An Overview of the Lower and Upper Solutions Method with Nonlinear Boundary Value Conditions

Boundary Value Problems20102011:893753

DOI: 10.1155/2011/893753

Received: 19 April 2010

Accepted: 7 July 2010

Published: 25 July 2010

Abstract

The aim of this paper is to point out recent and classical results related with the existence of solutions of second-order problems coupled with nonlinear boundary value conditions.

1. Introduction

The first steps in the theory of lower and upper solutions have been given by Picard in 1890 [1] for Partial Differential Equations and, three years after, in [2] for Ordinary Differential Equations. In both cases, the existence of a solution is guaranteed from a monotone iterative technique. Existence of solutions for Cauchy equations have been proved by Perron in 1915 [3]. In 1927, Müller extended Perron's results to initial value systems in [4].

Dragoni [5] introduces in 1931 the notion of the method of lower and upper solutions for ordinary differential equations with Dirichlet boundary value conditions. In particular, by assuming stronger conditions than nowadays, the author considers the second-order boundary value problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ1_HTML.gif
(1.1)

for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq1_HTML.gif a continuous function and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq2_HTML.gif .

The most usual form to define a lower solution is to consider a function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq3_HTML.gif that satisfies the inequality
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ2_HTML.gif
(1.2)
together with
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ3_HTML.gif
(1.3)
In the same way, an upper solution is a function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq4_HTML.gif that satisfies the reversed inequalities
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ4_HTML.gif
(1.4)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ5_HTML.gif
(1.5)

When http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq5_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq6_HTML.gif , the existence of a solution of the considered problem lying between http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq7_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq8_HTML.gif is proved.

In consequence, this method allows us to ensure the existence of a solution of the considered problem lying between the lower and the upper solution, that is, we have information about the existence and location of the solutions. So the problem of finding a solution of the considered problem is replaced by that of finding two well-ordered functions that satisfy some suitable inequalities.

Following these pioneering results, there have been a large number of works in which the method has been developed for different kinds of boundary value problems, thus first-, second- and higher-order ordinary differential equations with different type of boundary conditions such as, among others, periodic, mixed, Dirichlet, or Neumann conditions, have been considered. Also partial differential equations of first and second-order, have been treated in the literature.

In these situations, we have that for the Neumann problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ6_HTML.gif
(1.6)
a lower solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq9_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq10_HTML.gif -function that satisfies (1.2) coupled with the inequalities
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ7_HTML.gif
(1.7)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq11_HTML.gif is an upper solution of the Neumann problem if it satisfies (1.4) and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ8_HTML.gif
(1.8)
Analogously, for the periodic problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ9_HTML.gif
(1.9)
a lower solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq12_HTML.gif and an upper solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq13_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq14_HTML.gif -functions that satisfy (1.2) and (1.4), respectively, together with the inequalities
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ10_HTML.gif
(1.10)

In the classical books of Bernfeld and Lakshmikantham [6] and Ladde et al. [7] the classical theory of the method of lower and upper solutions and the monotone iterative technique are given. This gives the solution as the limit of a monotone sequence formed by functions that solve linear problems related to the nonlinear equations considered. We refer the reader to the classical works of Mawhin [811] and the surveys in this field of De Coster and Habets [1214] in which one can found historical and bibliographical references together with recent results and open problems.

It is important to point out that to derive the existence of a solution a growth condition on the nonlinear part of the equation with respect to the dependence on the first derivative is imposed. The most usual condition is the so-called Nagumo condition that was introduced for this author [15] in 1937. This condition imposes, roughly speaking, a quadratic growth in the dependence of the derivative. The most common form of presenting it is the following.

Definition 1.1.

We say that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq15_HTML.gif satisfies the Nagumo condition if there is http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq16_HTML.gif satisfying
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ11_HTML.gif
(1.11)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ12_HTML.gif
(1.12)

with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq17_HTML.gif .

The main importance of this condition is that it provides a priori bounds on the first derivative of all the possible solutions of the studied problem that lie between the lower and the upper solution. A careful proof of this property has been made in [6]. One can verify that in the proof the condition (1.12) can be replaced by the weaker one,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ13_HTML.gif
(1.13)

The usual tool to derive an existence result consists in the construction of a modified problem that satisfies the two following properties.

()The nonlinear part of the modified equation is bounded.

