The aim of this paper is to extend the theory of lower and upper solutions to the periodic problem associated with planar systems of differential equations. We generalize previously given definitions and we are able to treat both the well-ordered case and the non-well-ordered case. The proofs involve topological degree arguments, together with a detailed analysis of the solutions in the phase plane.
The method of lower and upper solutions for scalar second order differential equations of the type
can be dated back to the pioneering papers by Picard , Scorza Dragoni  and Nagumo , dealing with separated boundary conditions. Its full extension to the periodic problem is due to Knobloch . Further extensions to partial differential equations of elliptic or parabolic type have also been proposed, and there is nowadays a huge literature on this subject. For a rather complete historical and bibliographical account, we refer to the book .
Recently Toader , jointly with the first author, extended the main idea in the definition of lower and upper solutions to planar systems of ordinary differential equations, with the aim of finding bounded solutions through the method of Ważewski . As a by-product, the theorem of Massera  provided also the existence of periodic solutions. It is the aim of this paper to further develop this theory, concentrating on the periodic problem, by the use of topological degree methods.
We consider the periodic problem
where and are continuous functions, T-periodic in their first variable. Our purpose is to give a general definition of a lower and an upper solution with the aim of obtaining the existence of a solution to problem (P). In order to do this, let us first recall the definition of lower solution given in .
In , a continuously differentiable function is said to be a lower solution for problem (P) if it is T-periodic and the following properties hold:
there exists a unique function such that
is continuously differentiable, and
there are two positive constants such that, when ,
An analogous definition was provided for an upper solution, and an existence result was proved for problem (P) assuming , the so called well-ordered case.
We will generalize the above definition in two directions. First of all, condition (iii) will be removed. Moreover, the function α will not need to be differentiable on all its domain, and the function will be allowed to have some discontinuity points. The precise definition will be given in Section 2. Moreover, after having proved the existence of a solution of problem (P) in the well-ordered case, we will also deal with the non-well-ordered case. Assuming some growth conditions on f and g in order to avoid resonance, we will then be able to prove an existence result also in this case.
A natural application of our results is provided by the periodic problem associated with the scalar equation
which can be written in the form of problem (P), with and . Here, is an increasing homeomorphism between two intervals I and J containing 0, and . Typical examples in the applications involve the choice , leading to the so-called “scalar p-Laplacian” operator (cf. ), or , providing a “mean curvature” operator (cf. ), or , providing a “relativistic” operator (cf. ). (See  for a detailed discussion in this direction.) A lower solution for the periodic problem associated with (1.1) is usually defined as a continuously differentiable function such that for every t, with , and
Our definition to be given in Section 2 extends also this one, with the natural choice . Similarly for what concerns an upper solution.
Notice however that for our problem (P) we do not need any monotonicity assumption on . Indeed, even in the simpler case , the inequalities in (i) resemble some sign condition, which may be satisfied also if f is not an increasing function.
Organization of the Paper.
In Section 2 we introduce our main definitions and provide some remarks and preliminaries needed in the sequel.
In Section 3 we prove an existence result in the well-ordered case , assuming (like in ) the existence of some bounding curves, in order to control the solutions in the phase plane. The construction of these curves can be easily carried out in concrete examples, assuming a Nagumo-type condition (see  or Lemma 15 below).
In Section 4 we deal with the non-well-ordered case. Here we need to ask an extra technical condition on the lower and upper solutions; it remains an open question if it could possibly be avoided. Moreover, we assume the existence of a whole family of bounding curves. This assumption is again verified under some type of Nagumo conditions.
In Section 5 we present some variants of our main theorems and discuss on the possibility of further extending the theory to higher dimensional systems.
