##
**Two-point boundary value problems. Lower and upper solutions.**
*(English)*
Zbl 1330.34009

Mathematics in Science and Engineering 205. Amsterdam: Elsevier (ISBN 0-444-52200-X/hbk). xii, 489 p. (2006).

In this excellent monograph the authors present a survey of classical and recent results in the theory of lower and upper solutions applied to two-point boundary value problems. This theory consists of reducing the problem of finding a solution of a given equation to that of looking for two functions that satisfy suitable inequalities. This method was employed in the last century by a large number of researchers in the field of nonlinear analysis. It remains one of the most useful tools for ensuring the existence of solutions for ordinary and partial differential equations. Here the authors consider only ordinary differential equations, because the difficulties inherent to the method already appear in this framework and such equations allow a more intuitive interpretation of the theory.

The first question that arises in this method concerns the difficulty of finding the lower and the upper solutions, which the authors liken to that of finding a Lyapunov function in stability theory. Neither is an easy problem, but, as the authors say in the preface, “This replacement reminds us of Lyapunov’s second method. In both instances, working cases is the key to developing the intuition related to these auxiliary functions.” In the book are presented an important number of very interesting and nice examples in which the reader can see how to construct a lower and an upper solution. This is not a technical question and, as a consequence, in a lot of research papers it represents the main part of the investigation developed.

Another natural question that arises concerns what assumptions imply the existence of a pair of lower and upper solutions for a given problem.

With these two questions in mind, the authors divide the book into two parts. In the first one, which comprises the first five chapters, they present theoretical results. In the second part, which consists of the last five chapters, they describe some applications to different problems and show how to build lower and upper solutions.

In the introduction the authors present the history of the method coupled with the different kinds of subjects that will be considered throughout the book: several classes of boundary value problems, maximal and minimal solutions, the Nagumo condition, degree theory, non-well-ordered lower and upper solutions, variational methods and monotone iterative techniques.

In Chapter I they study the periodic boundary value problem. They distinguish two cases depending on the regularity of the nonlinear part of the equation. In both cases the definition of lower and upper solutions allows corners in the graphs of such functions, and one-sided Nagumo conditions are considered.

In Chapter II the separated boundary value conditions are considered. Now, theoretical results similar to the ones given in the previous chapter are presented. Moreover, the particular cases of Dirichlet and Neumann boundary conditions are studied, for which multiplicity results are given. Fourth-order ordinary differential equations and second-order partial differential equations with Dirichlet conditions are also treated.

In Chapter III some existence and multiplicity results for Dirichlet and periodic boundary value problems are given. In this case the lower and the upper solution appear well-ordered, in reverse order or without any order. The results obtained are deduced from degree theory.

In Chapter IV variational methods are used. The solutions of the problems considered are obtained as the critical points of suitable related functionals. By means of the minimax method the authors prove the existence of at least one, three, four or seven periodic solutions. By using the concept of weak lower and upper solutions and the reduction method, they give some ideas for constructing lower and upper solutions in the usual sense.

In Chapter V the authors develop monotone iterative techniques. These types of methods ensure the existence of minimal and maximal solutions lying between the lower and the upper solution. Moreover, such solutions can be obtained as the uniform limit of suitable related problems. To apply such methods to periodic, Dirichlet and Neumann problems, the authors impose a one-sided Lipschitz condition on the nonlinear part of the equation. Moreover, some maximum and anti-maximum principles of suitable operators are used.

In the second part of the book, in Chapter VI the authors study multiplicity results for parametric problems. Chapter VII is devoted to the existence of solutions of different boundary value problems depending on whether the slope of the nonlinear part at \(\pm \infty\) is smaller than the first eigenvalue.

In Chapter VIII are studied positive solutions of Dirichlet and separated boundary value problems depending on a parameter. In Chapter IX singular periodic boundary value problems are considered, and Chapter X is devoted to the study of singular perturbations of Dirichlet problems. In the majority of the situations, the construction of pairs of lower and upper solutions together with Wirtinger-type inequalities is the fundamental tool of the proofs.

