Abstract and Applied Analysis

VolumeΒ 2010Β (2010), Article IDΒ 234015, 26 pages

http://dx.doi.org/10.1155/2010/234015

## Boundary Value Problems for Systems of Second-Order Dynamic Equations on Time Scales with -CarathΓ©odory Functions

^{1}DΓ©partement de MathΓ©matiques et de Statistique, UniversitΓ© de MontrΓ©al, CP 6128, Succursale Centre-Ville, MontrΓ©al, QC, H3C 3J7, Canada^{2}DΓ©partement de MathΓ©matiques, CollΓ¨ge Γdouard-Montpetit, 945 Chemin de Chambly, Longueuil, QC, J4H 3M6, Canada

Received 31 August 2010; Accepted 30 November 2010

Academic Editor: J.Β Mawhin

Copyright Β© 2010 M. Frigon and H. Gilbert. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We establish the existence of solutions to systems of second-order dynamic equations on time scales with the right member , a -CarathΓ©odory function. First, we consider the case where the nonlinearity does not depend on the -derivative, (). We obtain existence results for Strum-Liouville and for periodic boundary conditions. Finally, we consider more general systems in which the nonlinearity depends on the -derivative and satisfies a linear growth condition with respect to (). Our existence results rely on notions of solution-tube that are introduced in this paper.

#### 1. Introduction

In this paper, we establish existence results for the following systems of second-order dynamic equations on time scales: Here, is a compact time scale where , , and is defined in (2.4). The map is -CarathΓ©odory (see Definition 2.9), and denotes one of the following boundary conditions: where , , and .

Problem (1.1) was mainly treated in the case where it has only one equation () and is continuous. In particular, the existence of a solution of (1.1) was established by AkΔ±n [1] for the Dirichlet boundary condition and by StehlΓk [2] for the periodic boundary condition. Equation (1.1) with nonlinear boundary conditions was studied by Peterson et al. [3]. In all those results, the method of lower and upper solutions was used. See also [4, 5] and the references therein for other results on the problem (1.1) when .

Very few existence results were obtained for the system (1.1) when . Recently, Henderson et al. [6] and Amster et al. [7] established the existence of solutions of (1.1) with Sturm-Liouville and nonlinear boundary conditions, respectively, assuming that is a continuous function satisfying the following condition:

The fact that the right member in the system (1.2) depends also on the -derivative, , increases considerably the difficulty of this problem. So, it is not surprising that there are almost no results for this problem in the literature. Atici et al. [8] studied this problem in the particular case, where there is only one equation () and is positive, continuous and satisfies a monotonicity condition. Assuming a growth condition of Wintner type and using the method of lower and upper solutions, they obtained the existence of a solution.

The system (1.2) with the Dirichlet boundary condition was studied by Henderson and Tisdell [9] in the general case where . They considered a continuous function and a regular time scale (i.e., or ). They established the existence of a solution of (1.2) under the following assumptions: (A1) there exists such that if , ,(A2) there exist such that and if .

In the third section of this paper, we establish an existence theorem for the system (1.1). To this aim, we introduce a notion of solution-tube of (1.1) which generalizes to systems the notions of lower and upper solutions introduced in [1, 2]. This notion generalizes also condition (1.5) used by Henderson et al. [6] and Amster et al. [7]. Our notion of solution-tube is in the spirit of the notion of solution-tube for systems of second-order differential equations introduced in [10]. Our notion is new even in the case of systems of second-order difference equations.

In the last section of this paper, we study the system (1.2). Again, we introduce a notion of solution-tube of (1.2) which generalizes the notion of lower and upper solutions used by Atici et al. [8]. This notion generalizes also condition (1.5) and the notion of solution-tube of systems of second-order differential equations introduced in [10]. In addition, we assume that satisfies a linear growth condition. It is worthwhile to mention that the time scale does not need to be regular, and we do not require the restriction as in assumption (A2) used in [9].

Moreover, we point out that the right members of our systems are not necessarily continuous. Indeed, we assume that the weaker condition: is a -CarathΓ©odory function. This condition is interesting in the case where the points of are not all right scattered. We obtain the existence of solutions to (1.1) and to (1.2) in the Sobolev space . To our knowledge, it is the first paper applying the theory of Sobolev spaces with topological methods to obtain solutions to (1.1) and (1.2). Solutions of second-order Hamiltonian systems on time scales were obtained in a Sobolev space via variational methods in [11]. Finally, let us mention that our results are new also in the continuous case and for systems of second-order difference equations.