()The nonlinear part of the modified equation coincides with the nonlinear part when the spatial variable is in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq20_HTML.gif .

When the Dirichlet problem (1.1) is studied, the usual truncated problem considered is
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ14_HTML.gif
(1.14)
Here
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ15_HTML.gif
(1.15)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ16_HTML.gif
(1.16)
with
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ17_HTML.gif
(1.17)

Notice that both http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq21_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq22_HTML.gif are continuous and bounded functions and, in consequence, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq23_HTML.gif is continuous, both properties are satisfied by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq24_HTML.gif .

In the proof it is deduced that all the solutions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq25_HTML.gif of the truncated problem (1.14) belong to the segment http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq26_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq27_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq28_HTML.gif . Notice that the constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq29_HTML.gif only depends on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq30_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq31_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq32_HTML.gif . The existence of solutions is deduced from fixed point theory.

It is important to point out that the a priori bound is deduced for all solutions of the truncated problem. The boundary data is not used. This property is fundamental when more general situations are considered.

In 1954, Nagumo [16] constructed an example in which the existence of well-ordered lower and upper solutions is not sufficient to ensure the existence of solutions of a Dirichlet problem, that is, in general this growth condition cannot be removed for the Dirichlet case. An analogous result concerning the optimality of the Nagumo condition for periodic and Sturm-Liouville conditions has been showed recently by Habets and Pouso in [17].

In 1967, Kiguradze [18] proved that it is enough to consider a one-sided Nagumo condition (by eliminating the absolute value in (1.11)) to deduce existence results for Dirichlet problems. Similar results have been given in 1968 by Schrader [19].

Other classical assumptions that impose some growth conditions on the nonlinear part of the equation are given in 1939 by Tonelli [20]. In this situation, considering the Dirichlet problem (1.1) with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq33_HTML.gif , the following one-sided growth condition is assumed:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ18_HTML.gif
(1.18)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq34_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq35_HTML.gif are sufficiently small numbers.

Different generalizations of these conditions have been developed by, among others, Epheser [21], Krasnoselskii [22], Kiguradze [23, 24], Mawhin [25], and Fabry and Habets [26].

In the case of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq36_HTML.gif being a Carathéodory function, the arguments to deduce the existence result are not a direct translation from the continuous case. This is due to the fact that in the proof the properties are fulfilled at every point of the interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq37_HTML.gif . In this new situation the equalities and inequalities hold almost everywhere and, in consequence, the arguments must be directed to positive measurable sets. Thus, a suitable truncated problem is the following:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ19_HTML.gif
(1.19)

coupled with the corresponding boundary value conditions.

This truncation has been introduced by Gao and Wang in [27] for the periodic problem and improves a previous one given by Wang et al. in [28]. Notice that the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq38_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq39_HTML.gif and it is measurable because of the following result proved in [28, Lemma http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq40_HTML.gif ].

Lemma 1.2.

For any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq41_HTML.gif , the two following properties hold:

(a) http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq42_HTML.gif exists for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq43_HTML.gif ;

(b)If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq44_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq45_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq46_HTML.gif for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq47_HTML.gif .

When a one-sided Lipschitz condition of the following type:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ20_HTML.gif
(1.20)

is assumed on function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq48_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq49_HTML.gif and all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq50_HTML.gif , it is possible to deduce the existence of extremal solutions in the sector http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq51_HTML.gif of the considered problem. By extremal solutions we mean the greatest and the smallest solutions in the set of all the solutions in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq52_HTML.gif . The deduction of such a property holds from an iterative technique that consists of solving related linear problems on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq53_HTML.gif and using suitable maximum principles which are equivalent to the constant sign of the associated Green's function. One can find in [7] a complete development of this theory for different kinds of boundary value conditions.

It is important to note that there are many papers that have tried to get existence results under weaker assumptions on the definition of lower and upper solutions. In particular, Scorza Dragoni proves in 1938 [29], an existence result for the Dirichlet problem by assuming the existence of two http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq54_HTML.gif functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq55_HTML.gif that satisfy
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ21_HTML.gif
(1.21)

are nondecreasing in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq56_HTML.gif .

