The lower and upper solutions method for boundary value problems.

*(English)*Zbl 1269.34001
Cañada, A. (ed.) et al., Handbook of differential equations: Ordinary differential equations. Vol. I. Amsterdam: Elsevier/North Holland (ISBN 0-444-51128-8/hbk). Handbook of Differential Equations, 69-160 (2004).

This is a masterly written comprehensive introduction to the method of lower and upper solutions in connection with second order boundary value problems. More specifically, the authors choose the framework of Carathéodory differential equations together with either Dirichlet or periodic boundary conditions.

De Coster and Habets prove that they are not only proficient in the topic but also concerned about rigorous and pleasant writing in mathematics. The clarity of the exposition and accuracy, the many superb examples, and the fine balance between generality and amount of details make this paper suitable as a textbook. On the other hand, researchers will find in it a good reference for many recent results and proofs and an exhaustive bibliographical discussion. Moreover, some relevant historical notes about the development of the methods are included.

The paper is organized in four main sections, which we will review separately.

Section 1 – Well-ordered lower and upper solutions. The core of the method of lower and upper solutions is described in this part. As a starting point, the authors choose the periodic problem \[ u''=f(t,u),\quad u(a)=u(b),\;u'(a)=u'(b), \] with a locally \(L^1\)-bounded Carathéodory function \(f\). In this setting, a lower solution is defined as a function \(\alpha\in\mathcal C([a,b])\) such that \(\alpha(a)=\alpha(b)\) whose periodic extension to the whole real line is such that, for any real \(t_0\), either \(D_-\alpha(t_0)<D^+\alpha(t_0)\) or there exists an open interval \(I_0\) such that \(t_0\in I_0\), \(\alpha\in W^{2,1}(I_0)\) and, for a.e. \(t\in I_0\), \[ \alpha''(t)\geq f(t,\alpha(t)). \] Among the many definitions that one can find in the literature on the topic the previous one is satisfactorily general, allowing corners in the graphs of the lower solution, and requires just a reasonably low level of technicality.

Derivative dependent periodic and Dirichlet problems with Nagumo conditions, and one-sided Nagumo conditions, are also considered. Finally, some existence results for derivative independent singular Dirichlet problems are derived.

Besides the fundamental existence and localization results obtained with well-ordered lower and upper solutions, the authors study some other important features such as existence of extremal solutions between the lower and the upper solutions and existence of a continuum of solutions.

Section 2 – Relation with degree theory. The authors introduce the concepts of strict lower and upper solutions for a derivative dependent periodic problem and present a fundamental result which connects strict lower and upper solutions with degree theory. After that, and for simplicity, a derivative independent problem is considered, and the authors establish a version of the three solutions theorem.

Several sufficient conditions for having strict lower and upper solutions for Dirichlet problems are provided, and, then, a result which associates a degree to a pair of well-ordered strictly lower and upper solutions is established. In this case, singular right-hand sides are also considered. This section concludes with some results on the existence of solutions in the presence of non-well-ordered lower and upper solutions for periodic problems. A usual assumption required in this case concerns the control of \(f(t,u)/u\) as \(|u|\) goes to infinity, in order to prevent the nonlinearity from interfering with the second eigenvalue of the associated linear periodic problem. This is generalized further in terms of the Fučík spectrum, and, then, a similar result for Dirichlet problems is stated.

Section 3 – Variational methods. A number of complementary results are valid when the boundary value problems can be studied in the frame of variational methods. A fundamental first result is that the corresponding functional attains a minimum between well-ordered lower and upper solutions, and this minimum is a solution.

Another important part of this section deals with the minus gradient flow technique, which is applied to derive existence results in the presence of non-well-ordered lower and upper solutions. Finally a four solution theorem and a five solution theorem are proven combining variational methods and degree theory.

Section 4 – Monotone methods. Theoretical approximation of solutions between lower and upper solutions by means of monotone iterative techniques is considered in this section. The exposition starts with abstract results on convergence of iterations of operators in Banach spaces which are then applied to the boundary value problems considered in the monograph. Specifically, the authors study both the situation in which the lower and upper solutions are well-ordered and that in which they are given in the reverse order. Finally, a mixed approximation scheme is analyzed.

Reviewer’s remark: Although this falls outside the scope of this paper, in the reviewer’s opinion it is worth mentioning that generalized monotone methods exist to deal with non-Carathéodory differential equations. The interested reader is referred to the monograph [S. Heikkilä and V. Lakshmikantham, Monotone iterative techniques for discontinuous nonlinear differential equations. New York: Marcel Dekker (1994; Zbl 0804.34001)].

