© Hugues Gilbert. 2010
Received: 14 June 2010
Accepted: 3 December 2010
Published: 15 December 2010
This paper concerns the existence of solutions for two kinds of systems of first-order equations on time scales. Existence results for these problems are obtained with new notions of solution tube adapted to these systems. We consider the general case where the right member of the system is -Carathéodory and, hence, not necessarily continuous.
In the literature, this kind of problem was mainly treated for . The existence of extremal solutions was established in [1, 2]. Moreover, some existence results in the particular case where the time scale is a discrete set (difference equation) were obtained with the lower and upper solution method as in [3, 4]. In this paper, we introduce a new notion which generalizes to systems of first-order equations on time scales the notions of lower and upper solutions. This notion called solution tube for system (1.1) (resp., (1.2)) will be useful to get a new existence result for (1.1) (resp., (1.2)). Our notion of solution tube is in the spirit of the notion of solution tube for systems of first-order differential equations introduced in . Our notion is new even in the case of systems of first-order difference equations. In this case, we generalize to the systems a result of  for equation (1.2).
Some papers treat the existence of solutions to systems of first-order equations on time scales. Existence results are obtained in [6, 7] under hypothesis different from ours. However, some particular cases obtained in  are corollaries of our existence result for problem (1.1). Also, our existence results treat the case where the right members in (1.1) and (1.2) are -Carathéodory functions which are more general than continuous functions used for systems studied in [6, 7]. Let us mention that existence of extremal solutions for infinite systems of first-order equations of time scale with -Carathéodory functions is established in .
This paper is organized as follows. The third section presents an existence result for the problem (1.1), and in the last section, we obtain an existence theorem for the problem (1.2). We start with some notations, definitions, and results on time scales equations which are used throughout this paper.
2. Preliminaries and Notations
In this section, we establish notations, definitions, and results on equations on time scales which are used throughout this paper. The reader may consult [9–11] and the references therein to find the proofs and to get a complete introduction to this subject.
Let be a time scale, which is a closed nonempty subset of . For , we define the forward jump operator (resp., the backward jump operator ) by (resp., by ). We suppose that if is the maximum of and that if is the minimum of . We say that is right-scattered (resp., left-scattered) if (resp., if ). We say that is isolated if it is right-scattered and left-scattered. Also, if and , we say that is right-dense. If and , we say that is left dense. Points that are right dense and left dense are called dense. The graininess function is defined by .
The next result is an adaptation of Theorem 1.87 in .
A function is called -continuous provided it is continuous at right-dense points in and its left-sided limits exist (finite) at left-dense points in . The set of -continuous functions is denoted by . The set of functions that are -differentiable and whose -derivative is -continuous is denoted by .
With this definition of complete measurable space for a bounded time scale , we can define the notions of -measurability and -integrability for functions following the same ideas of the theory of Lebesgue integral. We omit here these definitions that an interested reader can find in . We only present definitions and results which will be useful for this paper.
Many results of integration theory are established for measurable functions where is a complete measurable space. These results are in particular true for the measurable space . We recall two results of the theory of integration adapted to our situation.
Theorem 2.10 (Lebesgue-dominated convergence Theorem).
The following results were obtained in  where a useful relation between the -measure on (resp., -integral on ) and the Lebesgue measure ( ) on (resp., Lebesgue integral on ) is established. To establish these results, the authors of  prove that the set of right-scattered points of is at most countable. Then, there are a set of index and a set such that .
The three following results were obtained in .
We also recall the Banach Lemma.
Using the previous results, we now prove two propositions that will be used later.
We recall a notion of Sobolev's space of functions defined on a bounded time scale where . The definition and the result are from .
Suppose that and that (2.22) holds for a function . Then, there exists a unique function absolutely continuous such that -almost everywhere on , one has and . Moreover, if is -continuous on , then there exists a unique function such that -almost everywhere on and such that on .
We prove two maximum principles that will be useful to get a priori bounds for solutions of systems considered in this paper.
by hypothesis and by Theorem 2.17. Hence, we get a contradiction. The case is impossible if hypothesis (i) holds and if , we must have . If we take , by using previous steps of this proof, one can check that and, then, the lemma is proved.
by Theorem 2.17, which contradicts the fact that is a maximum. If , then by hypothesis, we must have . Thus, we can take , and by using the previous steps of this proof, one can check that . Then, the lemma is proved.
As direct consequences of Proposition 2.19 and Theorem 2.3, we get the following results.
We now define a notion of Carathéodory functions on a compact time scale.
3. Existence Theorem for the Problem (1.1)
In this section, we establish an existence result for the problem (1.1). A solution of this problem will be a function satisfying (1.1). Let us recall that is compact and . We introduce the notion of solution tube for the problem (1.1).
If is a real interval , our definition of solution tube is equivalent to the notion of solution tube introduced in .
The following result can be proved as the previous one.
Now, we can obtain the main theorem of this section.
By Proposition 3.2 (resp., Proposition 3.3), (resp., ) is compact. It has a fixed point by the Schauder fixed-point Theorem. Proposition 2.27 (resp., Proposition 2.28) implies that this fixed point is a solution for the problem (3.2). Then, it suffices to show that for every solution of (3.2), .