#### 2. Preliminaries and Notations

For sake of completeness, we recall some notations, definitions, and results concerning functions defined on time scales. The interested reader may consult [12, 13] and the references therein to find the proofs and to get a complete introduction to this subject.

Let be a compact time scale with . The *forward jump operator * (resp., the *backward jump operator *) is defined by
We say that is *right scattered* (resp., is *left scattered*) if (resp., ) otherwise, we say that is *right dense* (resp., *left dense*). The set of right-scattered points of is at most countable, see [14]. We denote it by
for some . The *graininess function * is defined by . We denote
So, if is left dense, otherwise . Since is also a time scale, we denote

In 1990, Hilger [15] introduced the concept of dynamic equations on time scales. This concept provides a unified approach to continuous and discrete calculus with the introduction of the notion of delta-derivative . This notion coincides with (resp., ) in the case where the time scale is an interval (resp., the discrete set ).

*Definition 2.1. *A map is *-differentiable* at if there exists (called the *-derivative* of at ) such that for all , there exists a neighborhood of such that
We say that is *-differentiable* if exists for every .

If is -differentiable and if is -differentiable at , we call the *second **-derivative of ** at *.

Proposition 2.2. *Let and . *(i)*If is -differentiable at , then is continuous at .*(ii)*If is continuous at , then
*(iii)*The map is -differentiable at if and only if *

Proposition 2.3. *If and are -differentiable at , then *(i)*if , for every ,*(ii)*if , ,*(iii)*if and , then
*(iv)*if is open and is differentiable at and , then .*

We denote the space of continuous maps on , and we denote the space of continuous maps on with continuous -derivative on . With the norm (resp., ), (resp., ) is a Banach space.

We study the second -derivative of the norm of a map.

Lemma 2.4. *Let be -differentiable. *(1)*On and exists},
*(2)* On and exists}, *

*Proof. *Denote and exists}. By Proposition 2.3, on the set , we have
If is such that , then by Propositions 2.2 and 2.3, we have
If is such that , then
and we conclude as in the previous case.

Let . The exponential function is defined by where It is the unique solution to the initial value problem

Here is a result on time scales, analogous to Gronwall's inequality. The reader may find the proof of this result in [13].

Theorem 2.5. *Let , , and . If
**
then
*

We recall some notions and results related to the theory of *-measure* and -Lebesgue integration introduced by Bohner and Guseinov in [12]. The reader is also referred to [14] for expressions of the -measure and the -integral in terms of the classical Lebesgue measure and the classical Lebesgue integral, respectively.

*Definition 2.6. *A set is said to be *-measurable* if for every set ,
where
The *-measure* on , denoted by , is the restriction of to . So, is a complete measurable space.

Proposition 2.7 (see [14]). * Let , then is -measurable if and only if is Lebesgue measurable. Moreover, if ,
**
where is the Lebesgue measure.*

The notions of -measurable and -integrable functions can be defined similarly to the general theory of Lebesgue integral.

Let be a -measurable set and a -measurable function. We say that provided The set is a Banach space endowed with the norm Here is an analog of the Lebesgue dominated convergence Theorem which can be proved as in the general theory of Lebesgue integration theory.

Theorem 2.8. *Let be a sequence of functions in . Assume that there exists a function such that -a.e. , and there exists a function such that -a.e. and for every , then in .*

In our existence results, we will consider -CarathΓ©odory functions.

*Definition 2.9. *A function is *-CarathΓ©odory* if the following conditions hold: (i) is -measurable for every ,(ii) is continuous for -almost every ,(iii)for every , there exists such that for -almost every and for every such that .

In this context, there is also a notion of absolute continuity introduced in [16].

*Definition 2.10. *A function is said to be *absolutely continuous on * if for every , there exists a such that if with is a finite pairwise disjoint family of subintervals satisfying

Proposition 2.11 (see [17]). *If is an absolutely continuous function, then the -measure of the set is zero.*

Proposition 2.12 (see [17]). *If and is the function defined by
**
then is absolutely continuous and -almost everywhere on .*

Proposition 2.13 (see [16]). * A function is absolutely continuous on if and only if is -differentiable -almost everywhere on , and
*

We also recall a notion of Sobolev space, see [18], where A function can be identified to an absolutely continuous map.

