# First-Order Singular and Discontinuous Differential Equations

- DanielC Biles
^{1}and - Rodrigo López Pouso
^{2}Email author

**2009**:507671

**DOI: **10.1155/2009/507671

© D. C. Biles and R. López Pouso. 2009

**Received: **10 March 2009

**Accepted: **4 May 2009

**Published: **9 June 2009

## Abstract

We use subfunctions and superfunctions to derive sufficient conditions for the existence of extremal solutions to initial value problems for ordinary differential equations with discontinuous and singular nonlinearities.

## 1. Introduction

It is well-known that Peano's theorem ensures the existence of local continuously differentiable solutions of (1.1) in case is continuous. Despite its fundamental importance, it is probably true that Peano's proof of his theorem is even more important than the result itself, which nowadays we know can be deduced quickly from standard fixed point theorems (see [1, Theorem ] for a proof based on the Schauder's theorem). The reason for believing this is that Peano's original approach to the problem in [2] consisted in obtaining the greatest solution as the pointwise infimum of strict upper solutions. Subsequently this idea was improved by Perron in [3], who also adapted it to study the Laplace equation by means of what we call today Perron's method. For a more recent and important revisitation of the method we mention the work by Goodman [4] on (1.1) in case is a Carathéodory function. For our purposes in this paper, the importance of Peano's original ideas is that they can be adapted to prove existence results for (1.1) under such weak conditions that standard functional analysis arguments are no longer valid. We refer to differential equations which depend discontinuously on the unknown and several results obtained in papers as [5–9], see also the monographs [10, 11].

On the other hand, singular differential equations have been receiving a lot of attention in the last years, and we can quote [7, 12–19]. The main objective in this paper is to establish an existence result for (1.1) with discontinuous and singular nonlinearities which generalizes in some aspects some of the previously mentioned works.

This paper is organized as follows. In Section 2 we introduce the relevant definitions together with some previously published material which will serve as a basis for proving our main results. In Section 3 we prove the existence of the greatest and the smallest Carathéodory solutions for (1.1) between given lower and upper solutions and assuming the existence of a -bound for on the sector delimited by the graphs of the lower and upper solutions (regular problems), and we give some examples. In Section 4 we show that looking for piecewise continuous lower and upper solutions is good in practice, but once we have found them we can immediately construct a pair of continuous lower and upper solutions which provide better information on the location of the solutions. In Section 5 we prove two existence results in case does not have such a strong bound as in Section 3 (singular problems), which requires the addition of some assumptions over the lower and upper solutions. Finally, we prove a result for singular quasimonotone systems in Section 6 and we give some examples in Section 7. Comparison with the literature is provided throughout the paper.

## 2. Preliminaries

In the following definition stands for the set of absolutely continuous functions on .

Definition 2.1.

A lower solution of (1.1) is a function such that and for almost all (a.a.) ; an upper solution is defined analogously reversing the inequalities. One says that is a (Carathéodory) solution of (1.1) if it is both a lower and an upper solution. On the other hand, one says that a solution is the least one if on for any other solution , and one defines the greatest solution in a similar way. When both the least and the greatest solutions exist, one calls them the extremal solutions.

*quasi*-

*semicontinuous*, namely, for all we have

A similar result was established in [20] assuming moreover that is superpositionally measurable, and the systems case was considered in [5, 8]. The term "quasi-semicontinuous" in connection with (2.1) was introduced in [5] for the first time and it appears to be conveniently short and descriptive. We note however that, rigorously speaking, we should say that is left upper and right lower semicontinuous.

The mappings and turn out to be the extremal solutions even under more general conditions. It is proven in [9] that solutions exist even if (2.1) fails on the points of a countable family of curves in the conditions of the following definition.

Definition 2.2.

Remark 2.3.

In this case (2.1) only fails over the line , but solutions coming from above that line collide with solutions coming from below and there is no way of continuing them to the right once they reach the level . Following Binding [21] we can say that the equation "jams" at .

An easily applicable sufficient condition for an absolutely continuous function to be an admissible nqsc curve is that either it is a solution or there exist and such that one of the following conditions hold:

(1) for a.a. and all ,

(2) for a.a. and all .

These conditions prevent the differential equation from exhibiting the behavior of the previous example over the line in several ways. First, if is a solution of then any other solution can be continued over once they contact each other and independently of the definition of around the graph of . On the other hand, if holds then solutions of can cross from above to below (hence at most once), and if holds then solutions can cross from below to above, so in both cases the equation does not jam over the graph of .