In Chapter XI the authors give a thorough survey of the bibliography used in the previous chapters. Finally, they present an appendix in which some classical definitions and results are outlined. These concern degree theory, variational methods, spectral theory, inequalities and maximum and anti-maximum principles. The bibliography is complete and covers both classical and recent references.

This monograph is essential for researchers in this field.

The first question that arises in this method concerns the difficulty of finding the lower and the upper solutions, which the authors liken to that of finding a Lyapunov function in stability theory. Neither is an easy problem, but, as the authors say in the preface, “This replacement reminds us of Lyapunov’s second method. In both instances, working cases is the key to developing the intuition related to these auxiliary functions.” In the book are presented an important number of very interesting and nice examples in which the reader can see how to construct a lower and an upper solution. This is not a technical question and, as a consequence, in a lot of research papers it represents the main part of the investigation developed.

Another natural question that arises concerns what assumptions imply the existence of a pair of lower and upper solutions for a given problem.

With these two questions in mind, the authors divide the book into two parts. In the first one, which comprises the first five chapters, they present theoretical results. In the second part, which consists of the last five chapters, they describe some applications to different problems and show how to build lower and upper solutions.

In the introduction the authors present the history of the method coupled with the different kinds of subjects that will be considered throughout the book: several classes of boundary value problems, maximal and minimal solutions, the Nagumo condition, degree theory, non-well-ordered lower and upper solutions, variational methods and monotone iterative techniques.

In Chapter I they study the periodic boundary value problem. They distinguish two cases depending on the regularity of the nonlinear part of the equation. In both cases the definition of lower and upper solutions allows corners in the graphs of such functions, and one-sided Nagumo conditions are considered.

In Chapter II the separated boundary value conditions are considered. Now, theoretical results similar to the ones given in the previous chapter are presented. Moreover, the particular cases of Dirichlet and Neumann boundary conditions are studied, for which multiplicity results are given. Fourth-order ordinary differential equations and second-order partial differential equations with Dirichlet conditions are also treated.

In Chapter III some existence and multiplicity results for Dirichlet and periodic boundary value problems are given. In this case the lower and the upper solution appear well-ordered, in reverse order or without any order. The results obtained are deduced from degree theory.

In Chapter IV variational methods are used. The solutions of the problems considered are obtained as the critical points of suitable related functionals. By means of the minimax method the authors prove the existence of at least one, three, four or seven periodic solutions. By using the concept of weak lower and upper solutions and the reduction method, they give some ideas for constructing lower and upper solutions in the usual sense.

In Chapter V the authors develop monotone iterative techniques. These types of methods ensure the existence of minimal and maximal solutions lying between the lower and the upper solution. Moreover, such solutions can be obtained as the uniform limit of suitable related problems. To apply such methods to periodic, Dirichlet and Neumann problems, the authors impose a one-sided Lipschitz condition on the nonlinear part of the equation. Moreover, some maximum and anti-maximum principles of suitable operators are used.

In the second part of the book, in Chapter VI the authors study multiplicity results for parametric problems. Chapter VII is devoted to the existence of solutions of different boundary value problems depending on whether the slope of the nonlinear part at \(\pm \infty\) is smaller than the first eigenvalue.

In Chapter VIII are studied positive solutions of Dirichlet and separated boundary value problems depending on a parameter. In Chapter IX singular periodic boundary value problems are considered, and Chapter X is devoted to the study of singular perturbations of Dirichlet problems. In the majority of the situations, the construction of pairs of lower and upper solutions together with Wirtinger-type inequalities is the fundamental tool of the proofs.

In Chapter XI the authors give a thorough survey of the bibliography used in the previous chapters. Finally, they present an appendix in which some classical definitions and results are outlined. These concern degree theory, variational methods, spectral theory, inequalities and maximum and anti-maximum principles. The bibliography is complete and covers both classical and recent references.

This monograph is essential for researchers in this field.

### MSC:

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

34Bxx | Boundary value problems for ordinary differential equations |

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

47J30 | Variational methods involving nonlinear operators |