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

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

(1.1)

for a continuous function and .

The most usual form to define a lower solution is to consider a function that satisfies the inequality

(1.2)

together with

(1.3)

In the same way, an upper solution is a function that satisfies the reversed inequalities

(1.4)
(1.5)

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

(1.6)

a lower solution is a -function that satisfies (1.2) coupled with the inequalities

(1.7)

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

(1.8)

Analogously, for the periodic problem

(1.9)

a lower solution and an upper solution are -functions that satisfy (1.2) and (1.4), respectively, together with the inequalities

(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 satisfies the Nagumo condition if there is satisfying

(1.11)
(1.12)

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,

(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 .

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

(1.14)

Here

(1.15)
(1.16)

with

(1.17)

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:

(1.18)

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:

(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 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:

(1.20)

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

(1.21)

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. :

(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 -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. ,

(1.23)

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. ,

(1.24)

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]).

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, or .

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 , defined by

(2.1)

where

(2.2)

is the Green's function related to the linear problem

(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 .

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 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 [6] studied the boundary conditions ; with and monotone nonincreasing in the second variable.

Erbe considers in [41] the three types of boundary value conditions:

(2.4)

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.

Mawhin studies in [11] the nonlinear separated boundary conditions

(2.5)

with and two nondecreasing functions in for all and .

In this case, he constructs the modified problem

(2.6)

with defined in (1.12) and in (1.15).

Virzhbitskiĭ and Sadyrbaev consider in [42] the conditions

(2.7)

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.

Fabry and Habets treat in [26] the two types of boundary value conditions

(2.8)
(2.9)

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 ;

()the functions

(2.10)

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.

To deduce existence results for (2.8) they consider a variant of the truncated problem (1.14) (by adding the term tanh) coupled with the following nonconstant Dirichlet boundary conditions:

(2.11)

with defined in (1.15).

When the conditions (2.9) are studied, the authors consider

(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

(2.13)

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.

For being a Carathéodory function Sadyrbaev studies in [44, 45] the first-order system , , coupled with boundary value conditions with 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

(2.14)

and is a -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 satisfies a Nagumo condition relative to the interval , where is a lower solution and is an upper solution if for

(2.15)

there exists a constant such that

(2.16)

and a continuous function such that

(2.17)

3. -Laplacian Problems and Functional Boundary Conditions

A more general framework of the second-order general equation is given by the so-called -laplacian equation. This kind of problems follow the expression

(3.1)

where for some . 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 ) 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 ;

().

As consequence, after this work authors considered the -laplacian equation

(3.2)

with an operator that satisfies the above mentioned properties.

After this paper, the method of lower and upper solutions has been applied to -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 being a Carathéodory function, is the direct translation to this case for the identity.

Definition 3.1.

A function is said to be a lower solution for (3.2) if and

(3.3)

A function is an upper solution for (3.2) if and

(3.4)

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.

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 satisfies a Nagumo condition with respect to continuous functions and , with , if there exist , and a continuous function , such that

(3.5)

where .

Furthermore

(3.6)

where

(3.7)

If , we replace .

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 , defined as

(3.8)

where is the unique solution of the equation

(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 .

Two point nonlinear boundary value conditions have been treated in [53]. 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).

Lepin et al. study in [54] the more general equation together with the two point nonlinear boundary conditions

(3.10)

The existence results are on the basis of the monotone properties of functions and .

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:

(3.11)

where , , and , have the same sign for all and .

Many other type of boundary conditions can be treated, is the case, for instance, of the following:

(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

(3.13)

with and a continuous functions that satisfy certain monotonicity conditions which include, as particular cases, the periodic ones.

More precisely, it is considered the equation

(3.14)

with and locally bounded (possibly discontinuous) function and measurable.

The discontinuity on can be eliminated by the use of the transformation , where is given by

(3.15)

In this case, the studied problem is translated to the usual -laplacian

(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

(4.1)

with such that .

We can formulate the previous equation in the form

(4.2)

with

(4.3)

()We refer to the usual diffusion equation

(4.4)

with such that and .

By defining

(4.5)

it is possible to rewrite this equation as

(4.6)

()We study a higher-order differential equation, for instance, the following third-order problem:

(4.7)

By means of the change of variables

(4.8)

we arrive at the following equivalent second-order Dirichlet functional equation:

(4.9)

()Consider the following third-order problem:

(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]:

(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 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

(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.

References

  1. Picard E: Mémoire sur la théorie des équations aux derivés partielles et las méthode des approximations succesives. Journal de Mathématiques Pures et Appliquées 1890, 6: 145–210.