Kiguradze uses in [24] regular lower and upper solutions and explain that it is possible to get the same results for lower and upper solutions whose first derivatives are not absolutely continuous functions. Ponomarev considers in [30] two continuous functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq57_HTML.gif with right Dini derivatives http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq58_HTML.gif , absolute semicontinuous from below in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq59_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq60_HTML.gif absolute semicontinuous from above in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq61_HTML.gif , that satisfy the following inequalities a.e. http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq62_HTML.gif :
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ22_HTML.gif
(1.22)

For further works in this direction see [3136].

Cherpion et al. prove in [37] the existence of extremal solutions for the Dirichlet problem without assuming the condition (1.20). In fact, they consider a more general problem: the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq63_HTML.gif -laplacian equation. In this case, they define a concept of lower and upper solutions in which some kind of angles are allowed. The definitions are the following.

Definition 1.3.

A function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq64_HTML.gif is a lower solution of problem (1.1) (with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq65_HTML.gif ), if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq66_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq67_HTML.gif , and for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq68_HTML.gif , either http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq69_HTML.gif , or there exists an open interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq70_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq71_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq72_HTML.gif and, for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq73_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ23_HTML.gif
(1.23)

Definition 1.4.

A function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq74_HTML.gif is a lower solution of problem (1.1) (with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq75_HTML.gif ), if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq76_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq77_HTML.gif , and for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq78_HTML.gif , either http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq79_HTML.gif , or there exists an open interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq80_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq81_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq82_HTML.gif and, for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq83_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ24_HTML.gif
(1.24)

Here http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq84_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq85_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq86_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq87_HTML.gif denote the usual Dini derivatives.

By means of a sophisticated argument, the authors construct a sequence of upper solutions that converges uniformly to the function defined at each point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq88_HTML.gif as the minimum value attained by all the solutions of problem (1.1) in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq89_HTML.gif at this point. Passing to the limit, they conclude that such function is a solution too. The construction of these upper solutions is valid only in the case that corners are allowed in the definition. The same idea is valid to get a maximal solution.

Similar results are deduced for the periodic boundary conditions in [12]. In this case the arguments follow from the finite intersection property of the set of solutions (see [38, 39]).

2. Two-Point Nonlinear Boundary Value Conditions

Two point nonlinear boundary conditions are considered with the aim of covering more complicated situations as, for instance, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq90_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq91_HTML.gif .

In general, the framework of linear boundary conditions cannot be directly translated to this new situation. For instance, as we have noticed in the previous section, to ensure existence results for linear boundary conditions, we make use of the fixed point theory. So, in the case of a Dirichlet problem, the set of solutions of (1.1) coincide with the set of the fixed points of the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq92_HTML.gif , defined by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ25_HTML.gif
(2.1)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ26_HTML.gif
(2.2)
is the Green's function related to the linear problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ27_HTML.gif
(2.3)

It is obvious that when nonlinear boundary value conditions are treated, the operator whose fixed points are the solutions of the considered problem must be modified.

Moreover the truncations that have to be made in the nonlinear part of the problem (1.14), for the continuous case, and in (1.19), for the Carathéodory one, must be extended to the nonlinear boundary conditions. This new truncation on the boundary conditions must satisfy similar properties to the ones of the nonlinear part of the equation, that is,

()the modified nonlinear boundary value conditions must be bounded,

()the modified nonlinear boundary value conditions coincide with the nonmodified ones in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq95_HTML.gif .

So, to deduce existence results for this new situation, it is necessary to make use of the qualitative properties of continuity and monotonicity of the functions that define the nonlinear boundary value conditions.

Perhaps the first work that considers nonlinear boundary value conditions coupled with lower and upper solutions is due to Bebernes and Fraker [40] in 1971. In this work, the equation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq96_HTML.gif coupled with the boundary conditions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq97_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq98_HTML.gif is considered. Here, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq99_HTML.gif is compact and connected and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq100_HTML.gif is closed and connected. Under some additional conditions on the two sets, that include as a particular case http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq101_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq102_HTML.gif , the existence result is deduced under the assumption that a pair of well-ordered lower and upper solutions exist and a Nagumo condition is satisfied.

Later Bernfeld and Lakshmikantham [6] studied the boundary conditions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq103_HTML.gif ; with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq104_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq105_HTML.gif monotone nonincreasing in the second variable.

