Existence Theorems for First-Order Equations on Time Scales with -Carathéodory Functions

Abstract

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.

1. Introduction

In this paper, we establish existence results for the following systems:

(1.1)

(1.2)

Here, is an arbitrary compact time scale where we note , , and . Moreover, is a -Carathéodory function and denotes one of the following boundary conditions:

(1.3)

(1.4)

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 [5]. 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 [3] 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 [7] 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 [8].

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 .

If has a left-scattered maximum, then . Otherwise, . In summary,

(2.1)

If is bounded, then where .

Definition 2.1.

Assume is a function and let . We say that is -differentiable at if there exists a vector such that for all , there exists a neighborhood of , where

(2.2)

for every . We call the -derivative of at . If is -differentiable at for every , then is called the -derivative of on .

Theorem 2.2.

Assume is a function and let . Then, we have the following.

• (i) If is -differentiable at , then is continuous at .

• (ii) If is continuous at and if is right-scattered, then is -differentiable at and

  

  (2.3)

• (iii) If is right dense, then is -differentiable at if and only if exists in . In this case, .

• (iv) If is -differentiable at , then .

Theorem 2.3.

If are -differentiable at , then

• (i) is -differentiable at and ,

• (ii) is -differentiable at for every and ,

• (iii) is -differentiable at and ,

• (iv) If , then is -differentiable at and

  

  (2.4)

The next result is an adaptation of Theorem 1.87 in [10].

Theorem 2.4.

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 .

Example 2.5.

Assume is- differentiable at. We know that is differentiable. If, by the previous theorem, we have.

Definition 2.6.

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 .

It is possible to define a theory of measure and integration for an arbitrary bounded time scale where . We recall the notion of -measure as introduced in chapter 5 of [9]. Define the family of intervals of of the form

(2.5)

where and . The interval is understood as the empty set. An outer measure on is defined as follows: for ,

(2.6)

Definition 2.7.

A set is said to be -measurable if for every set ,

(2.7)

Now, denote

(2.8)

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 [12]. We only present definitions and results which will be useful for this paper.

Definition 2.8.

Let be a -measurable set and be a -measurable function. We say that provided

(2.9)

We say that a -measurable function is in the set provided

(2.10)

for each of its components .

Proposition 2.9.

Assume . Then,

(2.11)

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 .

Theorem 2.11.

The set is a Banach space endowed with the norm .

The following results were obtained in [12] 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 [12] 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 .

Proposition 2.12.

Let . Then, is -measurable if and only if is Lebesgue measurable. In such a case, the following properties hold for every -measurable set A.

• (i) If , then

  

  (2.12)

• (ii) if and only if and A has no right-scattered points. Here, .

To establish the relation between -integration on and Lebesgue integration on a real compact interval, the function is extended to in the following way.

(2.13)

Theorem 2.13.

Let be a -measurable set such that and let . Let be a -measurable function and its extension on . Then, is -integrable on if and only if is Lebesgue integrable on . In such a case, one has

(2.14)

Also, the function can be extended on in another way. Define by

(2.15)

Definition 2.14.

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 [13].

Lemma 2.15.

If is differentiable at , then is -differentiable at and .

Theorem 2.16.

Consider a function and its extension . Then, is absolutely continuous on if and only if is absolutely continuous on .

Theorem 2.17.

A function is absolutely continuous on if and only if is -differentiable -almost everywhere on , and

(2.16)

We also recall the Banach Lemma.

Lemma 2.18.

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.

Proposition 2.19.

Let and the function defined by

(2.17)

Then, -almost everywhere on .

Proof.

By Theorem 2.13, remark that

(2.18)

We can also check that for right scattered,

(2.19)

Obviously,

(2.20)

It is well known that almost everywhere on . By Lemma 2.15, we have that except on a set such that . Since is continuous, for right scattered,

(2.21)

by Theorem 2.2(ii). Then, is -differentiable at . By Proposition 2.12(ii), and, then, the proposition is proved.

Proposition 2.20.

Let be an absolutely continuous function, then the -measure of the set and is zero.

Proof.

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 [14].

Definition 2.21.

We say that a function belongs to if and only if and there exists a function such that and

(2.22)

with

(2.23)

We say that a function is in the set if each of its components are in .

Theorem 2.22.

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.

Remark 2.23.

If, then its componentsare 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.

Lemma 2.24.

Let such that -a.e. . If one of the following conditions holds,

• (i) ,

• (ii) ,

then , for every .

Proof.

Suppose the conclusion is false. Then, there exists such that , since is continuous on . If , then exists, since and because . Then,

(2.24)

which is a contradiction since . If , then there exists an interval such that for all . Thus,

(2.25)

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.

Lemma 2.25.

Let be a function such that -a.e. on if , then , for every .

Proof.

If there exists such that , then there exists a such that . If and , then . Since , exists -almost everywhere. Then, we must have

(2.26)

which contradicts the hypothesis of the lemma. If and , there exists an interval such that for every . Then,

(2.27)

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.

Definition 2.26.

