First-Order Singular and Discontinuous Differential Equations

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

Let and be fixed and let be a given mapping. We are concerned with the existence of solutions of the initial value problem

(1.1)

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adapted to prove existence results for (1.1) under such weak conditions that standard functional analysis arguments are no longer valid.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.

It is proven in [8] that (1.1) has extremal solutions if is -bounded for all is measurable, and for a.a. is quasi-semicontinuous, namely, for all we have

(2.1)

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.

On the other hand, the above assumptions imply that the extremal solutions of (1.1) are given as the infimum of all upper solutions and the supremum of all lower solutions, that is, the least solution of (1.1) is given by

(2.2)

and the greatest solution is

(2.3)

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.

An admissible non-quasi-semicontinuity (nqsc) curve for the differential equation is the graph of an absolutely continuous function such that for a.a. one has either , or

(2.4)

(2.5)

Remark 2.3.

The condition (2.1) cannot fail over arbitrary curves. As an example note that (1.1) has no solution for and

(2.6)

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;

(2)there exists an integrable function , , such that

(2.7)

Then the mapping

(2.8)

is absolutely continuous on and satisfies and for a.a. , where and for all the set

(2.9)

contains no positive measure set.

Analogously, the mapping

(2.10)

is absolutely continuous on and satisfies and for a.a. , where and for all the set

(2.11)

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.

For each , is measurable, and for one has

(2.12)

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;

(ii)for every and all one has either (2.1) or

(2.13)

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

(iii)there exists an integrable function , , such that

(2.14)

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] actually asserts that , as defined in (2.8), is the least Carathéodory solution, but it is easy to prove that in that case , as defined in (2.2). Indeed, let be an arbitrary upper solution of (1.1), let and let

(2.15)

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 ;

() there exists an integrable function , , such that

(3.1)

Then (1.1) has extremal solutions in the set

(3.2)

Moreover the least solution of (1.1) in is given by

(3.3)

and the greatest solution of (1.1) in is given by

(3.4)

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).

For each we define

(3.5)

Claim 1.

The modified problem

(3.6)

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

To show that condition in Theorem 2.4 is satisfied with replaced by , let be fixed. The verification of (2.1) for at is trivial in the following cases: and satisfies (2.1) at , , and . Let us consider the remaining situations: we start with the case and satisfies (2.1) at , for which we have and

(3.7)

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

The previous argument shows that satisfies (2.1) at every except, at most, over the graphs of the countable family of admissible nonquasisemicontinuity curves in condition for . Therefore it remains to show that if is one of those admissible nqsc curves for then it is also an admissible nqsc curve for . As long as the graph of remains in the interior of we have nothing to prove because and are the same, so let us assume that on a positive measure set , . Since and are absolutely continuous there is a null measure set such that for all , thus for we have

(3.8)

so condition (2.5) with replaced by is satisfied on . On the other hand, we have to check whether for those at which we have

(3.9)

We distinguish two cases: and . In the first case (3.9) is equivalent to

(3.10)

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 .

By virtue of Claim 1 and Theorem 2.4 we can ensure that the functions and defined as

(3.11)

are absolutely continuous on and satisfy and for a.a. , where and for all the set

(3.12)

contains no positive measure set, and for a.a. , where and for all the set

(3.13)

contains no positive measure set.

Claim 2.

For all we have

(3.14)

(3.15)

Let be an upper solution of (3.6) and let us show that for all . Reasoning by contradiction, assume that there exist such that , and

(3.16)

For a.a. we have

(3.17)

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.

Let us show that the set is a null measure set. First, note that

(3.18)

and we can split , where and

Let us show that is a null measure set. Since and are absolutely continuous the set

(3.19)

is null. If then there is some such that and , but then the definitions of and yield

(3.20)

Therefore and thus is a null measure set.

The set can be expressed as , where for each

(3.21)

For , , we have , so the definition of implies that

(3.22)

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.

satisfies (3.3) and satisfies (3.4). Let be an upper solution of (1.1), let , and for all let

(3.23)

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.

Let us denote by the integer part of a real number . We are going to show that the problem

(3.24)

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.

In order to apply Theorem 3.1 we consider (1.1) with , , and

(3.25)

It is elementary matter to check that and , , are lower and upper solutions for the problem. Condition (2.1) only fails over the graphs of the functions

(3.26)

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

Finally note that

(3.27)

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.

Let us consider problem (1.1) with , and defined as

(3.28)

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.

