Higher order functional boundary value problems without monotone assumptions
 João F Fialho^{1, 3}Email author and
 Feliz Minhós^{2, 3}
DOI: 10.1186/16872770201381
© Fialho and Minhós; licensee Springer. 2013
Received: 15 December 2012
Accepted: 22 March 2013
Published: 10 April 2013
Abstract
In this paper, given $f:[a,b]\times {(C([a,b]))}^{n2}\times {\mathbb{R}}^{2}\to \mathbb{R}$ a ${L}^{1}$Carathéodory function, it is considered the functional higher order equation
together with the nonlinear functional boundary conditions, for $i=0,\dots ,n2$
Here, ${L}_{i}$, $i=0,\dots ,n1$, 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.
1 Introduction
where ${L}_{i}$, $i=0,\dots ,n1$, 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 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 $(n3)$. 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 [16].
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 ${L}_{i}$, $i=0,\dots ,n1$, by using adequate auxiliary functions and global arguments. This fact with the functional dependence on the unknown function and its derivatives till order $(n1)$ 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 $n=2$, where equation (1) loses its functional part.

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 $(n1)$th derivative of the corresponding solutions.
for spaces ${L}^{p}$, $1\le p\le \mathrm{\infty}$.
and for every $({y}_{0},{y}_{1},{y}_{2},{y}_{3})\in {(C(I))}^{(n2)}\times {\mathbb{R}}^{2}$ with ${y}_{i}\le M$, for $i=0,\dots ,n1$.
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 $i=0,\dots ,n4$.
The Nagumotype condition is given by next definition.
The next result gives an a priori estimate for the $(n1)$th derivative of all possible solutions of (1).
Moreover, the constant R depends only on the functions φ and${\gamma}_{i},{\mathrm{\Gamma}}_{i}$ ($i=0,\dots ,n2$) and not on the boundary conditions.
Proof The proof is similar to [[19], Lemma 2.1]. □
The upper and lower solution definition is then given by the following.
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 $(n1)$ order.
The arguments of the proof require the following lemma, given on [29].
 (a)
$\frac{d}{dx}q(x,u(x))$ exists for a.e. $x\in I$.
 (b)If $u,{u}_{m}\in {C}^{1}(I)$ and ${u}_{m}\to u$ in ${C}^{1}(I)$ then$\frac{d}{dx}q(x,{u}_{m}(x))\to \frac{d}{dx}q(x,u(x))\phantom{\rule{1em}{0ex}}\mathit{\text{for a.e.}}\phantom{\rule{1em}{0ex}}x\in I.$
Now, we are in a position to prove the main result of this paper.
Theorem 5 Assume that there exists a pair of lower and upper solutions$(\alpha ,\beta )$of problem (1)(2).
and$R>0$is given by (7).
with K given by (10).
The proof will follow the next steps:
and ${u}^{(n1)}(x)<K$, for every $x\in I$, with $K>0$ given in (10).
As ${(u\alpha )}^{(n1)}({x}_{0})=0$ and ${(u\alpha )}^{(n1)}$ is nonincreasing in $({x}_{1},{x}_{2})$, this contradicts the definitions of ${x}_{0}$ and ${x}_{2}$.
Analogously, it can be obtained ${u}^{(n3)}(x)\le {\beta}_{n3}(x)$, for $x\in I$.
The remaining inequalities are obtained by the same integration process.
for K given by (10), it is obtained, by Lemma 2, the a priori bound ${u}^{(n1)}(x)<K$, for $x\in I$. For details, see [[33], Lemma 2].
Step 2. Problem (12)(13) has at least one solution.
Since ${L}_{0},\dots ,{L}_{n1}$ are continuous and f is a ${L}^{1}$Carathéodory function, then, from Lemma 4, ${\mathcal{N}}_{\lambda}$ is continuous. Moreover, as ${\mathcal{L}}^{1}$ is compact, it can be defined the completely continuous operator ${\mathcal{T}}_{\lambda}:{C}^{n1}(I)\to {C}^{n1}(I)$ by ${\mathcal{T}}_{\lambda}u={\mathcal{L}}^{1}{\mathcal{N}}_{\lambda}(u)$.
It is obvious that the fixed points of operator ${\mathcal{T}}_{\lambda}$ coincide with the solutions of problem (16)(17).
for some $\overline{K}>0$ independent of λ.
In the set $\mathrm{\Omega}=\{u\in {C}^{n1}(I):{\parallel u\parallel}_{{C}^{n1}}<\overline{K}+1\}$, the degree $d(\mathcal{I}{\mathcal{T}}_{\lambda},\mathrm{\Omega},0)$ is well defined for every $\lambda \in [0,1]$ and, by the invariance under homotopy, $d(\mathcal{I}{\mathcal{T}}_{0},\mathrm{\Omega},0)=d(\mathcal{I}{\mathcal{T}}_{1},\mathrm{\Omega},0)$.
which has only the trivial solution, then $d(\mathcal{I}{\mathcal{T}}_{0},\mathrm{\Omega},0)=\pm 1$. So, by degree theory, the equation $x={\mathcal{T}}_{1}(x)$ 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).
□
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.
for all $x\in [0,1]$.
Declarations
Acknowledgements
Dedicated to Professor Jean Mawhin on the occasion of his 70th anniversary.
Authors’ Affiliations
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