In this paper, given a -Carathéodory function, it is considered the functional higher order equation
together with the nonlinear functional boundary conditions, for
Here, , , are continuous functions. It will be proved an existence and location result in presence of not necessarily ordered lower and upper solutions, without assuming any monotone properties on the boundary conditions and on the nonlinearity f.
In this paper, it is considered the functional higher order boundary value problem, for composed by the equation
for a.a. , where is a -Carathéodory function, and the function boundary conditions
where , , are continuous functions without assuming monotone conditions or another type of variation.
The functional differential equation (1) can be seen as a generalization of several types of full differential and integro-differential equations and allow to consider delays, maxima or minima arguments, or another kind of global variation on the unknown function or its derivatives until order . On the other hand, the functional dependence in (2) makes possible its application to a huge variety of boundary conditions, such as Lidstone, separated, multipoint, nonlocal and impulsive conditions, among others. As example, we mention the problems contained in [1-15]. A detailed list about the potentialities of functional problems and some applications can be found in .
Recently, functional boundary value problems have been studied by several authors following several approaches, as it can be seen, for example, in [17-24]. In this work, the lower and upper solutions method is applied together with topological degree theory, according some arguments suggested in [25-27].
The novelty of this paper consists in the following items:
• There is no monotone assumptions on the boundary functions , , by using adequate auxiliary functions and global arguments. This fact with the functional dependence on the unknown function and its derivatives till order will allow that problem (1)-(2) can include the periodic and antiperiodic cases, which were not covered by the existent literature on functional boundary value problems. In this sense, the results in this area, as for instance [28-32], are improved, even for , where equation (1) loses its functional part.
• No extra condition on the nonlinear part of (1) is considered, besides a Nagumo-type growth assumption. In fact, as far as we know, it is the first time where lower and upper solutions technique is used without such hypothesis on function f, by the use of stronger definitions for lower and upper solutions.
• No order between lower and upper solutions is assumed. Putting the ‘well ordered’ case on adequate auxiliary functions, it allows that lower and upper solutions could be well ordered, by reversed order or without a defined order.
The last section contains an example where the potentialities of the functional dependence on the equation and on the boundary conditions are explored.
2 Definitions and auxiliary functions
In this section, it will be introduced the notations and definitions needed forward together with some auxiliary functions useful to construct some ordered functions on the basis of the not necessarily ordered lower and upper solutions of the referred problem.
A Nagumo-type growth condition, assumed on the nonlinear part, will be an important tool to set an a priori bound for the th derivative of the corresponding solutions.
In the following, denotes the usual Sobolev Spaces in I, that is, the subset of functions, whose th derivative is absolutely continuous in I and the mth derivative belongs to and the usual norms
for spaces , .
The function is a -Carathéodory function, that is, is a continuous function for a.e. ; is measurable for ; and for every there is a real-valued function such that
and for every with , for .
The main tool to obtain the location part is the upper and lower solutions method. However, in this case, they must be defined as a pair, which means that it is not possible to define them independently from each other. Moreover, it is pointed out that lower and upper functions, and the correspondent first derivatives, are not necessarily ordered.
To introduce ‘some order’, some auxiliary functions must be defined.
For any define functions , , as it follows:
The Nagumo-type condition is given by next definition.
Definition 1 Consider , , such that , , and the set
A function is said to verify a Nagumo-type condition in E if there exists such that
for every , and
where is given by
The next result gives an a priori estimate for the th derivative of all possible solutions of (1).
Lemma 2There exists such that for every -Carathéodory function satisfying (4) and (5) and every solutionuof (1) such that
for , we have
Moreover, the constantRdepends only on the functionsφand ( ) and not on the boundary conditions.
Proof The proof is similar to [, Lemma 2.1]. □
The upper and lower solution definition is then given by the following.
Definition 3 The functions are a pair of lower and upper solutions for problem (1)-(2) if , on I, for all , and for every , for some , the following inequalities hold for a.e. ,
and for ,
3 Existence and location result
In this section, it is provided an existence and location theorem for the problem (1)-(2). More precisely, sufficient conditions are given for, not only the existence of a solution u, but also to have information about the location of u, and all its derivatives up to the order.
The arguments of the proof require the following lemma, given on .
Lemma 4For such that , for every , define
Then, for each the next two properties hold:
(a) exists for a.e. .
(b) If and in then
Now, we are in a position to prove the main result of this paper.
Theorem 5Assume that there exists a pair of lower and upper solutions of problem (1)-(2).
If is a -Carathéodory function, satisfying a Nagumo-type condition in
then problem (1)-(2) has at least one solutionusuch that
for every , and , , where
and is given by (7).
Proof Define the continuous functions, for ,
and the truncation, not necessarily continuous,
with K given by (10).
Consider the modified problem composed by the equation
and the boundary conditions, for ,
The proof will follow the next steps:
Step 1. Every solution u of problem (12)-(13), satisfies
and , for every , with given in (10).
Let u be a solution of the modified problem (12)-(13). Assume, by contradiction, that there exists such that and let be such that
As, by (13), and , then . So, there is such that
Now, since for all it is satisfied that , we deduce that
As and is nonincreasing in , this contradicts the definitions of and .
The inequality , in I, can be proved in same way and so,
By (13) and (3), the following inequalities hold for every :
Analogously, it can be obtained , for .
The remaining inequalities are obtained by the same integration process.
Applying previous bounds in Lemma 2, and remarking that
for K given by (10), it is obtained, by Lemma 2, the a priori bound , for . For details, see [, Lemma 2].
Step 2. Problem (12)-(13) has at least one solution.
For let us consider the homotopic problem given by
and the boundary conditions, for ,
Let us consider the norms in and in , respectively,
Define the operators by and, for , , by
Since are continuous and f is a -Carathéodory function, then, from Lemma 4, is continuous. Moreover, as is compact, it can be defined the completely continuous operator by .
It is obvious that the fixed points of operator coincide with the solutions of problem (16)-(17).
As is bounded in and uniformly bounded in , we have that any solution of the problem (16)-(17), verifies the following a priori bound
for some independent of λ.
In the set , the degree is well defined for every and, by the invariance under homotopy, .
As the equation is equivalent to the problem
which has only the trivial solution, then . So, by degree theory, the equation has at least one solution, that is, the problem (12)-(13) has at least a solution in Ω.
Step 3. Every solution u of problem (12)-(13) is a solution of (1)-(2).
Let u be a solution of the modified problem (12)-(13). By previous steps, function u fulfills equation (1). So, it will be enough to prove the following inequalities, for :
Then, by (13), . By previous steps, it is obtained the following contradiction with (18):
Applying similar arguments, it can be proved that
and analogously, for ,
Also, using the same arguments and the same techniques, it can be proved that
This section contains a problem composed by an integro-differential equation with some functional boundary conditions, whose solvability is proved in presence of nonordered lower and upper solutions. We remark that such fact was not possible with the results in the current literature. This example does not model any particular problem arising in real phenomena. Our purpose consists on emphasizing the powerful of the developed theory in this paper by showing what kind of problems we can deal with.
Consider, for , the fourth-order equation
coupled with the boundary value conditions
One can verify that functions
are, respectively, lower and upper solutions for the problem (19)-(20). Moreover, we deduce that
As the continuous function f verifies (4) and (5) for in
then, by Theorem 5, there is a nontrivial solution u for problem (19)-(20) such that
for all .
The authors declare that they have no competing interests.
The work presented here was carried out in collaboration between the authors. The authors contributed to every part of this study equally, read and approved the final version of the manuscript.
Dedicated to Professor Jean Mawhin on the occasion of his 70th anniversary.
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