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

for
a continuous function and
.

The most usual form to define a lower solution is to consider a function

that satisfies the inequality

In the same way, an upper solution is a function

that satisfies the reversed inequalities

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 these situations, we have that for the Neumann problem

a lower solution

is a

-function that satisfies (1.2) coupled with the inequalities

is an upper solution of the Neumann problem if it satisfies (1.4) and

Analogously, for the periodic problem

a lower solution

and an upper solution

are

-functions that satisfy (1.2) and (1.4), respectively, together with the inequalities

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 [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 [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

satisfies the Nagumo condition if there is

satisfying

with
.

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.

()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
.

When the Dirichlet problem (1.1) is studied, the usual truncated problem considered is

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 [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

, the following one-sided growth condition is assumed:

where
and
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

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

. 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:

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
is bounded in
and it is measurable because of the following result proved in [28, Lemma
].

Lemma 1.2.

For any
, the two following properties hold:

(a)
exists for a.e.
;

(b)If
and
, then
for a.e.
.

When a one-sided Lipschitz condition of the following type:

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 [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

functions

that satisfy

are nondecreasing in
.

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

with right Dini derivatives

, absolute semicontinuous from below in

, and

absolute semicontinuous from above in

, that satisfy the following inequalities a.e.

:

For further works in this direction see [31–36].

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
-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

is a lower solution of problem (1.1) (with

), if

,

, and for any

, either

, or there exists an open interval

such that

,

and, for a.e.

,

Definition 1.4.

A function

is a lower solution of problem (1.1) (with

), if

,

, and for any

, either

, or there exists an open interval

such that

,

and, for a.e.

,

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.

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]).