This book introduces the method of lower and upper solutions for ordinary differential equations. This method is known to be both easy and powerful to solve second order boundary value problems. Besides an extensive introduction to the method, the first half of the book describes some recent and more involved results on this subject. These concern the combined use of the method with degree theory, with variational methods and positive operators. The second half of the book concerns applications. This part exemplifies the method and provides the reader with a fairly large introduction to the problematic of boundary value problems. Although the book concerns mainly ordinary differential equations, some attention is given to other settings such as partial differential equations or functional differential equations. A detailed history of the problem is described in the introduction.
· Presents the fundamental features of the method · Construction of lower and upper solutions in problems · Working applications and illustrated theorems by examples · Description of the history of the method and Bibliographical notes
Any mathematicians and PHD students.
Preface Notations Introduction - The History I. The Periodic Problem II. The Separated BVP III. Relation with Degree Theory IV. Variational Methods V. Monotone Iterative Methods VI. Parametric Multiplicity Problems VII. Resonance and Nonresonance VIII. Positive Solutions IX. Problem with Singular Forces X. Singular Perturbations XI. Bibliographical Notes Appendix Bibliography Index
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- © Elsevier Science 2006
- 21st March 2006
- Elsevier Science
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Université du Littoral-Côte d'Opale, France
Université Catholique de Louvain, Belgium
"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 differential equations, because the difficulties inherent to the method already appear in this frame-work 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 lo