For the convenience of the reader we state the main results in [9]. The next result establishes the fact that we can have "weak" solutions in a sense just by assuming very general conditions over .

Theorem 2.4.

Suppose that there exists a null-measure set such that the following conditions hold:

(1)condition (2.1) holds for all except, at most, over a countable family of admissible non-quasi-semicontinuity curves;

contains no positive measure set.

contains no positive measure set.

Note that if the sets and are measurable then and immediately become the extremal Carathéodory solutions of (1.1). In turn, measurability of those sets can be deduced from some measurability assumptions on . The next lemma is a slight generalization of some results in [8] and the reader can find its proof in [9].

Lemma 2.5.

Assume that for a null-measure set the mapping satisfies the following condition.

Then the mappings and are measurable for each pair such that for all .

Remark 2.6.

A revision of the proof of [9, Lemma ] shows that it suffices to impose (2.12) for all such that . This fact will be taken into account in this paper.

As a consequence of Theorem 2.4 and Lemma 2.5 we have a result about existence of extremal Carathéodory solutions for (1.1) and -bounded nonlinearities. Note that the assumptions in Lemma 2.5 include a restriction over the type of discontinuities that can occur over the admissible nonqsc curves, but remember that such a restriction only plays the role of implying that the sets and in Theorem 2.4 are measurable. Therefore, only using the axiom of choice one can find a mapping in the conditions of Theorem 2.4 which does not satisfy the assumptions in Lemma 2.5 and for which the corresponding problem (1.1) lacks the greatest (or the least) Carathéodory solution.

Theorem 2.7 ([9, Theorem ]).

Suppose that there exists a null-measure set such that the following conditions hold:

(i)for every , is measurable;

and (2.1) can fail, at most, over a countable family of admissible nonquasisemicontinuity curves;

Then the mapping defined in (2.2) is the least Carathéodory solution of (1.1) and the mapping defined in (2.3) is the greatest one.

Remark 2.8.

Theorem in [9] implies that also is the least Carathéodory solution of (1.1), thus on . Hence .

Analogously we can prove that can be replaced by in the statement of [21, Theorem ].

## 3. Existence between Lower and Upper Solutions

Condition in Theorem 2.7 is rather restrictive and can be relaxed by assuming boundedness of between a lower and an upper solution.

In this section we will prove the following result.

Theorem 3.1.

Suppose that (1.1) has a lower solution and an upper solution such that for all and let .

Suppose that there exists a null-measure set such that the following conditions hold:

() for every , the mapping with domain is measurable;

() for every , , one has either (2.1) or (2.13), and (2.1) can fail, at most, over a countable family of admissible non-quasisemicontinuity curves contained in ;

Proof.

Without loss of generality we suppose that and exist and satisfy , , , and on . We also may (and we do) assume that every admissible nqsc curve in condition , say , satisfies for all either or (2.4)-(2.5).

Claim 1.

satisfies conditions and in Theorem 2.4 with replaced by . First we note that is an immediate consequence of and the definition of .

and an analogous argument is valid when and satisfies (2.1).

and therefore either or condition (2.4) holds, yielding . If then we have .

Analogous arguments show that either or (2.4)-(2.5) hold for at almost every point where coincides with , so we conclude that is an admissible nqsc curve for .

contains no positive measure set.

Claim 2.

which together with imply on , a contradiction with (3.16). Therefore every upper solution of (3.6) is greater than or equal to , and, on the other hand, is an upper solution of (3.6) with a.e., thus satisfies (3.14).

One can prove by means of analogous arguments that satisfies (3.15).

Claim 3.

is the least solution of (1.1) in and is the greatest one. From (3.14) and (3.15) it suffices to show that and are actually solutions of (3.6). Therefore we only have to prove that and are null measure sets.

and we can split , where and

Therefore and thus is a null measure set.

which is a measurable set by virtue of Lemma 2.5 and Remark 2.6.

Since contains no positive measure subset we can ensure that is a null measure set for all , , and since increases with and , we conclude that is a null measure set. Finally is null because it is the union of countably many null measure sets.

Analogous arguments show that is a null measure set, thus the proof of Claim 3 is complete.

Claim 4.

Repeating the previous arguments we can prove that also is the least Carathéodory solution of (1.1) in , thus on . Hence satisfies (3.3).

