- Research Article
- Open Access

# Existence of Solutions for a Class of Damped Vibration Problems on Time Scales

- Yongkun Li
^{1}Email author and - Jianwen Zhou
^{1}

**2010**:727486

https://doi.org/10.1155/2010/727486

© Yongkun Li and Jianwen Zhou. 2010

**Received:**3 June 2010**Accepted:**24 November 2010**Published:**6 December 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.

## Keywords

- Hilbert Space
- Cauchy Sequence
- Functional Differential Equation
- Critical Point Theory
- Positive Periodic Solution

## 1. Introduction

where denotes the delta (or Hilger) derivative of at , , is the forward jump operator, is a positive constant, , , and satisfies the following assumption.

for all and , where denotes the gradient of in .

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.

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]).

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]).

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]).

Definition 2.8 (see [7, Definition 8.2.18]).

is called the exponential function and denoted by .

The exponential function has some important properties.

Lemma 2.9 (see [3, Theorem 2.36]).

The -measure and -integration are defined as those in [10].

Definition 2.10 (see [23, Definition 2.7]).

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 .

We have the following lemma.

Lemma 2.12 (see [23, Theorem 2.1]).

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]).

Lemma 2.16 (see [23, Theorem 2.3]).

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]).

Remark 2.18 (see [23, Remark 2.2]).

is true for every with .

Lemma 2.19 (see [23, Theorem 2.5]).

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 .

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.

Proof.

We will derive some properties of the Banach space .

Lemma 2.22 (see [10, Theorem A.2]).

Theorem 2.23.

holds for all , where .

Proof.

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.

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.

Proof.

is continuous.

and has a directional derivative at and given by (2.44).

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.

is Hilbert space.

we prove the following theorem.

Theorem 3.1.

The norm and are equivalent.

Proof.

Consequently, the norm and are equivalent.

We have the following facts.

Theorem 3.2.

for all .

Proof.

for all .

Theorem 3.3.

If is a critical point of in , that is, , then is a solution of problem (1.1).

Proof.

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.

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]).

## 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.

for all and -a.e. .

(ii) as .

Then problem (1.1) has at least one solution which minimizes the function .

Proof.

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.

where .

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.

From (4.14), (4.15), (4.16), and ( ), it follows that in . Thus, satisfies P.S. condition.

Now, we prove Theorem 4.3.

Proof.

on . Hence (4.18) follows from (4.20).

as and . By Theorem 3.3, Lemmas 3.9 and 4.4, problem (1.1) has at least one solution.

Example 4.5.

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.

Then problem (1.1) has at least one solution which minimizes the function .

Proof.

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.

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.

## Declarations

### Acknowledgment

This work is supported by the National Natural Sciences Foundation of People's Republic of China under Grant 10971183.

## Authors’ Affiliations

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