Proposition 2.14 (see [18]). *Suppose that with some satisfying (2.27), then there exists absolutely continuous such that
**
Moreover, if is , then there exists such that
*

Sobolev spaces of higher order can be defined inductively as follows: With the norm (resp., ), (resp., ) is a Banach space.

*Remark 2.15. *By Proposition 2.7, we know that for every . From this fact and the previous proposition, one realizes that there is no interest to look for solutions to (1.1) and (1.2) in and to consider -CarathΓ©odory maps in the case where the time scale is such that . In particular, this is the case for difference equations. Let us point out that we consider more general time scales. Nevertheless, the results that we obtained are new in both cases.

As in the classical case, some embeddings have nice properties.

Proposition 2.16 (see [18]). *The inclusion is continuous.*

Proposition 2.17. *The inclusion is a continuous, compact, linear operator.*

*Proof. *Arguing as in the proof of the ArzelΓ -Ascoli Theorem, we can show that the inclusion is linear, continuous, and compact. The conclusion follows from the previous proposition since .

We obtain a maximum principle in this context. To this aim, we use the following result.

Lemma 2.18 (see [19]). *Let be a function with a local maximum at . If exists, then provided is not at the same time left dense and right scattered.*

Theorem 2.19. *Let be a function such that -almost everywhere on . If one of the following conditions holds: *(i)* and (where , , , and are defined as in (1.4)),*(ii)* and , ** then , for every .*

*Proof. *If the conclusion is false, there exists such that

In the case where , then exists since and . By Lemma 2.18, , which is a contradiction since .

If , there exists such that for every . On the other hand, since is a maximum, , and there exists such that , thus,
by hypothesis and Proposition 2.13, which is a contradiction. Observe that the same argument applies if and . Notice also that if and , we get a contradiction with an analogous argument for a suitable .

If and , we argue as in the first case replacing by to obtain a contradiction.

In the case where and , then . Since is continuous, there exists such that on an interval , which contradicts the maximality of .

Observe that and could happen if or if and . In this case, we get a contradiction if satisfies condition (i). On the other hand, if satisfies condition (ii), then and . So since . This contradicts the maximality of .

On the other hand, the case where and satisfies condition (i) can be treated similarly to the previous case.

Finally, we define the -differential operator associated to the problems that we will consider where with denoting the periodic boundary condition (1.3) or the Sturm-Liouville boundary condition (1.4).

Proposition 2.20. *The operator is invertible and is affine and continuous.*

*Proof. *If denotes the Sturm-Liouville boundary condition (1.4), consider the associated homogeneous boundary condition
Denote
Notice that is a Banach space. Define
It is obvious that is linear and continuous. It follows directly from Theorem 2.19 that is injective.

If denotes (1.4) (resp., (1.3)), let be the Green function given in [13, Theorem β4.70] (resp., [13, Theorem β4.89]). Arguing as in [13, Theorem β4.70] (resp., [13, Theorem β4.89]), one can verify that for any ,
is a solution of . So, is bijective and, hence, invertible with continuous by the inverse mapping theorem.

Finally, if denotes (1.3), since , we have the conclusion. On the other hand, if denotes (1.4), let be given in [13, Theorem β4.67] such that
then .

*Remark 2.21. *We could have considered the operator
when the boundary condition is (1.4) with suitable constants , , , such that is injective. For simplicity, we prefer to use only the operator .

#### 3. Nonlinearity Not Depending on the Delta-Derivative

In this section, we establish existence results for the problem where denotes the periodic boundary condition or the Sturm-Liouville boundary condition where , , and . We look for solutions in .

We introduce the notion of solution-tube for the problem (3.1).

*Definition 3.1. *Let . We say that is a *solution-tube* of (3.1) if (i) for -almost every and for every such that ,(ii) and for -almost every such that ,(iii)(a)if denotes (3.2), then , , and , (b)if denotes (3.3), , . We denote

We state the main theorem of this section.

Theorem 3.2. *Let be a -CarathΓ©odory function. If is a solution-tube of (3.1), then the system (3.1) has a solution .*

In order to prove this result, we consider the following modified problem: where

We define the operator by

Proposition 3.3. *Under the assumptions of Theorem 3.2, the operator defined above is continuous and bounded.*

*Proof. *First of all, we show that the set is bounded. Fix . Let be given by Definition 2.9(iii). Thus, for every ,

To prove the continuity of , we consider a sequence of converging to . We already know that for every ,
One can easily check that for all . It follows from Definition 2.9(ii) that
Theorem 2.8 implies that in .