2 Main Definitions and Preliminaries
For any function we use the notation
A continuous function is said to be a lower solution for problem (P) if it is T-periodic and there exist a T-periodic function and a finite number of points such that the following properties hold:
the restriction of α [resp. ] to each open interval , with , is continuously differentiable [resp. differentiable],
and exist in for every , with(2.1)
for every ,(2.2)
A continuous function is said to be an upper solution for problem (P) if it is T-periodic and there exist a T-periodic function and a finite number of points such that the following properties hold:
the restriction of β [resp. ] to each open interval , with , is continuously differentiable [resp. differentiable],
and exist in for every , with(2.4)
for every ,(2.5)
In what follows, when dealing with a couple of a lower and an upper solution, we will assume, without loss of generality, that the points provided in the previous definitions are the same, both for α and β. Moreover, since we are dealing with T-periodic functions, it is worth defining the sets
Therefore, (2.1), (2.4) hold with replaced by any , and (2.2), (2.3), (2.5), (2.6) hold for every .
When dealing with the periodic problem associated with the scalar equation (1.1), the usual definitions of lower/upper solutions are contained in the above ones, taking
Indeed, the conditions , permit to continuously extend the functions to the whole real line , and the conditions , are included in (2.1), (2.4). The possibility of having some discontinuity points can be useful in the applications, e.g., when taking as a lower solution the maximum of two or more smooth lower solutions, and as an upper solution the minimum of two or more smooth upper solutions.
From (2.2) we have that
and is the only value for which this identity holds. Similarly, from (2.5) we have
and is uniquely defined on by this identity.
It is well known in the case of scalar second order equations that if a function is at the same time a lower and an upper solution, then it is a solution. Let us write the analogous statement in our situation.
Let be at the same time a lower and an upper solution for problem (P). Then there exists a function such that is a solution of problem (P).
Denote by and the functions provided by Definitions 1 and 2 taking and , respectively. From (2.7) and (2.8) we deduce that
Then, from (2.1) and (2.4) we first see that , thus implying that is continuously differentiable; moreover, on one hand we have , and on the other hand
showing that for every k. We can thus define
a continuous function.
Since is continuously differentiable, and are continuous, from (2.7) we deduce that for every . Moreover, by (2.3) and (2.6) we get for every ; since and are continuous, we first see that is continuously differentiable, and then also that for every , thus completing the proof. ∎
We will need the following estimates involving our lower and upper solutions, where we adopt the usual definition of the Dini derivatives:
If α is a lower solution for problem (P), then
If β is an upper solution for problem (P), then
Let us fix k and consider the restrictions of the functions and to the interval , redefining the two functions at the extremes in such a way to make them continuous. Then, since both and are differentiable in the interval , by [6, Corollary 3.7] we have
Similarly, we have
thus ending the proof. ∎
3 Well-Ordered Lower and Upper Solutions
We will say that is a well-ordered couple of lower/upper solutions of problem (P) if α and β are respectively a lower and an upper solution of problem (P), and for every . The following result generalizes that part of [8, Theorem 2.5] concerning the existence of periodic solutions.
Assume the existence of a well-ordered couple of lower/upper solutions of problem (P). Set and , with . Let there exist two continuously differentiable functions such that, for every and ,
Then there exists at least one solution of problem (P) such that
for every .
Some remarks are in order.
We will provide in Lemma 15 some Nagumo-type conditions which guarantee the existence of the curves .
The assumption is inessential, since if we have that , hence by Proposition 4 we immediately get a solution.
3.1 Proof of Theorem 6
3.1.1 An Auxiliary Problem
Let be defined as
Fix such that
We interpolate the vector field on with a constant vector field on . Precisely, we define as
We will write .
By the use of the auxiliary functions
we define, for every ,
so to introduce the modified problem
We will write . In the space
we introduce the open set
where, see Figure 1,
Our aim is to prove that there exists a solution of problem () belonging to . Since and on the set , then u will solve also (P).
3.1.2 No Solutions of () Outside
We show that all the solutions of system () are such that for every .
Let us start proving a preliminary lemma.
For every , the following inequalities hold:
Moreover, for every ,
Let us prove the first inequality in (3.7). Suppose , and . We have that
We need to consider three different cases.
Case 1. If , then
Case 2. If , then
Case 3. If , then, by (3.4),
Hence, the first inequality in (3.7) is proved. The second one can be proved analogously, as well as the inequalities in (3.8).