Erbe considers in [41] the three types of boundary value conditions:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ28_HTML.gif
(2.4)

Here functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq106_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq107_HTML.gif satisfy suitable monotonicity conditions. Such monotonicity properties include, as particular cases, the periodic problem in the first situation and the separated conditions in the second and third cases.

The proofs follow from the study of the Dirichlet problem (1.1) with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq108_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq109_HTML.gif . From the monotonicity assumptions it is proved, by a similar argument to the shooting method, that there is at least a pair http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq110_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq111_HTML.gif for which the boundary conditions hold.

Mawhin studies in [11] the nonlinear separated boundary conditions
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ29_HTML.gif
(2.5)

with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq112_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq113_HTML.gif two nondecreasing functions in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq114_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq115_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq116_HTML.gif .

In this case, he constructs the modified problem
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ30_HTML.gif
(2.6)

with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq117_HTML.gif defined in (1.12) and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq118_HTML.gif in (1.15).

Virzhbitskiĭ and Sadyrbaev consider in [42] the conditions
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ31_HTML.gif
(2.7)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq119_HTML.gif is a continuously parametrized curve in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq120_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq121_HTML.gif is a continuous function. The proof is based on reducing the problem to another one with divided boundary conditions and applying the Bol'-Brauer theorem.

Fabry and Habets treat in [26] the two types of boundary value conditions
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ32_HTML.gif
(2.8)
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ33_HTML.gif
(2.9)

with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq122_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq123_HTML.gif monotone functions in some of their variables.

The first case covers the periodic case and the second one separated boundary conditions.

In the proofs, a more general definition of lower and upper solutions is used. In particular, they replace the definitions (1.2) and (1.4) by the following ones.

http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq124_HTML.gif are continuous functions with right Dini derivatives http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq125_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq126_HTML.gif continuous from the right and left Dini derivatives http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq127_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq128_HTML.gif such that

()for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq130_HTML.gif it is satisfied that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq131_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq132_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq133_HTML.gif ;

()the functions
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ34_HTML.gif
(2.10)

are nondecreasing in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq135_HTML.gif .

It is clear that if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq136_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq137_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq138_HTML.gif -functions, this definition reduces to (1.2) and (1.4). Moreover, they assume a more general condition than the Nagumo one.

To deduce existence results for (2.8) they consider a variant of the truncated problem (1.14) (by adding the term tanh http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq139_HTML.gif ) coupled with the following nonconstant Dirichlet boundary conditions:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ35_HTML.gif
(2.11)

with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq140_HTML.gif defined in (1.15).

When the conditions (2.9) are studied, the authors consider
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ36_HTML.gif
(2.12)

The proofs follow from oscillation theory and boundedness of the boundary conditions.

In [43], by using degree theory, Rachůnková proves the existence of at least two different solutions with boundary conditions
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ37_HTML.gif
(2.13)

Here, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq141_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq142_HTML.gif satisfy some suitable monotonicity conditions that cover as a particular case the separated ones.

In all of the previous works, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq143_HTML.gif is considered a continuous function.

For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq144_HTML.gif being a Carathéodory function Sadyrbaev studies in [44, 45] the first-order system http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq145_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq146_HTML.gif , coupled with boundary value conditions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq147_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq148_HTML.gif some suitable sets.

Lepin et al. generalize in [31, 34, 35] some of the results proved by Erbe in [41].

Adje generalizes in [46] the results obtained by Fabry and Habets in [26] for problem (2.8) and proves the existence of solution by considering the boundary value conditions
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ38_HTML.gif
(2.14)

and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq149_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq150_HTML.gif -Carathéodory function.

Franco and O'Regan, by avoiding some monotonicity assumptions on the boundary data, introduce in [47] a new definition of coupled lower and upper solutions for the boundary value conditions (2.9). In this case, the definition of such functions concerns both of the functions together. Under this definition they cover, under the same notation, periodic, antiperiodic, and Dirichlet boundary value conditions. Moreover they introduce a new concept of Nagumo condition as follows.

Definition 2.1.

