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

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

In the same way, an upper solution is a function β ∈ C 2 I that satisfies the reversed inequalities β t ≤ f t, β t , β t , When α ≤ β on I, the existence of a solution of the considered problem lying between α and β 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 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  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 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.
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, The usual tool to derive an existence result consists in the construction of a modified problem that satisfies the two following properties.
1 The nonlinear part of the modified equation is bounded.
2 The nonlinear part of the modified equation coincides with the nonlinear part when the spatial variable is in α, β .
When the Dirichlet problem 1.1 is studied, the usual truncated problem considered is Notice that both p and q are continuous and bounded functions and, in consequence, if f is continuous, both properties are satisfied by g.
In the proof it is deduced that all the solutions u of the truncated problem 1.14 belong to the segment α, β and |u | ≤ K on I. Notice that the constant K only depends on α, β and h. 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 A B 0, the following one-sided growth condition is assumed: where ψ ∈ L 1 I and σ 1 , σ 2 ≥ 0 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 f 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 I. 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: Boundary Value Problems 5 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 F is bounded in L 1 I and it is measurable because of the following result proved in 28, Lemma 2 . Lemma 1.2. For any u ∈ C 1 I , the two following properties hold: When a one-sided Lipschitz condition of the following type: 1.20 is assumed on function f for some M > 0 and all t ∈ I and z ∈ R, it is possible to deduce the existence of extremal solutions in the sector α, β of the considered problem. By extremal solutions we mean the greatest and the smallest solutions in the set of all the solutions in α, β . The deduction of such a property holds from an iterative technique that consists of solving related linear problems on u 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 C 1 functions α ≤ β that satisfy 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 α, β : I → R with right Dini derivatives D r α, absolute semicontinuous from below in I, and D r β absolute semicontinuous from above in I, that satisfy the following inequalities a.e. t ∈ I:

1.22
For further works in this direction see 31-36 . 6 Boundary Value Problems 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 ϕ-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. 1.23 Here D , D , D − , and D − 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 t ∈ I as the minimum value attained by all the solutions of problem 1.1 in α, β 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 .

Two-Point Nonlinear Boundary Value Conditions
Two point nonlinear boundary conditions are considered with the aim of covering more complicated situations as, for instance, u b u 3 a or u b − u a arctan u a u b . 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 T :

2.2
Boundary Value Problems 7 is the Green's function related to the linear problem u t 0, ∀t ∈ I, u a u b 0.

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, 1 the modified nonlinear boundary value conditions must be bounded, 2 the modified nonlinear boundary value conditions coincide with the nonmodified ones in α, β .
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 u f t, u, u coupled with the boundary conditions 0, u 0 , u 0 ∈ S 1 and 1, u 1 , u 1 ∈ S 2 is considered. Here, S 1 is compact and connected and S 2 is closed and connected. Under some additional conditions on the two sets, that include as a particular case u 0 0; u 1 −L 1 u 1 , 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 g u a , u a 0 z u a , u a ; with g and z monotone nonincreasing in the second variable.
Erbe considers in 41 the three types of boundary value conditions: x u a , u a 0 y u a , u b , u a , u b , r u b , u b 0 w u a , u b , u a , u b .

2.4
Here functions g, z, x, y, r and w 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 A ∈ α a , β a and B ∈ α b , β b . From the monotonicity assumptions it is proved, by a similar argument to the shooting method, that there is at least a pair A, B for which the boundary conditions hold.
Mawhin studies in 11 the nonlinear separated boundary conditions g a u a , u a g b u b , u b 0, 2.5 with g a x, · and g b y, · two nondecreasing functions in R for all x ∈ α a , β a and y ∈ α b , β b .

Boundary Value Problems
In this case, he constructs the modified problem

u t f t, p t, u t , u t h u t u t − p t, u t ,
u a g a p a, u a , u a p a, u a , with g, z, w, and r 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. It is clear that if α and β are C 2 -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 u t − p t, u t coupled with the following nonconstant  , u a w u a , u a , u b , u b ,   u b  p b, u a r u a , u a , u b , u b .

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 Here, g 1 and g 2 satisfy some suitable monotonicity conditions that cover as a particular case the separated ones.
In all of the previous works, f is considered a continuous function. 2.17

ϕ-Laplacian Problems and Functional Boundary Conditions
A more general framework of the second-order general equation u t f t, u t , u t is given by the so-called p-laplacian equation. This kind of problems follow the expression where ϕ p x x|x| p−2 for some p > 1. 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 u coupled with Dirichlet conditions. Moreover, she treat, a more general operator ϕ that includes, as a particular case, the p-laplacian operator. To be concise, operator ϕ conserves the two main qualitative properties of operator ϕ p : 1 ϕ is a strictly increasing homeomorphism from R onto R, such that ϕ R R; 2 ϕ 0 0.
As consequence, after this work authors considered the ϕ-laplacian equation with ϕ an operator that satisfies the above mentioned properties. After this paper, the method of lower and upper solutions has been applied to ϕlaplacian 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 f being a Carathéodory function, is the direct translation to this case for ϕ the identity. Definition 3.1. A function α ∈ C 1 I is said to be a lower solution for 3.2 if ϕ α ∈ W 1,1 I and ϕ α t ≥ f t, α t , α t , for a.e. t ∈ I.

3.3
A function β ∈ C 1 I is an upper solution for 3.2 if ϕ β ∈ W 1,1 I and ϕ β t ≤ f t, β t , β t , for a.e. t ∈ I. 3.4 We will say that u ∈ C 1 I 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 f satisfies a Nagumo condition with respect to continuous functions α and β, with α ≤ β, if there exist k ∈ L p I , 1 ≤ p ≤ ∞, and a continuous function θ : 0, ∞ → 0, ∞ , such that
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 where a < τ 1 < τ 2 < · · · < τ m < b, a < η 1 < η 2 < · · · < η n < b, and a i , b j ∈ R have the same sign for all i ∈ {1, . . . , m} and j ∈ {1, . . . , n}. Many other type of boundary conditions can be treated, is the case, for instance, of the following:

3.12
Boundary Value Problems

13
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   L 1 u a , u b , u a , u b , u 0 L 2 u a , u b , 3.13 with L 1 and L 2 a continuous functions that satisfy certain monotonicity conditions which include, as particular cases, the periodic ones. More precisely, it is considered the equation ϕ u t r u t f t, u t , u t , for a.e. t ∈ I, 3.14 with r : R → 0, ∞ and 1/r locally bounded possibly discontinuous function and r • ϕ −1 measurable. The discontinuity on u can be eliminated by the use of the transformation ϕ Q • ϕ, where Q is given by In this case, the studied problem is translated to the usual ϕ-laplacian ϕ u f t, u, u , for a.e. t ∈ I, 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.

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.

4.10
Using the same change of variable as above, we arrive at the following second-order differential equation with functional boundary conditions:

4.11
In order to include under the same formulation all the previous problems, the following equation is considered in 56 :

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 tubesolutions 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.