This last equality follows from Definition 3.1(iii) and Proposition 2.20.
If we set , then -almost everywhere on . Moreover, since is a solution tube of (1.1) and satisfies (1.3) (resp., satisfies (1.4)), then (resp., ). Lemma 2.24 implies that . Therefore, and, hence, the theorem is proved.
Existence theorems are obtained in  for the problem (1.1), (1.3) when is continuous by using a hypothesis different of ours. When is bounded, we can directly use the Schauder fixed-point Theorem to deduce the existence of a solution to (1.1), (1.3). We now show that in the case where is unbounded, Theorems 4.7 and 4.8 of  become corollaries of our existence theorem.
4. Existence Theorem for the Problem (1.2)
In this section, we establish an existence result for the problem (1.2). A solution of this problem will be a function for which (1.2) is satisfied. As before, is compact and . We introduce the notion of solution tube for the problem (1.2). Conditions of this definition are slightly different than conditions in Definition 3.1.
The following result can be proved as Proposition 3.2.
Here is the main existence theorem for problem (1.2).
then the problem (1.2) has a solution.
for every , then the difference equation (1.2) has one solution.
Observe that Theorem 4.3 is valid for every arbitrary time scale . Here is an example where (4.7) and (4.9) are not satisfied, but where Theorem 4.3 can be applied to deduce the existence of a solution.
for every , where , , , and . Taking the limit as , we get a contradiction. Similarly, if , it can be shown that (4.9) is not satisfied. On the other hand, it is easy to verify that is a solution tube of (4.10). By Theorem 4.3, this problem has a solution such that for every .
Remark that if are, respectively, lower and upper solutions of (4.14) such that for every , then is a solution tube for this problem. Conversely, if is a solution tube of (4.14), then and are, respectively, lower and upper solution for the same problem if, in addition, condition (ii) of Definition 4.7 is satisfied. Then, Theorem 5 of  becomes a corollary of Theorem 4.3.
The author would like to thank Professor Marlene Frigon for useful discussion and comments and the FQRNT for financial support.
- Otero-Espinar V, Vivero DR: Existence of extremal solutions by approximation to a first-order initial dynamic equation with Carathéodory's conditions and discontinuous non-linearities. Journal of Difference Equations and Applications 2006,12(12):1225-1241. 10.1080/10236190600949808MathSciNetView ArticleMATHGoogle Scholar
- Otero-Espinar V, Vivero DR: The existence and approximation of extremal solutions to several first-order discontinuous dynamic equations with nonlinear boundary value conditions. Nonlinear Analysis: Theory, Methods & Applications 2008,68(7):2027-2037. 10.1016/j.na.2007.01.030MathSciNetView ArticleMATHGoogle Scholar
- Bereanu C, Mawhin J: Existence and multiplicity results for periodic solutions of nonlinear difference equations. Journal of Difference Equations and Applications 2006,12(7):677-695. 10.1080/10236190600654689MathSciNetView ArticleMATHGoogle Scholar
- Franco D, O'Regan D, Perán J: Upper and lower solution theory for first and second order difference equations. Dynamic Systems and Applications 2004,13(2):273-282.MathSciNetMATHGoogle Scholar
- Mirandette B: Résultats d'Existence pour des Systèmes d'Équations Différentielles du Premier Ordre avec Tube-Solution. Mémoire de Maîtrise, Université de Montréal, Montréal, Canada; 1996.Google Scholar
- Dai Q, Tisdell CC: Existence of solutions to first-order dynamic boundary value problems. International Journal of Difference Equations 2006,1(1):1-17.MathSciNetMATHGoogle Scholar
- Tisdell CC, Zaidi A: Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling. Nonlinear Analysis: Theory, Methods & Applications 2008,68(11):3504-3524. 10.1016/j.na.2007.03.043MathSciNetView ArticleMATHGoogle Scholar
- 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.2007.06.031MathSciNetView ArticleMATHGoogle Scholar
- Bohner M, Peterson A: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2003:xii+348.View ArticleMATHGoogle Scholar
- Bohner M, Peterson A: Dynamic Equations on Time Scales. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleMATHGoogle Scholar
- Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results in Mathematics. Resultate der Mathematik 1990,18(1-2):18-56.MathSciNetView ArticleMATHGoogle Scholar
- Cabada A, Vivero DR:Expression of the Lebesgue -integral on time scales as a usual Lebesgue integral: application to the calculus of -antiderivatives. Mathematical and Computer Modelling 2006,43(1-2):194-207. 10.1016/j.mcm.2005.09.028MathSciNetView ArticleMATHGoogle Scholar
- Cabada A, Vivero DR: Criterions for absolute continuity on time scales. Journal of Difference Equations and Applications 2005,11(11):1013-1028. 10.1080/10236190500272830MathSciNetView ArticleMATHGoogle Scholar
- 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.Google Scholar
- Tisdell CC: On first-order discrete boundary value problems. Journal of Difference Equations and Applications 2006,12(12):1213-1223. 10.1080/10236190600949790MathSciNetView ArticleMATHGoogle Scholar
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