An Overview of the Lower and Upper Solutions Method with Nonlinear Boundary Value Conditions
© Alberto Cabada. 2011
Received: 19 April 2010
Accepted: 7 July 2010
Published: 25 July 2010
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© Alberto Cabada. 2011
Received: 19 April 2010
Accepted: 7 July 2010
Published: 25 July 2010
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.
The first steps in the theory of lower and upper solutions have been given by Picard in 1890  for Partial Differential Equations and, three years after, in  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 . In 1927, Müller extended Perron's results to initial value systems in .
for a continuous function and .
When on , the existence of a solution of the considered problem lying between and 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 the classical books of Bernfeld and Lakshmikantham  and Ladde et al.  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 [8–11] and the surveys in this field of De Coster and Habets [12–14] 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  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.
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 .
Notice that both and are continuous and bounded functions and, in consequence, if is continuous, both properties are satisfied by .
In the proof it is deduced that all the solutions of the truncated problem (1.14) belong to the segment and on . Notice that the constant only depends on , and . 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  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 .
In 1967, Kiguradze  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 .
where and are sufficiently small numbers.
coupled with the corresponding boundary value conditions.
This truncation has been introduced by Gao and Wang in  for the periodic problem and improves a previous one given by Wang et al. in . Notice that the function is bounded in and it is measurable because of the following result proved in [28, Lemma ].
For any , the two following properties hold:
(a) exists for a.e. ;
(b)If and , then for a.e. .
is assumed on function for some and all , it is possible to deduce the existence of extremal solutions in the sector of the considered problem. By extremal solutions we mean the greatest and the smallest solutions in the set of all the solutions in . The deduction of such a property holds from an iterative technique that consists of solving related linear problems on and using suitable maximum principles which are equivalent to the constant sign of the associated Green's function. One can find in  a complete development of this theory for different kinds of boundary value conditions.
are nondecreasing in .
Cherpion et al. prove in  the existence of extremal solutions for the Dirichlet problem without assuming the condition (1.20). In fact, they consider a more general problem: the -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.
Here , , , and 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 as the minimum value attained by all the solutions of problem (1.1) in 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.
Two point nonlinear boundary conditions are considered with the aim of covering more complicated situations as, for instance, or .
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 .
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  in 1971. In this work, the equation coupled with the boundary conditions and is considered. Here, is compact and connected and is closed and connected. Under some additional conditions on the two sets, that include as a particular case ; , 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  studied the boundary conditions ; with and monotone nonincreasing in the second variable.
Here functions and 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 and . From the monotonicity assumptions it is proved, by a similar argument to the shooting method, that there is at least a pair , for which the boundary conditions hold.
with and two nondecreasing functions in for all and .
with defined in (1.12) and in (1.15).
where is a continuously parametrized curve in and 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.
with , and 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.
are continuous functions with right Dini derivatives and continuous from the right and left Dini derivatives and such that
()for all it is satisfied that , and ;
are nondecreasing in .
It is clear that if and are -functions, this definition reduces to (1.2) and (1.4). Moreover, they assume a more general condition than the Nagumo one.
with defined in (1.15).
The proofs follow from oscillation theory and boundedness of the boundary conditions.
Here, and satisfy some suitable monotonicity conditions that cover as a particular case the separated ones.
In all of the previous works, is considered a continuous function.
and is a -Carathéodory function.
Franco and O'Regan, by avoiding some monotonicity assumptions on the boundary data, introduce in  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.
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 , who considers (3.1) (without dependence on ) coupled with Dirichlet conditions. Moreover, she treat, a more general operator that includes, as a particular case, the -laplacian operator. To be concise, operator conserves the two main qualitative properties of operator :
() is a strictly increasing homeomorphism from onto , such that ;
with an operator that satisfies the above mentioned properties.
In this case, the definition of a lower and an upper solution, for being a Carathéodory function, is the direct translation to this case for the identity.
We will say that 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.
If , we replace .
In the previous mentioned papers, the existence of solutions lying between a pair of well-ordered lower and upper solutions was shown. In , 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 .
Two point nonlinear boundary value conditions have been treated in . In this case and have been studied. The monotonicity assumptions on and cover the periodic conditions. The proof follows from a truncated problem on as in the nonlinear part of (1.19) and a truncated boundary conditions as in (2.11).
The existence results are on the basis of the monotone properties of functions and .
where , , and , have the same sign for all and .
with and a continuous functions that satisfy certain monotonicity conditions which include, as particular cases, the periodic ones.
with and locally bounded (possibly discontinuous) function and measurable.
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  to get the extremal solutions.
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.
with such that .
with such that and .
coupled with the nonlinear functional boundary conditions (3.13).
covering in this case the Sturm-Liouville and the multipoint boundary conditions as particular cases.
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  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  or the Frigon's tube-solutions  give some generalizations of the concept of lower and upper solutions that ensure the existence of solutions of nonlinear boundary problems under weaker assumptions.
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.
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