    Google Scholar 

  2. Picard E: Sur l'application des métodes d'approximations succesives à l'étude de certains équations différentielles ordinaires. Journal de Mathématiques Pures et Appliquées 1893, 9: 217–271.

    Google Scholar 

  3. Perron O: Ein neuer existenzbeweis für die integrale der differentialgleichung . Mathematische Annalen 1915, 76(4):471–484. 10.1007/BF01458218

    Article  MathSciNet  Google Scholar 

  4. Müller M: Über das fundamentaltheorem in der theorie der gewöhnlichen differentialgleichungen. Mathematische Zeitschrift 1927, 26(1):619–645. 10.1007/BF01475477

    Article  MathSciNet  Google Scholar 

  5. Dragoni GS: II problema dei valori ai limiti studiato in grande per le equazioni differenziali del secondo ordine. Mathematische Annalen 1931, 105(1):133–143. 10.1007/BF01455811

    Article  MathSciNet  Google Scholar 

  6. Bernfeld SR, Lakshmikantham V: An Introduction to Nonlinear Boundary Value Problems, Mathematics in Science and Engineering, Vol. 10. Academic Press, New York, NY, USA; 1974:xi+386.

    Google Scholar 

  7. Ladde GS, Lakshmikantham V, Vatsala AS: Monotone Iterative Techniques for Nonlinear Differential Equations. Pitman, Boston, Mass, USA; 1985.

    Google Scholar 

  8. Mawhin J: Bounded solutions of nonlinear ordinary differential equations. In Non-Linear Analysis and Boundary Value Problems for Ordinary Differential Equations (Udine), CISM Courses and Lectures. Volume 371. Springer, Vienna, Austria; 1996:121–147.

    Chapter  Google Scholar 

  9. Mawhin J: Boundary value problems for nonlinear ordinary differential equations: from successive approximations to topology. In Development of Mathematics 1900–1950 (Luxembourg, 1992). Birkhäuser, Basel, Switzerland; 1994:443–477.

    Chapter  Google Scholar 

  10. Mawhin J: Twenty years of ordinary differential equations through twelve Oberwolfach meetings. Results in Mathematics 1992, 21(1–2):165–189.

    Article  MathSciNet  Google Scholar 

  11. Mawhin J: Points fixes, points critiques et problèmes aux limites, Séminaire de Mathématiques Supérieures. Volume 92. Presses de l'Université de Montréal, Montreal, Canada; 1985:162.

    Google Scholar 

  12. De Coster C, Habets P: Two-Point Boundary Value Problems: Lower and Upper Solutions, Mathematics in Science and Engineering. Volume 205. Elsevier, Amsterdam, The Netherlands; 2006:xii+489.

    Google Scholar 

  13. De Coster C, Habets P: The lower and upper solutions method for boundary value problems. In Handbook of Differential Equations. Elsevier/North-Holland, Amsterdam, The Netherlands; 2004:69–160.

    Google Scholar 

  14. De Coster C, Habets P: An overview of the method of lower and upper solutions for ODEs. In Nonlinear Analysis and Its Applications to Differential Equations, Progr. Nonlinear Differential Equations Appl.. Volume 43. Birkhäuser, Boston, Mass, USA; 2001:3–22.

    Chapter  Google Scholar 

  15. Nagumo M: Uber die differentialgleichung . Proceedings of the Physico-Mathematical Society of Japan 1937, 19: 861–866.

    Google Scholar 

  16. Nagumo M: On principally linear elliptic differential equations of the second order. Osaka Journal of Mathematics 1954, 6: 207–229.

    MathSciNet  Google Scholar 

  17. Habets P, Pouso RL: Examples of the nonexistence of a solution in the presence of upper and lower solutions. The ANZIAM Journal 2003, 44(4):591–594. 10.1017/S1446181100012955

    Article  MathSciNet  Google Scholar 

  18. Kiguradze IT: A priori estimates for the derivatives of bounded functions satisfying second-order differential inequalities. Differentsial'nye Uravneniya 1967, 3: 1043–1052.

    MathSciNet  Google Scholar 

  19. Schrader KW: Solutions of second order ordinary differential equations. Journal of Differential Equations 1968, 4: 510–518. 10.1016/0022-0396(68)90002-8

    Article  MathSciNet  Google Scholar 

  20. Tonelli L: Sull'equazione differenziale . Annali della Scuola Normale Superiore di Pisa 1939, 8(1):75–88.

    MathSciNet  Google Scholar 

  21. Epheser H: Über die Existenz der Lösungen von Randwertaufgaben mit gewöhnlichen, nichtlinearen Differentialgleichungen zweiter Ordnung. Mathematische Zeitschrift 1955, 61: 435–454.