Lemma 3.4. *Under the assumptions of Theorem 3.2, every solution of (3.5) is in .*

*Proof. *Since (resp., , ), (resp., , ), there exists -almost everywhere on . Denote
By Lemma 2.4(1),
We claim that
Indeed, we deduce from the fact that is a solution-tube of (3.5) and from (3.12) that -almost everywhere on ,
and -almost everywhere on ,

Observe that if ,
Indeed, this follows from Proposition 2.3 when . For ,
Similarly, if ,

If denotes (3.2),
We deduce from (3.16), (3.18), and Definition 3.1 that
or

If denotes (3.3), we deduce from (3.16), (3.18), and Definition 3.1 that
or
and similarly
or

Finally, it follows from (3.13), (3.20), (3.21), (3.22), (3.23), (3.24), (3.25), and Theorem 2.19 applied to that for every .

Now, we can prove the main theorem of this section.

*Proof of Theorem 3.2. *A solution of (3.5) is a fixed point of the operator
where and are defined in (2.34) and Proposition 2.17, respectively. By Propositions 2.17, 2.20, and 3.3, the operator is compact. So, the Schauder fixed point theorem implies that has a fixed point and, hence, Problem (3.5) has a solution . By Lemma 3.4, this solution is in . Thus, is a solution of (3.1).

In the particular case where , as corollary of Theorem 3.2, we obtain a generalization of results of AkΔ±n [1] and StehlΓk [2] for the Dirichlet and the periodic boundary conditions, respectively.

Corollary 3.5. *Let be a -CarathΓ©odory function. Assume that there exists such that *(i)* for -almost every ,*(ii)* and for -almost every ,*(iii)*(a) if denotes (3.2), then , , , and ,(b)if denotes (3.3), then and , *

*then (3.1) has a solution such that for every .*

*Proof. *Observe that is a solution-tube of (3.1). The conclusion follows from Theorem 3.2.

Theorem 3.2 generalizes also a result established by Henderson et al. [6] for systems of second-order dynamic equations on time scales. Let us mention that they considered a continuous map and they assumed a strict inequality in (3.27).

Corollary 3.6. *Let be a -CarathΓ©odory function. Assume that there exists a constant such that
**
Moreover, if denotes (3.3), assume that and , then the system (3.1) has a solution such that for every .*

Here is an example in which one cannot find a solution-tube of the form .

*Example 3.7. *Consider the system
where and is such that . One can check that is a solution-tube of (3.28) with , . By Theorem 3.2, this problem has at least one solution such that . Observe that there is no such that (3.27) is satisfied.

#### 4. Nonlinearity Depending on

In this section, we study more general systems of second-order dynamic equations on time scales. Indeed, we allow the nonlinearity to depend also on . We consider the problem where and .

We also introduce a notion of solution-tube for this problem.

*Definition 4.1. *Let . We say that is a *solution-tube* of (4.1) if (i)for -almost every ,
for every such that and ,(ii)for every ,
for every such that ,(iii), .

If is the real interval , condition (ii) of the previous definition becomes useless, and we get the notion of solution-tube introduced by the first author in [10] for a system of second-order differential equations.

Here is the main result of this section.

Theorem 4.2. *Let be a -CarathΓ©odory function. Assume that *(H1)* there exists a solution-tube of (4.1),*(H2)*there exist constants such that for -almost every and for every such that , ** then (4.1) has a solution .*

To prove this existence result, we consider the following modified problem: where is defined by where is defined as in (3.6), with a constant which will be fixed later.

*Remark 4.3. *(1) Remark that

(2) If ,

(3) If and ,

(4) Since is -CarathΓ©odory, by (1), there exists such that for every ,

We associate to the operator defined by

Proposition 4.4. *Let be a -CarathΓ©odory function. Assume that (H1) is satisfied, then is continuous.*

*Proof. *Let be a sequence of converging to . It is clear that

On , we have
since is -CarathΓ©odory.

Similarly, -almost everywhere on ,
since and, for sufficiently large, .

Denote and . As before, it is easy to check that -almost everywhere on ,
On the other hand, Proposition 2.11 implies that
So, -almost everywhere on ,