We now prove the first inequality of (3.9). Let . Since , we have
The second inequality in (3.9) follows analogously, and a similar computation proves the ones in (3.10). ∎
Let us define the sets
(see Figure 1).
For every solution of
the following assertions hold true:
We will prove only the validity of the first assertion, since the others follow similarly. We argue by contradiction and assume the existence of and of a solution of (3.11) such that for every and , where (see Figure 2)
Without loss of generality we can assume the existence of such that . We define
We have and, from (3.7),
for every . Hence we have . We conclude that for every . So, being , recalling (3.12), we necessarily have .
If , then the function is continuous in the interval with and for all . Recalling that for all , by (3.9) or (3.10) we have
leading again to a contradiction.
The case could arise only if . However, such a situation is not possible, indeed we would have the existence of such that for every which gives a contradiction, since we have assumed for every . ∎
We have thus proved that the sets are invariant in the past, while the sets are invariant in the future. We also define the sets
(see Figure 1).
If is a solution of (3.11) such that , then there exists such that
Similarly, if is a solution of (3.11) such that , then there exists such that
We give the proof of the first part of the statement, the second one being similar. Let be a solution of (3.11) such that . If , then, defining as above the function ,
using inequalities (3.9) or (3.10). So, there exists such that and for every .
On the other hand, if , then and the strict inequalities and provide the same conclusion as before by a continuity argument.
We now give the proof for . If , then
and we get the existence of such that and for every . On the other hand, if , we reach the same conclusion, by continuity. ∎
If is a solution of (), then
Suppose that there exists a solution of () such that for a certain . If , then, from Lemma 8, we have that for every . Moreover, from (3.7) we get
a contradiction, since is a periodic function.
The same reasoning can be adopted if . Finally, if belongs to , Lemma 9 brings us to the previous contradicting situations.
A similar argument can be adopted in order to show that there are no solutions of () such that . ∎
If is a solution of (), then
We already know from Lemma 10 that any solution of () is such that for every . We claim that for every . Indeed, if the function y has minimum at such that , then we would have
a contradiction. Similarly, must hold.
We now define the periodic function . Let such that . If , we get the following contradiction:
where we have used both (3.1) and (3.5). So, . Similarly we can prove that , where , thus concluding the proof. ∎
3.1.3 A Topological Degree Argument
We define the operators
So, a solution
of problem () corresponds to a solution of
In the previous subsection we have found the a priori bound for all the possible solutions of problem (). In order to apply the degree theory we need to consider an open ball containing . By the above arguments, we can deduce that if u solves (3.16), then , so that the coincidence degree is well defined. We refer to  for more details on this topic.
Since (3.13) and (3.14) hold, we can rewrite system (3.11) as
are bounded functions. We now introduce the functions
and the problems
We define the Nemytskii operator related to the family of problem () as
Since the function is bounded, by a classical argument we can find a sufficiently large such that, for every , all the periodic solutions of () satisfy
Since in the case we have an autonomous linear problem ruled by the function , by [4, Lemma 1] we can conclude that
where denotes the Brouwer degree of the function on the ball and is the set of continuous functions having image in . We have so found a solution of problem () belonging to the set . However, such a solution belongs indeed to the a priori bound , and so it is also a solution of problem (P), thus concluding the proof of Theorem 6.
3.2 An Important Consequence of the Proof
We first recall the definition (3.6) of the open set
Let us introduce the Nemytskii operator related to problem (P) as
Under the assumptions of Theorem 6, if there are no solutions of (P) in , then
Since on , and so on , the additional assumption permits us to evaluate the coincidence degree also on the set . Recalling that all the solutions of problem () satisfy the a priori bounds (3.13) and (3.14), by the excision property we have
and the proof is completed. ∎
4 Non-Well-Ordered Lower and Upper Solutions
We still consider the periodic problem
where and are continuous functions, T-periodic in their first variable.