    Article  MathSciNet  Google Scholar 

  22. Krasnoselskii M: On some boundary value problems. Izvestiya Academy of Sciences USSR 1956, 20: 241–252.

    MathSciNet  Google Scholar 

  23. Kiguradze IT: Some Singular Boundary Value Problems for Ordinary Differential Equations. Izdat. Tbilis. Univ., Tbilisi, Russia; 1975.

    Google Scholar 

  24. Kiguradze IT: Some singular boundary value problems for second order nonlinear ordinary differential equations. Differentsial'nye Uravneniya 1968, 4: 1753–1773.

    MathSciNet  Google Scholar 

  25. Mawhin J: The Bernstein-Nagumo problem and two-point boundary value problems for ordinary differential equations. In Qualitative Theory of Differential Equations, Vol. I, II (Szeged, 1979), Colloq. Math. Soc. János Bolyai. Volume 30. North-Holland, Amsterdam, The Netherlands; 1981:709–740.

    Google Scholar 

  26. Fabry C, Habets P: Upper and lower solutions for second-order boundary value problems with nonlinear boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 1986, 10(10):985–1007. 10.1016/0362-546X(86)90084-2

    Article  MathSciNet  Google Scholar 

  27. Gao WJ, Wang JY: On a nonlinear second order periodic boundary value problem with Carathéodory functions. Annales Polonici Mathematici 1995, 62(3):283–291.

    MathSciNet  Google Scholar 

  28. Wang MX, Cabada A, Nieto JJ: Monotone method for nonlinear second order periodic boundary value problems with Carathéodory functions. Annales Polonici Mathematici 1993, 58(3):221–235.

    MathSciNet  Google Scholar 

  29. Scorza Dragoni S: Elementi uniti di transformazioni funzionali e problemi di valori ai limiti Rend. Sem. Mat. Univ. Roma 1938, 2(4):255–275.

    Google Scholar 

  30. Ponomarev VD: The existence of the solution of the simplest boundary value problem for a second-order differential equation. Latviĭskiĭ Matematicheskiĭ Ezhegodnik 1978, (22):69–74.

    Google Scholar 

  31. Lepin AYa: Existence of solutions of boundary value problems for a second-order equation. Differential Equations 1990, 26(10):1279–1286.

    MathSciNet  Google Scholar 

  32. Lepin LA: On the concepts of lower and upper functions. Differential Equations 1980, 16(10):1750–1759.

    MathSciNet  Google Scholar 

  33. Lepin LA: Generalized lower and upper functions and their properties. Latviĭskiĭ Matematicheskiĭ Ezhegodnik 1980, (24):113–123.

    Google Scholar 

  34. Lepin LA: Generalized solutions and the solvability of boundary value problems for a second-order differential equation. Differential Equations 1982, 18(8):1323–1330.

    MathSciNet  Google Scholar 

  35. Lepin AYa, Lepin LA, Ponomarev VD: Generalized solvability of nonlinear boundary value problems. Differential Equations 1986, 22(2):198–203.

    MathSciNet  Google Scholar 

  36. Lepin AJa, Sadyrbaev FZh: The upper and lower functions method for second order systems. Zeitschrift für Analysis und ihre Anwendungen 2001, 20(3):739–753.

    Article  MathSciNet  Google Scholar 

  37. Cherpion M, De Coster C, Habets P: Monotone iterative methods for boundary value problems. Differential and Integral Equations 1999, 12(3):309–338.

    MathSciNet  Google Scholar 

  38. Cid JÁ: On extremal fixed points in Schauder's theorem with applications to differential equations. Bulletin of the Belgian Mathematical Society. Simon Stevin 2004, 11(1):15–20.

    MathSciNet  Google Scholar 

  39. Kelley JL: General Topology, Graduate Texts in Mathematics, No. 27. Springer, Berlin, Germany; 1975:xiv+298.

    Google Scholar 

  40. Bebernes JW, Fraker R: A priori bounds for boundary sets. Proceedings of the American Mathematical Society 1971, 29: 313–318. 10.1090/S0002-9939-1971-0277807-9

    Article  MathSciNet  Google Scholar 

  41. Erbe LH: Existence of solutions to boundary value problems for second order differential equations. Nonlinear Analysis: Theory, Methods & Applications 1982, 6(11):1155–1162. 10.1016/0362-546X(82)90027-X

    Article  MathSciNet  Google Scholar 

  42. Virzhbitskiĭ YaV, Sadyrbaev FZh: A two-point boundary value problem for a second-order ordinary differential equation. Latviĭskiĭ Matematicheskiĭ Ezhegodnik 1986, (30):39–42.

