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

Consider the damped vibration problem on time-scale

(1.1)

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

(1.2)

for all and , where denotes the gradient of in .

Problem (1.1) covers the second-order damped vibration problem (for when )

(1.3)

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

(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 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

(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 , 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.

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

(2.1)

while the backward jump operator is defined by

(2.2)

(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

(2.3)

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

(2.4)

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

(2.5)

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

(2.6)

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

(2.7)

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

If and , then the unique solution of the initial value problem

(2.8)

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

(2.9)

Throughout this paper, we will use the following notations:

(2.10)

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

(2.11)

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

(2.12)

with the norm

(2.13)

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

(2.14)

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

(2.15)

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:

(2.16)

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

(2.17)

For , we denote

(2.18)

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

(2.19)

are satisfied and

(2.20)

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

(2.21)

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

(2.22)

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

(2.23)

Proof.

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

(2.24)

Thus, by Lemma 2.19, there exists such that

(2.25)

Combining (2.24) and (2.25), we obtain

(2.26)

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

(2.27)

(2.28)

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

(2.29)

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

(2.30)

From (2.26) and (2.30), one has

(2.31)

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

(2.32)

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

(2.33)

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

(2.34)

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

(2.35)

Theorem 2.23.

There exists such that the inequality

(2.36)

holds for all , where .

Moreover, if , then

(2.37)

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

(2.38)

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

(2.39)

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

(2.40)

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

(2.41)

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

(2.42)

where , then the functional defined as

(2.43)

is continuously differentiable on and

(2.44)

Proof.

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

(2.45)

is continuous.

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

(2.46)

From (2.42), one has

(2.47)

where

(2.48)

thus, . Since , , , one has

(2.49)

(2.50)

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

(2.51)

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

(2.52)

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

Moreover, (2.42) implies that the mapping from into defined by

(2.53)

is continuous, so that is continuous from into .

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

(3.1)

and the induced norm

(3.2)

is Hilbert space.

Since , by Theorem 2.44 in [3], one has that

(3.3)

in , we also consider the inner product

(3.4)

and the induced norm

(3.5)

we prove the following theorem.

Theorem 3.1.

The norm and are equivalent.

Proof.

Since is continuous on and

(3.6)

there exist two positive constants and such that

(3.7)

Hence, one has

(3.8)

Consequently, the norm and are equivalent.

Consider the functional defined by

(3.9)

We have the following facts.

Theorem 3.2.

The functional is continuously differentiable on and

(3.10)

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

(3.11)

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

(3.12)

for all , that is,

(3.13)

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

(3.14)

(3.15)

By (3.14), one has

(3.16)

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

(3.17)

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

(3.18)

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

(3.19)

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

(3.20)

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

(4.1)

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

(4.2)

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

(4.3)

for all , where ,   , . Therefore, one has

(4.4)

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

(4.5)

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

(4.6)

where .

Since, , , , ,

(4.7)

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

(4.8)

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

(4.9)

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

(4.10)

The inequalities (4.9) and (4.10) imply that

(4.11)

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

(4.12)

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

(4.13)

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

(4.14)

By Theorem 2.25, one has

(4.15)

On the other hand, one has

(4.16)

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

(4.17)

then, . We show that

(4.18)

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

(4.19)

It follows from (4.19) that

(4.20)

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

(4.21)

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

On the other hand, by , one has

(4.22)

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

(4.23)

where and

(4.24)

Since, , , , , ,

(4.25)

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

(4.26)

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

Proof.

By assumption, the function defined by

(4.27)

has a minimum at some point for which

(4.28)

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

(4.29)

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

(4.30)

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

(4.31)

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

(4.32)

By , one has

(4.33)

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

(4.34)

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

(4.35)

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

(4.36)

where and

(4.37)

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.