For the entire collection see [Zbl 1067.34001].

De Coster and Habets prove that they are not only proficient in the topic but also concerned about rigorous and pleasant writing in mathematics. The clarity of the exposition and accuracy, the many superb examples, and the fine balance between generality and amount of details make this paper suitable as a textbook. On the other hand, researchers will find in it a good reference for many recent results and proofs and an exhaustive bibliographical discussion. Moreover, some relevant historical notes about the development of the methods are included.

The paper is organized in four main sections, which we will review separately.

Section 1 – Well-ordered lower and upper solutions. The core of the method of lower and upper solutions is described in this part. As a starting point, the authors choose the periodic problem \[ u''=f(t,u),\quad u(a)=u(b),\;u'(a)=u'(b), \] with a locally \(L^1\)-bounded Carathéodory function \(f\). In this setting, a lower solution is defined as a function \(\alpha\in\mathcal C([a,b])\) such that \(\alpha(a)=\alpha(b)\) whose periodic extension to the whole real line is such that, for any real \(t_0\), either \(D_-\alpha(t_0)<D^+\alpha(t_0)\) or there exists an open interval \(I_0\) such that \(t_0\in I_0\), \(\alpha\in W^{2,1}(I_0)\) and, for a.e. \(t\in I_0\), \[ \alpha''(t)\geq f(t,\alpha(t)). \] Among the many definitions that one can find in the literature on the topic the previous one is satisfactorily general, allowing corners in the graphs of the lower solution, and requires just a reasonably low level of technicality.

Derivative dependent periodic and Dirichlet problems with Nagumo conditions, and one-sided Nagumo conditions, are also considered. Finally, some existence results for derivative independent singular Dirichlet problems are derived.

Besides the fundamental existence and localization results obtained with well-ordered lower and upper solutions, the authors study some other important features such as existence of extremal solutions between the lower and the upper solutions and existence of a continuum of solutions.

Section 2 – Relation with degree theory. The authors introduce the concepts of strict lower and upper solutions for a derivative dependent periodic problem and present a fundamental result which connects strict lower and upper solutions with degree theory. After that, and for simplicity, a derivative independent problem is considered, and the authors establish a version of the three solutions theorem.

Several sufficient conditions for having strict lower and upper solutions for Dirichlet problems are provided, and, then, a result which associates a degree to a pair of well-ordered strictly lower and upper solutions is established. In this case, singular right-hand sides are also considered. This section concludes with some results on the existence of solutions in the presence of non-well-ordered lower and upper solutions for periodic problems. A usual assumption required in this case concerns the control of \(f(t,u)/u\) as \(|u|\) goes to infinity, in order to prevent the nonlinearity from interfering with the second eigenvalue of the associated linear periodic problem. This is generalized further in terms of the Fučík spectrum, and, then, a similar result for Dirichlet problems is stated.

Section 3 – Variational methods. A number of complementary results are valid when the boundary value problems can be studied in the frame of variational methods. A fundamental first result is that the corresponding functional attains a minimum between well-ordered lower and upper solutions, and this minimum is a solution.

Another important part of this section deals with the minus gradient flow technique, which is applied to derive existence results in the presence of non-well-ordered lower and upper solutions. Finally a four solution theorem and a five solution theorem are proven combining variational methods and degree theory.

Section 4 – Monotone methods. Theoretical approximation of solutions between lower and upper solutions by means of monotone iterative techniques is considered in this section. The exposition starts with abstract results on convergence of iterations of operators in Banach spaces which are then applied to the boundary value problems considered in the monograph. Specifically, the authors study both the situation in which the lower and upper solutions are well-ordered and that in which they are given in the reverse order. Finally, a mixed approximation scheme is analyzed.

Reviewer’s remark: Although this falls outside the scope of this paper, in the reviewer’s opinion it is worth mentioning that generalized monotone methods exist to deal with non-Carathéodory differential equations. The interested reader is referred to the monograph [S. Heikkilä and V. Lakshmikantham, Monotone iterative techniques for discontinuous nonlinear differential equations. New York: Marcel Dekker (1994; Zbl 0804.34001)].

For the entire collection see [Zbl 1067.34001].

Reviewer: Rodrigo L. Pouso (MR2166490)

##### MSC:

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

34A45 | Theoretical approximation of solutions to ordinary differential equations |

47N20 | Applications of operator theory to differential and integral equations |