Existence of Solutions for a Class of Damped Vibration Problems on Time Scales
© Yongkun Li and Jianwen Zhou. 2010
Received: 3 June 2010
Accepted: 24 November 2010
Published: 6 December 2010
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.
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 ). 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 , 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  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].
Definition 2.1 (see [3, Definition 1.1]).
Definition 2.2 (see [3, Definition 1.10]).
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]).
Definition 2.5 (see [23, Definition 2.5]).
Definition 2.6 (see [23, Definition 2.6]).
Definition 2.7 (see [3, Definition 2.25]).
Definition 2.8 (see [7, Definition 8.2.18]).
The exponential function has some important properties.
Lemma 2.9 (see [3, Theorem 2.36]).
The -measure and -integration are defined as those in .
Definition 2.10 (see [23, Definition 2.7]).
Definition 2.11 (see [13, Definition 2.3]).
We have the following lemma.
Lemma 2.12 (see [23, Theorem 2.1]).
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]).
Definition 2.14 (see [23, Definition 2.11]).
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 . 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]).
Lemma 2.19 (see [23, Theorem 2.5]).
Lemma 2.20 (see [3, Theorem 1.16]).
Lemma 2.22 (see [10, Theorem A.2]).
From (2.40), (2.36) holds.
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 .
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.
We have the following facts.
To prove the existence of solutions for problem (1.1), we need the following definitions.
Definition 3.5 (see [23, page 81]).
Definition 3.6 (see [23, page 81]).
We also need the following result to prove our main results of this paper.
Lemma 3.8 (see [24, Theorem 1.1]).
Lemma 3.9 (see [24, Theorem 4.7]).
Lemma 3.10 (see [24, Proposition 1.4]).
4. Existence of Solutions
Assume that (A) and the following conditions are satisfied.
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.
Suppose that assumption (A) and the condition (i) of Theorem 4.1 hold. Assume that
Then problem (1.1) has at least one solution.
Firstly, we prove the following lemma.
Now, we prove Theorem 4.3.
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.
Suppose that assumption (A) and the following condition are satisfied.
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 .
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.
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.
This work is supported by the National Natural Sciences Foundation of People's Republic of China under Grant 10971183.
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