Analogous arguments show that satisfies (3.4).

Remark 3.2.

Problem (3.6) may not satisfy condition in Theorem 2.7 as the compositions and need not be measurable. That is why we used Theorem 2.4, instead of Theorem 2.7, to establish Theorem 3.1.

Next we show that even singular problems may fall inside the scope of Theorem 3.1 if we have adequate pairs of lower and upper solutions.

Example 3.3.

has positive solutions. Note that the limit of the right hand side as tends to the origin does not exist, so the equation is singular at the initial condition.

which are a countable family of admissible nqsc curves at which condition (2.13) holds.

so condition is satisfied.

Theorem 3.1 ensures that our problem has extremal solutions between and which, obviously, are different from zero almost everywhere. Therefore (3.24) has positive solutions.

The result of Theorem 3.1 may fail if we assume that condition is satisfied only in the interior of the set . This is shown in the following example.

Example 3.4.

It is easy to check that and for all are lower and upper solutions for this problem and that all the assumptions of Theorem 3.1 are satisfied in the interior of . However this problem has no solution at all.

In order to complete the previous information we can say that condition in the interior of is enough if we modify the definitions of lower and upper solutions in the following sense.

Theorem 3.5.

and let .

Suppose that there exists a null-measure set such that conditions and hold and, moreover,

() for every , , one has either (2.1) or (2.13), and (2.1) can fail, at most, over a countable family of admissible non-quasisemicontinuity curves contained in .

Then the conclusions of Theorem 3.1 hold true.

Proof (Outline)

Note that condition (2.1) with replaced by is immediately satisfied over the graphs of and thanks to the definition of .

Remarks

- (ii)
When satisfies (2.1) everywhere or almost all then every couple of lower and upper solutions satisfies the conditions in Theorem 3.5, so this result is not really new in that case (which includes the Carathéodory and continuous cases).

## 4. Discontinuous Lower and Upper Solutions

Another modification of the concepts of lower and upper solutions concerns the possibility of allowing jumps in their graphs. Since the task of finding a pair of lower and upper solutions is by no means easy in general, and bearing in mind that constant lower and upper solutions are the first reasonable attempt, looking for lower and upper solutions "piece by piece" might make it easier to find them in practical situations. Let us consider the following definition.

Definition 4.1.

One says that is a piecewise continuous lower solution of (1.1) if there exist such that

- (a)for all , one has and for a.a.(4.1)

and .

A piecewise continuous upper solution of (1.1) is defined reversing the relevant inequalities.

The existence of a pair of well-ordered piecewise continuous lower and upper solutions implies the existence of a better pair of continuous lower and upper solutions. We establish this more precisely in our next proposition. Note that the proof is constructive.

Proposition 4.2.

Assume that all the conditions in Theorem 3.1 hold with piecewise continuous lower and upper solutions and .

Then the following statements hold:

(ii)if is an upper solution of (1.1) with then , and if is a lower solution with then .

In particular, the conclusions of Theorem 3.1 remain valid and, moreover, every solution of (1.1) between and lies between and .

Proof.

We will only prove the assertions concerning because the proofs for are analogous.

To construct we simply have to join the points , with the graph of by means of an absolutely continuous curve with derivative less than or equal to a.e., being the function given in . It can be easily proven that this is a lower solution of (1.1) that lies between and .

so it cannot go below .

Piecewise continuous lower and upper solutions in the sense of Definition 4.1 were already used in [15, 22]. It is possible to generalize further the concept of lower and upper solutions, as a piecewise continuous lower solution is a particular case of a bounded variation function that has a nonincreasing singular part. Bounded variation lower and upper solutions with monotone singular parts were used in [23, 24], but it is not clear whether Theorem 3.1 is valid with this general type of lower and upper solutions. Anyway, piecewise continuous lower and upper solutions are enough in practical situations, and since these can be transformed into continuous ones which provide better information we will only consider from now on continuous lower and upper solutions as defined in Definition 2.1.

## 5. Singular Differential Equations

It is the goal of the present section to establish a theorem on existence of solutions for (1.1) between a pair of well-ordered lower and upper solutions and in lack of a local bound. Solutions will be weak, in the sense of the following definition. By we denote the set of functions such that for all , and in a similar way we define .

Definition 5.1.

We say that is a weak lower solution of (1.1) if and for a.a. . A weak upper solution is defined analogously reversing inequalities. A weak solution of (1.1) is a function which is both a weak lower solution and a weak upper solution.