We will say that is a non-well-ordered couple of lower/upper solutions of problem (P) if α and β are respectively a lower and an upper solution of problem (P) such that there exists satisfying
Let us set
Notice that , by (4.1).
Let us introduce our assumptions.
There are a continuous function and a constant such that(4.2)(4.3)
There exist two continuous functions , continuously differentiable with respect to the first variable, such that
for every , and . (Here we denote by the derivative with respect to the first variable.)
Assume the existence of a non-well-ordered couple of lower/upper solutions of problem (P) with the additional property that there exists a constant such that, for every ,
If (H1) and (H2) hold, there exists at least one solution of problem (P) such that, for some , one has and .
This theorem extends some classical results for scalar second order differential equations of the type (1.1). We will show below two examples of applications. Conditions (H1) and (H2) will be necessary in order to avoid resonance phenomena, and to obtain a priori bounds. Notice that (2.2) and (2.5) imply a weaker form of (4.6) and (4.7), i.e., with only weak inequalities. It remains an open problem if these additional assumptions can be omitted.
We will discuss in Section 5 on the possibility of reversing the inequalities in (4.4) and (4.5). Concerning the existence of the functions , let us prove the following lemma.
Let the following assumptions hold:
There are a constant and two continuous functions and such that
There is a positive continuous function such that
The above functions are such that(4.8)
Then there exist four continuous functions , continuously differentiable with respect to the first variable, such that
for every , and .
For every , we introduce the continuous strictly increasing function defined as
We can easily verify that and, from (4.8),
Construction of . For every and for every there exists a unique such that . Hence, we can define , for , as the unique solution of equation
In particular, since , we get
which provides the validity of (4.9) for the function . Differentiating in (4.14), we see that for every , and
thus proving (4.10).
Construction of . Arguing similarly as above, for every and for every there exists a unique such that . Hence we can define by
In particular, since , we get
so that (4.9) holds for the function . Differentiating in (4.15),
thus proving (4.11).
The construction of the functions and satisfying (4.12) and (4.13) is similar. ∎
Let us illustrate how our result applies to two classical scalar second order differential equations of type (1.1), involving a scalar p-Laplacian and a mean curvature operator, with and , respectively.
Consider first the problem
with , which is equivalent to problem (P), taking and , with .
Assume the existence of a non-well-ordered couple of lower/upper solutions of problem (4.16), and of a constant for which
Then there exists at least one solution of problem (4.16) such that, for some , one has and .
Notice that (4.17) implies (4.3). We can use Lemma 15 with to construct the curves . Then Theorem 14 applies. ∎
Consider now the problem
which is equivalent to problem (P), taking and . Notice that these functions are now only defined on .
Assume the existence of a non-well-ordered couple of lower/upper solutions of problem (4.18), and of a positive continuous function such that
Then there exists at least one solution of problem (4.18) such that, for some , one has and .
Recalling that , by (4.20) there is a such that
We define the functions and as
and we consider the system
Using Lemma 15, we see that all the assumptions of Theorem 14 hold, so that problem (4.22) has a T-periodic solution .
We now show that for every t, implying that is indeed a solution of problem (4.18). By contradiction, assume that , or . Let us treat the first case, the other one being similar. By periodicity, there exists a such that and, correspondingly, . Let and be such that , , , and for every . (When , we write and .) For every , by (4.19) we have
so that, by (4.21),
a contradiction. ∎
The above corollary generalizes [13, Proposition 3.7], where is a constant function with positive value
4.1 Proof of Theorem 14
4.1.1 An Auxiliary Problem
Let us set
where, for all these functions, the norm can be defined as in (3.3).
We recall here a classical result, which is a straightforward consequence of the Gronwall Lemma, often mentioned as elastic property.
For every constant we can define a function with the following property: given a differentiable function satisfying
if for a certain , then for every .
For example, we can take .
Using the notation introduced in the previous lemma, let us now set
where M and have been introduced respectively in (4.3) and (4.23).
By assumption (H2), we can find a sufficiently large constant such that
Let us introduce the sets
(see Figure 3).
There are two constants and with the following property: if is a solution of