One says that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq151_HTML.gif satisfies a Nagumo condition relative to the interval http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq152_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq153_HTML.gif is a lower solution and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq154_HTML.gif is an upper solution if for
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ39_HTML.gif
(2.15)
there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq155_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ40_HTML.gif
(2.16)
and a continuous function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq156_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ41_HTML.gif
(2.17)

3. http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq157_HTML.gif -Laplacian Problems and Functional Boundary Conditions

A more general framework of the second-order general equation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq158_HTML.gif is given by the so-called http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq159_HTML.gif -laplacian equation. This kind of problems follow the expression
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ42_HTML.gif
(3.1)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq160_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq161_HTML.gif . This type of equations appears in the study of nonNewtonian fluid mechanics [48, 49].

As far as the author is aware, the first reference in which this problem has been studied in combination with the method of lower and upper solutions is due to De Coster in [50], who considers (3.1) (without dependence on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq162_HTML.gif ) coupled with Dirichlet conditions. Moreover, she treat, a more general operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq163_HTML.gif that includes, as a particular case, the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq164_HTML.gif -laplacian operator. To be concise, operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq165_HTML.gif conserves the two main qualitative properties of operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq166_HTML.gif :

() http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq168_HTML.gif is a strictly increasing homeomorphism from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq169_HTML.gif onto http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq170_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq171_HTML.gif ;

() http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq173_HTML.gif .

As consequence, after this work authors considered the http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq174_HTML.gif -laplacian equation
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ43_HTML.gif
(3.2)

with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq175_HTML.gif an operator that satisfies the above mentioned properties.

After this paper, the method of lower and upper solutions has been applied to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq176_HTML.gif -laplacian problems with Mixed boundary conditions in [51] and for Neumann and periodic boundary conditions in [52].

In this case, the definition of a lower and an upper solution, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq177_HTML.gif being a Carathéodory function, is the direct translation to this case for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq178_HTML.gif the identity.

Definition 3.1.

A function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq179_HTML.gif is said to be a lower solution for (3.2) if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq180_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ44_HTML.gif
(3.3)
A function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq181_HTML.gif is an upper solution for (3.2) if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq182_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ45_HTML.gif
(3.4)

We will say that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq183_HTML.gif is a solution of (3.2) if it is both a lower and an upper solution.

Of course, some additional assumptions are needed depending on the considered boundary conditions, that is, we assume (1.3) and (1.5) for the Dirichlet case, (1.7) and (1.8) for the Neumann case, or (1.10) for the periodic case.

The definition of a Nagumo condition, see [52], is the direct adaptation of the one used by Adje in [46]. Note that such condition does not depend on the boundary data of the problem.

Definition 3.2.

One says that the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq184_HTML.gif satisfies a Nagumo condition with respect to continuous functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq185_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq186_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq187_HTML.gif , if there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq188_HTML.gif , and a continuous function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq189_HTML.gif , such that
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ46_HTML.gif
(3.5)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq190_HTML.gif .

Furthermore
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ47_HTML.gif
(3.6)
where
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ48_HTML.gif
(3.7)

If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq191_HTML.gif , we replace http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq192_HTML.gif .

It is obvious that, to construct an operator whose fixed points coincide with the solutions of the considered problem, there is no possibility of constructing a Green's function, so no operator analogous to (2.1) can be given. In this case, one can see [52] that the solutions of (3.2) coupled with Dirichlet boundary conditions are the fixed points of the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq193_HTML.gif , defined as
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ49_HTML.gif
(3.8)
where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq194_HTML.gif is the unique solution of the equation
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ50_HTML.gif
(3.9)

In the previous mentioned papers, the existence of solutions lying between a pair of well-ordered lower and upper solutions was shown. In [37], the existence of extremal solutions for the Dirichlet problem is proved. As we have noted earlier, in that paper, a new definition of lower and upper solutions with corners is used that allows one to construct a sequence of upper solutions over the function minimum of the solutions in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq195_HTML.gif .

Two point nonlinear boundary value conditions have been treated in [53]. In this case http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq196_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq197_HTML.gif have been studied. The monotonicity assumptions on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq198_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq199_HTML.gif cover the periodic conditions. The proof follows from a truncated problem on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq200_HTML.gif as in the nonlinear part of (1.19) and a truncated boundary conditions as in (2.11).

Lepin et al. study in [54] the more general equation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq201_HTML.gif together with the two point nonlinear boundary conditions
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ51_HTML.gif
(3.10)

The existence results are on the basis of the monotone properties of functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq202_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq203_HTML.gif .