We will also refer to *extremal weak solutions* with obvious meaning.

and none of them is absolutely continuous on . Another example, which uses lower and upper solutions, can be found in [15, Remark ].

However weak (lower/upper) solutions are of Carathéodory type provided they have bounded variation. We establish this fact in the next proposition.

Proposition 5.2.

Let be such that and let be continuous on and locally absolutely continuous on .

A necessary and sufficient condition for to be absolutely continuous on is that be of bounded variation on .

Proof.

Next we present our main result on existence of weak solutions for (1.1) in absence of integrable bounds.

Theorem 5.3.

Suppose that (1.1) has a weak lower solution and a weak upper solution such that for all and .

Suppose that there is a null-measure set such that conditions and in Theorem 3.1 hold for and assume moreover that the following condition holds:

Proof.

We will only prove that (5.6) defines the greatest weak solution of (1.1) in , as the arguments to show that (5.5) is the least one are analogous.

First note that is a weak lower solution between and , so is well defined.

has extremal Carathéodory solutions between and . Let denote the greatest solution of (5.7) between and . By virtue of Theorem 2.7 we also know that is the greatest lower solution of (5.7) between and .

Next we prove in several steps that on for each .

Step 1 ( on for each ).

The restriction to of each weak lower solution between and is a lower solution of (5.7) between and , thus is, on the interval , greater than or equal to any weak lower solution of (1.1) between and . The definition of implies then that on .

Step 2 ( on for all ).

would be a solution of (5.7) (with replaced by ) between and which is greater than on , a contradiction.

Step 3 ( ).

and therefore . Since was fixed arbitrarily in the previous arguments, we conclude that .

Step 4 ( is a weak lower solution of (1.1)).

Let where and .

For and a.a. we have , thus for a.a. .

Let be such that (5.15) holds. We have two possibilities: either (2.1) holds for at and then from (5.15) we deduce , or , where is an admissible curve of non quasisemicontinuity. In the last case we have that either belongs to a null-measure set or , which, in turn, yields two possibilities: either and then , or and then (5.15), with and , and the definition of admissible curve of non quasisemicontinuity imply that .

The above arguments prove that a.e. on , and since was fixed arbitrarily, the proof of Step 4 is complete.

Conclusion

The construction of and Step 1 imply that and the definition of and Step 4 imply that . Therefore for all we have on and then is a weak solution of (1.1). Since every weak solution is a weak lower solution, is the greatest weak solution of (1.1) in .

The assumption in Theorem 5.3 can be replaced by other types of conditions. The next theorem generalizes the main results in [7, 12–14] concerning existence of solutions of singular problems of the type of (1.1).

Theorem 5.4.

Suppose that (1.1) with has a weak lower solution and a weak upper solution such that for all and on .

Suppose that there is a null-measure set such that conditions and in Theorem 3.1 hold for and assume moreover that the following condition holds:

() for every there exists such that for all , , and one has .

Then the conclusions of Theorem 5.3 hold true.

Proof.

Claim [ exists]

and with respect to the lower solution and the upper solution , so there exists the greatest solution between and of (5.17). Notice that if is a solution of (5.17) then , so is also the greatest solution between and of (5.17).

with respect to the lower solution and the upper solution (both functions restricted to ). Hence there exists the greatest solution of (5.18) between and .

Analogous arguments to those in the proof of Theorem 5.3 show that is a weak upper solution and it is clear that .

Finally we show that holds with replaced by . We consider a decreasing sequence such that and . As and are positive on , we can find such that on . We deduce then from the existence of so that for a.e. and all . The function defined by for works.

Theorem 5.3 implies that (1.1) has extremal weak solutions in which, moreover, satisfy (5.6) and (5.5) with replaced by . Furthermore if is a weak solution of (1.1) in then on . Assume, on the contrary, that for some , then there would exist such that and then would be a solution of (5.16) between and which is strictly greater than on some subinterval, a contradiction. Hence (1.1) has extremal weak solutions in which, moreover, satisfy (5.6) and (5.5).

## 6. Systems

where , , , , and .

Our goal is to extend Theorem 5.3 to this multidimensional setting, which, as usual, requires the right hand side to be quasimonotone, as we will define later.

We start extending to the vector case the definitions given before for scalar problems. To do so, let denote the set of functions such that for each the component is absolutely continuous on for each . Also, stands for the class of -valued functions which are defined and continuous on .

