- Research Article
- Open Access
Existence Theorems for First-Order Equations on Time Scales with -Carathéodory Functions
© 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.
- Existence Result
- Existence Theorem
- Extremal Solution
- Nonnegative Constant
- Lebesgue Integration
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.
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 .
If is bounded, then where .
for every . We call the -derivative of at . If is -differentiable at for every , then is called the -derivative of on .
If is -differentiable at , then is continuous at .
If is right dense, then is -differentiable at if and only if exists in . In this case, .
If is -differentiable at , then .
is -differentiable at and ,
is -differentiable at for every and ,
is -differentiable at and ,
The next result is an adaptation of Theorem 1.87 in .
Let be an open set of and a right-dense point. If is -differentiable at and if is differentiable at , then is -differentiable at and .
Assume is - differentiable at . We know that is differentiable. If , by the previous theorem, we have .
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 .
The Lebesgue -measure on , denoted by , is the restriction of to . We get a complete measurable space with .
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.
for each of its components .
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).
Let be a sequence of functions in . If there exists a function such that -a.e. and if there exists a function such that -a.e. and for every , then in .
The set is a Banach space endowed with the norm .
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 .
if and only if and A has no right-scattered points. Here, .
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 of satisfying , then .
The three following results were obtained in .
If is differentiable at , then is -differentiable at and .
Consider a function and its extension . Then, is absolutely continuous on if and only if is absolutely continuous on .
We also recall the Banach Lemma.
Let be a Banach space and an absolutely continuous function, then the measure of the set and is zero.
Using the previous results, we now prove two propositions that will be used later.
Then, -almost everywhere on .
by Theorem 2.2(ii). Then, is -differentiable at . By Proposition 2.12(ii), and, then, the proposition is proved.
Let be an absolutely continuous function, then the -measure of the set and is zero.
It suffices to consider the extension of on and successively apply Theorem 2.16, Lemmas 2.18, 2.15, and the Proposition 2.12(ii).
We recall a notion of Sobolev's space of functions defined on a bounded time scale where . The definition and the result are from .
We say that a function is in the set if each of its components are in .
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 .
By the previous theorem, we can conclude that is also continuous.
If , then its components are in . By Theorems 2.22 and 2.17, is - differentiable - a.e. on . From Example 2.5, we obtain - a.e. on .
We prove two maximum principles that will be useful to get a priori bounds for solutions of systems considered in this paper.
then , for every .
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.
Let be a function such that -a.e. on if , then , for every .
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.
(C-i)The map is -measurable for every .
(C-ii)The map is continuous -a.e. .
(C-iii)For every , there exists a function such that -a.e. and for every such that .
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).
-a.e. and for every such that ,
-a.e. such that ,
for every such that ,
If (BC) denotes (1.3), ; if (BC) denotes (1.4), then .
If is a real interval , our definition of solution tube is equivalent to the notion of solution tube introduced in .
If is a solution tube of (1.1), (1.3), then is compact.
We first observe that from Definitions 2.30 and 3.1, there exists a function such that -a.e. for every .
We can easily check that for every and, then, by (C-ii) of Definition 2.30, -a.e. . Using also the fact that -a.e. , we deduce that in by Theorem 2.10. This prove the continuity of .
By an analogous version of the Arzelà-Ascoli Theorem adapted to our context, is relatively compact. Hence, is compact.
The following result can be proved as the previous one.
If is a solution tube of (1.1), (1.4), then the operator is compact.
Now, we can obtain the main theorem of this section.
If is a solution tube of (1.1), then the problem (1.1) has a solution .
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.
for every and every , then the problem (1.1), (1.3) has at least one solution.
for every and every . Then, if we take , we get a solution tube for our problem and by Theorem 3.4, the problem has a solution such that for every .
for every and every , then the problem (1.1), (1.3) has at least one solution.
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.
-a.e. and for every such that ,
and , -a.e. such that ,
where is defined in (3.3).
The following result can be proved as Proposition 3.2.
If is a solution tube of (1.2), then the operator is compact.
Here is the main existence theorem for problem (1.2).
If is a solution tube of (1.2), then the problem (1.2) has a solution .
By Proposition 4.2, is compact. Then, by the Schauder fixed-point Theorem, has a fixed point which is a solution of (4.1) by Proposition 2.29. It suffices to show that for every solution of (4.1), .
If we set , then -almost everywhere on . Moreover, since is a solution tube of (1.2) and satisfies (1.4), . Lemma 2.25 implies that . So, and the theorem is proved.
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.
where are real positive constants such that and where is a continuous function such that for every .
for every .
for every .
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 .
for every , (resp., ),
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.
If are, respectively, lower and upper solutions of (4.14) such that for every , then this equation has a solution such that for every .
The author would like to thank Professor Marlene Frigon for useful discussion and comments and the FQRNT for financial support.
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