Functional boundary conditions allow us to consider dependence on some intermediate points of the interval of definition. This is the case of the multipoint boundary conditions:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ52_HTML.gif
(3.11)

where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq204_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq205_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq206_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq207_HTML.gif have the same sign for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq208_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq209_HTML.gif .

Many other type of boundary conditions can be treated, is the case, for instance, of the following:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ53_HTML.gif
(3.12)
In this situation, it is necessary to consider functions that are not only defined at the extremes of the interval, but also in its interior. Such kind of problems have been treated in [55]. There the authors consider nonlinear functional boundary conditions of the form
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ54_HTML.gif
(3.13)

with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq210_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq211_HTML.gif a continuous functions that satisfy certain monotonicity conditions which include, as particular cases, the periodic ones.

More precisely, it is considered the equation
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ55_HTML.gif
(3.14)

with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq212_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq213_HTML.gif locally bounded (possibly discontinuous) function and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq214_HTML.gif measurable.

The discontinuity on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq215_HTML.gif can be eliminated by the use of the transformation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq216_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq217_HTML.gif is given by
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ56_HTML.gif
(3.15)
In this case, the studied problem is translated to the usual http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq218_HTML.gif -laplacian
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ57_HTML.gif
(3.16)

and the existence of extremal solutions lying between a pair of well-ordered lower and upper solutions is obtained. The results follows from an appropriate truncated problem and the extension to this case of the arguments used in [37] to get the extremal solutions.

4. General Functional Equations

In this last section, we mention some kind of problems that model different real phenomena that, as we will see, can be presented under the same formulation.

()We consider the classical self-adjoint equation
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ58_HTML.gif
(4.1)

with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq220_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq221_HTML.gif .

We can formulate the previous equation in the form
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ59_HTML.gif
(4.2)
with
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ60_HTML.gif
(4.3)
()We refer to the usual diffusion equation
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ61_HTML.gif
(4.4)

with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq223_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq224_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq225_HTML.gif .

By defining
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ62_HTML.gif
(4.5)
it is possible to rewrite this equation as
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ63_HTML.gif
(4.6)
()We study a higher-order differential equation, for instance, the following third-order problem:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ64_HTML.gif
(4.7)
By means of the change of variables
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ65_HTML.gif
(4.8)
we arrive at the following equivalent second-order Dirichlet functional equation:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ66_HTML.gif
(4.9)
()Consider the following third-order problem:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ67_HTML.gif
(4.10)
Using the same change of variable as above, we arrive at the following second-order differential equation with functional boundary conditions:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ68_HTML.gif
(4.11)
In order to include under the same formulation all the previous problems, the following equation is considered in [56]:
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ69_HTML.gif
(4.12)

coupled with the nonlinear functional boundary conditions (3.13).

The definitions of the lower and upper solutions cover all the usual cases but the Nagumo condition does not generalize the case of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_IEq228_HTML.gif depending only on the first derivative given in Definition 3.2. This gap has been covered in the definition given in [57], where (4.12) is considered coupled with the boundary conditions
http://static-content.springer.com/image/art%3A10.1155%2F2011%2F893753/MediaObjects/13661_2010_Article_65_Equ70_HTML.gif
(4.13)

covering in this case the Sturm-Liouville and the multipoint boundary conditions as particular cases.

5. Final Remarks

It is important to note that in some of the previous results some kind of discontinuities on the spatial variable are assumed. In this case, some techniques developed by Heikkilä and Lakshmikantham in [58] are used.

There is large bibliography on papers related with lower and upper solutions with nonlinear boundary value conditions for first- and higher-order equations.

Problems with impulses, difference equations, and partial differential equations have been studied under this point of view for an important number of researchers.

Some theories as the Thompson's notion of compatibility [59] or the Frigon's tube-solutions [60] give some generalizations of the concept of lower and upper solutions that ensure the existence of solutions of nonlinear boundary problems under weaker assumptions.

Declarations

Acknowledgments

The author is grateful for professors Jean Mawhin and Felix Ž. Sadyrbaev, their interesting comments have been of great importance in the development of this work. He is also grateful for the anonymous referees whose useful remarks have made the paper more readable. This paper was partially supported by Ministerio de Educación y Ciencia, Spain, Project MTM2007-61724.

Authors’ Affiliations

(1)
Departamento de Análise Matemática, Facultade de Matemáticas, Universidade de Santiago de Compostela

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© Alberto Cabada. 2011

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