    Google Scholar 

  43. Rachůnková I: Upper and lower solutions and multiplicity results. Journal of Mathematical Analysis and Applications 2000, 246(2):446–464. 10.1006/jmaa.2000.6798

    Article  MathSciNet  Google Scholar 

  44. Sadyrbaev FZh: Level surfaces of Lyapunov functions and solvability of a two-point boundary value problem. Latviĭskiĭ Matematicheskiĭ Ezhegodnik 1980, (24):172–177.

    Google Scholar 

  45. Sadyrbaev FŽ: A two-point boundary value problem for a system of first-order ordinary differential equations. Latviĭskiĭ Matematicheskiĭ Ezhegodnik 1979, (23):131–136.

    Google Scholar 

  46. Assohoun A: Sur et sous-solutions généralisées et problèmes aux limites du second ordre. Bulletin de la Société Mathématique de Belgique. Série B 1990, 42(3):347–368.

    MathSciNet  Google Scholar 

  47. Franco D, O'Regan D: Existence of solutions to second order problems with nonlinear boundary conditions. Discrete and Continuous Dynamical Systems. Series A 2003, (supplement):273–280.

    Google Scholar 

  48. Esteban JR, Vázquez JL: On the equation of turbulent filtration in one-dimensional porous media. Nonlinear Analysis: Theory, Methods & Applications 1986, 10(11):1303–1325. 10.1016/0362-546X(86)90068-4

    Article  MathSciNet  Google Scholar 

  49. Herrero MA, Vázquez JL: On the propagation properties of a nonlinear degenerate parabolic equation. Communications in Partial Differential Equations 1982, 7(12):1381–1402. 10.1080/03605308208820255

    Article  MathSciNet  Google Scholar 

  50. De Coster C: Pairs of positive solutions for the one-dimensional -Laplacian. Nonlinear Analysis: Theory, Methods & Applications 1994, 23(5):669–681. 10.1016/0362-546X(94)90245-3

    Article  MathSciNet  Google Scholar 

  51. Wang J, Gao W: Existence of solutions to boundary value problems for a nonlinear second order equation with weak Carathéodory functions. Differential Equations and Dynamical Systems 1997, 5(2):175–185.

    MathSciNet  Google Scholar 

  52. Cabada A, Pouso RL: Existence result for the problem with periodic and Neumann boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 1997, 30(3):1733–1742. 10.1016/S0362-546X(97)00249-6

    Article  MathSciNet  Google Scholar 

  53. Cabada A, Pouso RL: Existence results for the problem with nonlinear boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 1999, 35(2):221–231. 10.1016/S0362-546X(98)00009-1

    Article  MathSciNet  Google Scholar 

  54. Lepin AJa, Lepin LA, Sadyrbaev FZh: Two-point boundary value problems with monotonically boundary conditions for one-dimensional φ -Laplacian equations. Functional Differential Equations 2005, 12(3–4):347–363.

    MathSciNet  Google Scholar 

  55. Cabada A, Pouso RL: Extremal solutions of strongly nonlinear discontinuous second-order equations with nonlinear functional boundary conditions. Nonlinear Analysis: Theory, Methods & Applications 2000, 42(8):1377–1396. 10.1016/S0362-546X(99)00158-3

    Article  MathSciNet  Google Scholar 

  56. Cabada A, Pouso RL: Existence theory for functional -Laplacian equations with variable exponents. Nonlinear Analysis: Theory, Methods & Applications 2003, 52(2):557–572. 10.1016/S0362-546X(02)00122-0

    Article  MathSciNet  Google Scholar 

  57. Cabada A, O'Regan D, Pouso RL: Second order problems with functional conditions including Sturm-Liouville and multipoint conditions. Mathematische Nachrichten 2008, 281(9):1254–1263. 10.1002/mana.200510675

    Article  MathSciNet  Google Scholar 

  58. Heikkilä S, Lakshmikantham V: Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics. Volume 181. Marcel Dekker, New York, NY, USA; 1994:xii+514.

    Google Scholar 

  59. Thompson HB, Tisdell CC: Nonlinear multipoint boundary value problems for weakly coupled systems. Bulletin of the Australian Mathematical Society 1999, 60(1):45–54. 10.1017/S0004972700033311

    Article  MathSciNet  Google Scholar 

  60. Frigon M, Montoki E: Systems of differential inclusions with maximal monotone terms. Journal of Mathematical Analysis and Applications 2006, 323(2):1134–1151. 10.1016/j.jmaa.2005.11.023

    Article  MathSciNet  Google Scholar 

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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.

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Cabada, A. An Overview of the Lower and Upper Solutions Method with Nonlinear Boundary Value Conditions. Bound Value Probl 2011, 893753 (2011). https://doi.org/10.1155/2011/893753

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