A weak lower solution of (6.1) is a function such that for each we have and for a.a. we have Weak upper solutions are defined similarly by reversing the relevant inequalities, and weak solutions of (6.1) are functions which are both weak lower and weak upper solutions.

Extremal (least and greatest) weak solutions of (6.1) in a certain subset of are defined in the obvious way considering the previous ordering.

Now we are ready to extend Theorem 5.3 to the vector case. We will denote by the th canonical vector. The proof follows the line of that of [8, Theorem ].

Theorem 6.1.

Suppose that (6.1) has weak lower and upper solutions and such that , , and let .

Suppose that is quasimonotone nondecreasing in , that is, for and the relations and imply .

Suppose, moreover, that for each the following conditions hold:

() the function is measurable;

and (6.3) fails, at most, over a countable family of admissible nqsc curves of the scalar differential equation contained in the sector ;

() there exists such that for and a.a. one has .

Proof.

In particular, . Further, every possible solution of (6.1) in is less than or equal to by (6.7) and , independently of .

Claim 1 ( ).

and, therefore, . Further is continuous at because for all , and are continuous at and .

Claim 2.

*.*For each weak lower solution such that a.e., the quasimonotonicity of yields

and then Theorem 5.3 implies that , where is the greatest weak solution of (6.11) in . Then .

hence is a weak lower solution of (6.1) in with a.e. on , thus . Therefore is a weak solution of (6.1), and, by (6.7) and , it is the greatest one in . In particular, the greatest weak solution of (6.1) in exists and it is greater than or equal to .

Claim 3.

The greatest weak solution of (6.1) in , , satisfies (6.6). The weak lower solution was fixed arbitrarily, so is greater than or equal to any weak lower solution in . On the other hand, is a weak lower solution.

Analogously, the least weak solution of (6.1) in is given by (6.5).

## 7. Examples

Example 7.1.

Here square brackets mean integer part, and by positive solution we mean a solution which is positive on .

First note that (7.1) has at most one positive weak solution because the right hand side in the differential equation is nonincreasing with respect to the unknown on , thus at no point can solutions bifurcate.

On the other hand it is easy to check that for we have and then , , is a lower solution.

The function is continuous between the graphs of and except over the lines , , which are admissible nqsc curves for all , (note that is not an admissible nqsc curve but it does not lie between and ).

where is such that .

Therefore Theorem 5.4 implies the existence of a weak solution of (7.1) between and . Moreover, this weak solution between and is increasing, so Proposition 5.2 ensures that it is, in fact, a Carathéodory solution on .

It is possible to extend the solution on the right of to some where the solution will assume the value . The solution cannot be extended further on the right of , as (7.1) with replaced by has no solution on the right of .

we deduce that the solution is defined at least on , where and .

Example 7.2.

has a unique positive Carathéodory solution.

Note that the equation is not separable and assumes positive and negative values on every neighborhood of the initial condition. Moreover the equation is singular at the initial condition with respect to both of its variables.

Once again the right hand side in the differential equation is nonincreasing with respect to the unknown on , thus we have at most one positive weak solution.

Lower and upper solutions are given by, respectively, and for .

This shows that is positive between and and, moreover, we can say that for it suffices to take such that to have for all between the graphs of and and .

Therefore Theorem 5.4 implies the existence of a weak solution of (7.7) between and . Moreover, since is positive between and the solution is increasing and, therefore, it is a Carathéodory solution.

The previous two examples fit the conditions of Theorems 5.3 and 5.4. Next we show an example where Theorem 5.3 can be used but it is not clear whether or not we can also apply Theorem 5.4.

Example 7.3.

Lower and upper solutions are given by and , . Since is nonnegative between and the lines , , are admissible nqsc curves for the differential equation. Finally it is easy to check that if and , thus one can construct such that for a.a. and .

Theorem 5.3 ensures that (7.11) has extremal weak solutions between and . Moreover (7.11) has a unique solution between and as is nonincreasing with respect to the unknown. Further, the unique solution is monotone and therefore it is a Carathéodory solution.

## Declarations

### Acknowledgments

The research of Rodrigo López Pouso is partially supported by Ministerio de Educación y Ciencia, Spain, Project MTM2007-61724, and by Xunta de Galicia, Spain, Project PGIDIT06PXIB207023PR.

## Authors’ Affiliations

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