Suppose that and are absolutely continuous functions on such that for all , ,

(3.29)

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)

It follows the same steps as the proof of Theorem 3.1 but replacing by

(3.30)

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

Remarks

(i)The function in Example 3.4 does not satisfy the conditions in Theorem 3.5.

• (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)

(b), for all

(4.2)

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:

(i)there exist a lower solution and an upper solution such that

(4.3)

(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 .

Moreover, if is an upper solution of (1.1) between and then we have

(4.4)

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.

Note that (lower/upper) solutions, as defined in Definition 2.1, are weak (lower/upper) solutions but the converse is false in general. For instance the singular linear problem

(5.1)

has exactly the following weak solutions:

(5.2)

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.

The necessary part is trivial. To estalish the sufficiency of our condition we use Banach-Zarecki's theorem, see [18, Theorem ]. Let be a null measure set, we have to prove that is also a null measure set. To do this let be such that . Since is absolutely continuous on the set is a null measure set for each . Therefore is also a null measure set because

(5.3)

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:

() there exists such that for all , , one has . Then (1.1) has extremal weak solutions in the set

(5.4)

Moreover the least weak solution of (1.1) in is given by

(5.5)

and the greatest weak solution of (1.1) in is given by

(5.6)

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.

Let be a decreasing sequence in such that . Theorem 2.7 ensures that for every the problem

(5.7)

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 ).

First, since on we have . Reasoning by contradiction, assume that there exists such that . Then there is some such that and on , but then the mapping

(5.8)

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

The above properties of imply that the following function is well defined:

(5.9)

Step 3 ().

Let be fixed. Condition implies that for all such that we have

(5.10)

with . Hence for , , we have

(5.11)

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

The continuity of at follows from the continuity of and at , the assumption , and the relation

(5.12)

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

For and such that we have (5.10) with , hence , and for , , Fatou's lemma yields

(5.13)

Hence for a.a. we have

(5.14)

Let where and .

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

On the other hand, for a.a. the relation (5.14) and the increasingness of yield

(5.15)

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.

We start observing that there exists a weak upper solution such that on and . If then it suffices to take as . If we proceed as follows in order to construct : let be a decreasing sequence in such that and for every let be the greatest solution between and of

(5.16)

Claim [ exists]

Let be so small that for all . Condition implies that there exists such that for a.a. and all we have . Let , and let be such that on . We can apply Theorem 2.7 to the problem

(5.17)

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).

Now condition ensures that Theorem 2.7 can be applied to the problem

(5.18)

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 .

Obviously we have

(5.19)

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

Let us consider the following system of ordinary differential equations:

(6.1)

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.

In the set we define a partial ordering as follows: let , we write if every component of is less than or equal to the corresponding component of on the whole of . If are such that then we define

(6.2)

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;

() for all and a.a. one has either

(6.3)

or

(6.4)

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 .

Then (6.1) has extremal weak solutions in . Moreover the least weak solution is given by

(6.5)

and the greatest weak solution is given by

(6.6)

Proof.

Let be a weak lower solution of (6.1), and let be as in and such that a.e. on for all . Now let be defined for as

(6.7)

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

Claim 1 ().

If is a weak lower solution in with a.e. on then for , , we have

(6.8)

which implies

(6.9)

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

Claim 2.

is the greatest weak solution of (6.1) in . For each weak lower solution such that a.e., the quasimonotonicity of yields

(6.10)

Hence is a weak lower solution between and of the scalar problem

(6.11)

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

On the other hand, we have

(6.12)

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.

Let us show that the following singular and non-quasisemicontinous problem has a unique positive Carathéodory solution:

(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.

For all we have and therefore , , is an upper solution of (7.1) as it solves the majorant problem

(7.2)

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 ).

Finally, for we have

(7.3)

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 owe to the anonymous referee the following remarks. Problem (7.1) is autonomous, so it falls inside the scope of the results in [21], which ensure that if we find such that

(7.4)

then (7.1) has a positive absolutely continuous solution defined implicitly by

(7.5)

Since

(7.6)

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

Example 7.2.

Let be measurable and for a.a. . We will prove that for each , , the problem

(7.7)

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 .

For each the function is continuous between the graphs of and except over the lines , , where , . Let us show that is positive between and , thus will be an admissible nqsc curve for each . For and , , we have

(7.8)

and if, moreover, we restrict our attention to those such that then we have which implies

(7.9)

and thus for , , and , we have

(7.10)

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.

Let be fixed and consider the problem

(7.11)

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.