- 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
- 1.
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.
- 2.
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.
- 3.
Bohner M, Peterson A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston, Mass, USA; 2001:x+358.
- 4.
Agarwal RP, O'Regan D: Infinite Interval Problems for Differential Difference and Integral Equations. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001:x+341.
- 5.
Bohner M, Peterso A (Eds): Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.
- 6.
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.
- 7.
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.
- 8.
Agarwal RP, Bohner M: Basic calculus on time scales and some of its applications. Results in Mathematics 1999,35(1–2):3–22.
- 9.
Anderson DR: Eigenvalue intervals for a second-order mixed-conditions problem on time scales. International Journal of Differential Equations 2002, 7: 97–104.
- 10.
Guseinov GSh: Integration on time scales. Journal of Mathematical Analysis and Applications 2003,285(1):107–127. 10.1016/S0022-247X(03)00361-5
- 11.
Bohner M, Guseinov GSh: Improper integrals on time scales. Dynamic Systems and Applications 2003,12(1–2):45–65.
- 12.
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
- 13.
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.
- 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
- 15.
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
- 16.
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
- 17.
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 - 18.
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
- 19.
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
- 20.
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
- 21.
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 - 22.
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 - 23.
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
- 24.
Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences. Volume 74. Springer, New York, NY, USA; 1989:xiv+277.
-Laplacian boundary value problems on time scales. Journal of Differential Equations 2007,240(2):217–248. 10.1016/j.jde.2007.06.004
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
spaces for boundary value problems on time-scales. Journal of Mathematical Analysis and Applications 2007,335(2):1038–1051. 10.1016/j.jmaa.2007.01.100Acknowledgment
This work is supported by the National Natural Sciences Foundation of People's Republic of China under Grant 10971183.
Author information
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:
Keywords
- Hilbert Space
- Cauchy Sequence
- Functional Differential Equation
- Critical Point Theory
- Positive Periodic Solution
