8) and (i), one has

for all large . It follows from (3.5) and (4.2) that

The inequalities (4.9) and (4.10) imply that

for all large and some positive constants
and
. Similar to the proof of Theorem 4.1, one has

for all . By the boundedness of
, (4.11) and (4.12), there exists constant
such that

for all large and some constant
. It follows from (4.13) and
that
is bounded. Hence
is bounded in
by (4.10) and (4.11). Therefore, there exists a subsequence of
(for simplicity denoted again by
) such that

By Theorem 2.25, one has

On the other hand, one has

From (4.14), (4.15), (4.16), and (), it follows that
in
. Thus,
satisfies P.S. condition.
Now, we prove Theorem 4.3.
Proof.
Let be the subspace of
given by

then, . We show that

Indeed, for , then
, similar to the proof of Theorem 4.1, one has

It follows from (4.19) that

for all . By Theorem 2.23 and Theorem 3.1, one has

on . Hence (4.18) follows from (4.20).
On the other hand, by , one has

as and
. By Theorem 3.3, Lemmas 3.9 and 4.4, problem (1.1) has at least one solution.
Example 4.5.
Let . Consider the damped vibration problem on time-scale

where and

Since, ,
,
,
,
,

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.
Theorem 4.6.
Suppose that assumption (A) and the following condition are satisfied.
(iv) is convex for
-a.e.
and that

Then problem (1.1) has at least one solution which minimizes the function .
Proof.
By assumption, the function defined by

has a minimum at some point for which

Let be a minimizing sequence for
. From Lemma 3.10 and (4.28), one has

where ,
. By (4.29),
and Theorem 2.23, we obtain

for some positive constants and
. Thus, by (4.30), there exists
such that

Theorem 2.23 and (4.31) imply that there exists such that

By , one has

for-a.e.
and all
. It follows from (3.9) and (4.33) that

Combining (4.32) and (4.34), there exists such that

Therefore, by (4.35) and ,
is bounded. Hence
is bounded in
by Theorem 2.23 and (4.31). By Lemma 3.8 and Theorem 3.4,
has a minimum point on
, which is a critical point of
. Hence, problem (1.1) has at least one solution which minimizes the function
.
Example 4.7.
Let . Consider the damped vibration problem on time-scale

where and

Since, , 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.
References
Aulbach B, Hilger S: Linear dynamic processes with inhomogeneous time scale. In Nonlinear Dynamics and Quantum Dynamical Systems (Gaussig, 1990), Math. Res.. Volume 59. Akademie, Berlin, Germany; 1990:9–20.
Lakshmikantham V, Sivasundaram S, Kaymakcalan B: Dynamic Systems on Measure Chains, Mathematics and Its Applications. Volume 370. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1996:x+285.
Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.
Agarwal RP, O'Regan D: Infinite Interval Problems for Differential Difference and Integral Equations. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001:x+341.
Bohner M, Peterso A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.
Agarwal RP, Bohner M, Řehák P: Half-linear dynamic equations. In Nonlinear Analysis and Applications: to V. Lakshmikantham on His 80th Birthday. Vol. 1, 2. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:1–57.
Agarwal RP, Bohner M, Li W-T: Nonoscillation and Oscillation: Theory for Functional Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics. Volume 267. Marcel Dekker, New York, NY, USA; 2004:viii+376.
Agarwal RP, Bohner M: Basic calculus on time scales and some of its applications. Results in Mathematics 1999,35(1–2):3–22.
Anderson DR: Eigenvalue intervals for a second-order mixed-conditions problem on time scales. International Journal of Differential Equations 2002, 7: 97–104.
Guseinov GSh: Integration on time scales. Journal of Mathematical Analysis and Applications 2003,285(1):107–127. 10.1016/S0022-247X(03)00361-5
Bohner M, Guseinov GSh: Improper integrals on time scales. Dynamic Systems and Applications 2003,12(1–2):45–65.
Kaufmann ER, Raffoul YN: Periodic solutions for a neutral nonlinear dynamical equation on a time scale. Journal of Mathematical Analysis and Applications 2006,319(1):315–325. 10.1016/j.jmaa.2006.01.063
Agarwal RP, Otero-Espinar V, Perera K, Vivero DR: Basic properties of Sobolev's spaces on time scales. Advances in Difference Equations 2006, 2006:-14.
Zhang H, Li Y: Existence of positive periodic solutions for functional differential equations with impulse effects on time scales. Communications in Nonlinear Science and Numerical Simulation 2009,14(1):19–26. 10.1016/j.cnsns.2007.08.006
Otero-Espinar V, Vivero DR: Existence and approximation of extremal solutions to first-order infinite systems of functional dynamic equations. Journal of Mathematical Analysis and Applications 2008,339(1):590–597. 10.1016/j.jmaa.2007.06.031
Agarwal RP, Otero-Espinar V, Perera K, Vivero DR: Multiple positive solutions of singular Dirichlet problems on time scales via variational methods. Nonlinear Analysis: Theory, Methods & Applications 2007,67(2):368–381. 10.1016/j.na.2006.05.014
Sun H-R, Li W-T: Existence theory for positive solutions to one-dimensional
-Laplacian boundary value problems on time scales. Journal of Differential Equations 2007,240(2):217–248. 10.1016/j.jde.2007.06.004
Hao Z-C, Xiao T-J, Liang J: Existence of positive solutions for singular boundary value problem on time scales. Journal of Mathematical Analysis and Applications 2007,325(1):517–528. 10.1016/j.jmaa.2006.01.083
Jiang L, Zhou Z: Existence of weak solutions of two-point boundary value problems for second-order dynamic equations on time scales. Nonlinear Analysis: Theory, Methods & Applications 2008,69(4):1376–1388. 10.1016/j.na.2007.06.034
Agarwal RP, Bohner M, Wong PJY: Sturm-Liouville eigenvalue problems on time scales. Applied Mathematics and Computation 1999,99(2–3):153–166. 10.1016/S0096-3003(98)00004-6
Rynne BP:
spaces and boundary value problems on time-scales. Journal of Mathematical Analysis and Applications 2007,328(2):1217–1236. 10.1016/j.jmaa.2006.06.008
Davidson FA, Rynne BP: Eigenfunction expansions in
spaces for boundary value problems on time-scales. Journal of Mathematical Analysis and Applications 2007,335(2):1038–1051. 10.1016/j.jmaa.2007.01.100
Zhou J, Li Y: Sobolev's spaces on time scales and its applications to a class of second order Hamiltonian systems on time scales. Nonlinear Analysis: Theory, Methods & Applications 2010,73(5):1375–1388. 10.1016/j.na.2010.04.070
Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences. Volume 74. Springer, New York, NY, USA; 1989:xiv+277.
Acknowledgment
This work is supported by the National Natural Sciences Foundation of People's Republic of China under Grant 10971183.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Li, Y., Zhou, J. Existence of Solutions for a Class of Damped Vibration Problems on Time Scales. Adv Differ Equ 2010, 727486 (2010). https://doi.org/10.1155/2010/727486
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/727486
Keywords
- Hilbert Space
- Cauchy Sequence
- Functional Differential Equation
- Critical Point Theory
- Positive Periodic Solution