For , the exponential function may be defined as the unique solution of the initial value problem

(2.28)

More explicitly, the exponential function is given by the formula

(2.29)

where for , we define as

(2.30)

As direct consequences of Proposition 2.19 and Theorem 2.3, we get the following results.

Proposition 2.27.

If , the function defined by

(2.31)

is a solution of the problem

(2.32)

Proposition 2.28.

If , then the function defined by

(2.33)

is a solution of the problem

(2.34)

Proposition 2.29.

If , then the function defined by

(2.35)

is a solution of the problem

(2.36)

We now define a notion of Carathéodory functions on a compact time scale.

Definition 2.30.

A function is called a -Carathéodory function if the three following conditions hold.

• (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 .

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).

Definition 3.1.

Let . We say that is a solution tube of (1.1) if

• (i) -a.e. and for every such that ,

• (ii) -a.e. such that ,

• (iii) for every such that ,

• (iv) If (BC) denotes (1.3), ; if (BC) denotes (1.4), then .

We denote

(3.1)

If is a real interval , our definition of solution tube is equivalent to the notion of solution tube introduced in [5].

We consider the following problem.

(3.2)

where

(3.3)

Let us define the operator by

(3.4)

Proposition 3.2.

If is a solution tube of (1.1), (1.3), then is compact.

Proof.

We first observe that from Definitions 2.30 and 3.1, there exists a function such that -a.e. for every .

Let be a sequence of converging to . By Proposition 2.9,

(3.5)

where .

Then, we must show that the sequence defined by

(3.6)

converges to the function in where

(3.7)

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 .

For the second part of the proof, we have to show that the set is relatively compact. Let . Therefore,

(3.8)

So, is uniformly bounded. This set is also equicontinuous since for every ,

(3.9)

By an analogous version of the Arzelà-Ascoli Theorem adapted to our context, is relatively compact. Hence, is compact.

We now define the operator by

(3.10)

The following result can be proved as the previous one.

Proposition 3.3.

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.

Theorem 3.4.

If is a solution tube of (1.1), then the problem (1.1) has a solution .

Proof.

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), .

Consider the set . By Remark 2.23, -a.e. on the set , we have

(3.11)

If is right scattered, then and

(3.12)

Therefore, since is a solution tube of (1.1), we have -a.e. on that

(3.13)

On the other hand, we have -a.e. on that

(3.14)

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 [7] 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 [7] become corollaries of our existence theorem.

Corollary 3.5.

Let be an unbounded continuous function. If there exist non-negative constants and such that

(3.15)

for every and every , then the problem (1.1), (1.3) has at least one solution.

Proof.

Observe that since is unbounded. By hypothesis, there exists a constant such that . Let us define by

(3.16)

Then, for every and, thus,

(3.17)

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 .

Corollary 3.6.

Let be an unbounded continuous function. If there exists a nonnegative constant such that

(3.18)

for every and every , then the problem (1.1), (1.3) has at least one solution.

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.

Definition 4.1.

Let . We say that is a solution tube of (1.2) if

• (i) -a.e. and for every such that ,

• (ii) and , -a.e. such that ,

• (iii) .

We consider the following problem.

(4.1)

where is defined in (3.3).

Let us define the operator by

(4.2)

The following result can be proved as Proposition 3.2.

Proposition 4.2.

If is a solution tube of (1.2), then the operator is compact.

Here is the main existence theorem for problem (1.2).

Theorem 4.3.

If is a solution tube of (1.2), then the problem (1.2) has a solution .

Proof.

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), .

Let us consider the set . By Remark 2.23, -a.e. on the set we have

(4.3)

If is right scattered, then and

(4.4)

Since is a solution tube of (1.2), we have -a.e. on that

(4.5)

On the other hand, we have -a.e. on that

(4.6)

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.

Let us observe that the following results obtained in [6] and [15], respectively, are different from ours.

Theorem 4.4.

Let be a continuous function with such that . If there exist nonnegative constants and such that

(4.7)

for every , where is defined by with

(4.8)

then the problem (1.2) has a solution.

Theorem 4.5.

Let , and a continuous function. If there exist nonnegative constants and such that

(4.9)

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.

Example 4.6.

Consider the system

(4.10)

where are real positive constants such that and where is a continuous function such that for every .

We first show that this system do not satisfy (4.7). Suppose there exist non-negative constants and such that

(4.11)

for every .

If we define , then

(4.12)

for every .

Then,

(4.13)

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 .

Definition 4.1 generalizes the notions of lower and upper solutions and introduced in [3] in the particular case where the problem (1.2) is considered with , for some and with depending only on . We recall these definitions. Consider the problem

(4.14)

where .

Definition 4.7.

A vector (resp., ) is called an upper solution (resp., a lower solution) of (4.14) if

• (i) for every , (resp., ),

• (ii) (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 [3] becomes a corollary of Theorem 4.3.

Corollary 4.8.

If are, respectively, lower and upper solutions of (4.14) such that for every , then this equation has a solution such that for every .