Abstract
In this paper, given
together with the nonlinear functional boundary conditions, for
Here,
1 Introduction
In this paper, it is considered the functional higher order boundary value problem,
for
for a.a.
where
The functional differential equation (1) can be seen as a generalization of several
types of full differential and integrodifferential 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
Recently, functional boundary value problems have been studied by several authors following several approaches, as it can be seen, for example, in [1724]. In this work, the lower and upper solutions method is applied together with topological degree theory, according some arguments suggested in [2527].
The novelty of this paper consists in the following items:
• There is no monotone assumptions on the boundary functions
• No extra condition on the nonlinear part of (1) is considered, besides a Nagumotype 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 Nagumotype growth condition, assumed on the nonlinear part, will be an important
tool to set an a priori bound for the
In the following,
for spaces
The function
and for every
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
for
The Nagumotype condition is given by next definition.
Definition 1 Consider
A function
for every
where
The next result gives an a priori estimate for the
Lemma 2There exists
for
Moreover, the constantRdepends only on the functionsφand
Proof The proof is similar to [[19], Lemma 2.1]. □
The upper and lower solution definition is then given by the following.
Definition 3 The functions
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
The arguments of the proof require the following lemma, given on [29].
Lemma 4For
Then, for each
(a)
(b) If
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
If
then problem (1)(2) has at least one solutionusuch that
for every
and
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
Let u be a solution of the modified problem (12)(13). Assume, by contradiction, that there
exists
As, by (13),
Therefore,
and
Now, since for all
As
The inequality
By (13) and (3), the following inequalities hold for every
Analogously, it can be obtained
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
Step 2. Problem (12)(13) has at least one solution.
For
and the boundary conditions, for
Let us consider the norms in
and
Define the operators
Since
It is obvious that the fixed points of operator
As
for some
In the set
As the equation
which has only the trivial solution, then
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
and
Assume that
Then, by (13),
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
□
4 Example
This section contains a problem composed by an integrodifferential 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
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
and
As the continuous function f verifies (4) and (5) for
then, by Theorem 5, there is a nontrivial solution u for problem (19)(20) such that
for all
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
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.
Acknowledgements
Dedicated to Professor Jean Mawhin on the occasion of his 70th anniversary.
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