First-Order Singular and Discontinuous Differential Equations
© D. C. Biles and R. López Pouso. 2009
Received: 10 March 2009
Accepted: 4 May 2009
Published: 9 June 2009
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.
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  consisted in obtaining the greatest solution as the pointwise infimum of strict upper solutions. Subsequently this idea was improved by Perron in , 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  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adapted to prove existence results for (1.1) under such weak conditions that standard functional analysis arguments are no longer valid.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.
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.
A similar result was established in  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  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  that solutions exist even if (2.1) fails on the points of a countable family of curves in the conditions of the following definition.
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  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:
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 . The next result establishes the fact that we can have "weak" solutions in a sense just by assuming very general conditions over .
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  and the reader can find its proof in .
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 ]).
and (2.1) can fail, at most, over a countable family of admissible nonquasisemicontinuity curves;
Theorem in  implies that also is the least Carathéodory solution of (1.1), thus on . Hence .
3. Existence between Lower and Upper Solutions
In this section we will prove the following result.
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).
contains no positive measure set.
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).
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.
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.
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.
which are a countable family of admissible nqsc curves at which condition (2.13) holds.
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.
Then the conclusions of Theorem 3.1 hold true.
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.
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.
Then the following statements hold:
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 .
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 .
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.
Next we present our main result on existence of weak solutions for (1.1) in absence of integrable bounds.
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 .
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 .
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 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).
Then the conclusions of Theorem 5.3 hold true.
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).
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).
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.
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 ].
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 .
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.
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.
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 .
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.
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.
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.
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.
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