Solvability of nonlinear singular problems for ordinary differential equations.

*(English)*Zbl 1228.34003
Contemporary Mathematics and Its Applications 5. New York, NY: Hindawi Publishing Corporation (ISBN 978-977-454-040-0/pbk). x, 268 p. (2008).

The topic of singular boundary value problems has been of substantial and rapidly growing interest for many scientists and engineers. This book is devoted to singular boundary value problems for ordinary differential equations. It presents the existence theory for a variety of problems having unbounded nonlinearities in regions where their solutions are searched for. The importance of a thorough investigation of the analytical solvability is emphasized by the fact that numerical simulations of solutions to such problems usually break down near singular points. The content of the monograph is mainly based on results obtained by the authors during the last few years. Nevertheless, most of the more advanced results achieved to date in this field can be found here. Besides, some known results are presented in a new way. The selection of topics reflects the particular interests of the authors. The book is addressed to researchers in related areas, to graduate students or advanced undergraduates, and, in particular, to those interested in singular and nonlinear boundary value problems. It can serve as a reference book on the existence theory for singular boundary value problems of ordinary differential equations as well as a textbook for graduate or undergraduate classes. The readers need basic knowledge of real analysis, linear and nonlinear functional analysis, the theory of the Lebesgue measure and integral, the theory of ordinary differential equations (including Carathédory theory and boundary value problems) on the graduate level.

The monograph deals with boundary value problems which are considered in the frame of Carathéodory theory. If nonlinearities in differential equations fulfil the Carathéodory conditions, the boundary value problems are called regular, while, if the Carathéodory conditions are not fulfilled on the whole region, the problems are called singular. Two types of singularities are distinguished – time and space ones. For singular boundary value problems, the notions of a solution and of a \(w\)-solution are introduced. Solutions of \(n\)th-order differential equations are understood as functions having absolutely continuous derivatives up to order \(n - 1\) on the whole basic compact interval. On the other hand, \(w\)-solutions have these derivatives only locally absolutely continuous on a noncompact subset of the basic interval. The main attention is paid to the existence of solutions of singular problems. The proofs are mostly based on regularization and the sequential technique. The impact of the theoretical results is demonstrated by illustrative examples. Essentially, the book is divided into two parts and four appendices.

Part I consists of 6 chapters and is devoted to scalar higher-order singular boundary value problems. In Chapter 1, time and space singularities are defined, three existence principles for problems with time singularities and two for problems with space singularities are formulated and proved. Chapter 2 presents existence results for focal problems with a time singularity and for focal problems having space singularities in all variables. Chapters 3–6 investigate other higher-order boundary value problems having only space singularities which appear most frequently in literature. They provide existence results for \((n,p)\)-problems, conjugate problems, Sturm-Liouville problems, and Lidstone problems.

Part II consists of Chapters 7–11 and deals with scalar second-order singular boundary value problems with one-dimensional \(\varphi\)-Laplacian. The exposition is focused mainly on Dirichlet and periodic problems which are considered in Chapters 7 and 8, respectively. Section 7.1 is fundamental for the further investigation. The operator representation of the regular Dirichlet problem with \(\varphi\)-Laplacian is derived here and the techniques of a priori estimates and of lower and upper functions are developed. In Sections 7.2–7.4, three existence principles are presented. These principles together with the principles of Chapter 1 are then specialized to important particular cases and existence theorems and criteria extending as well as supplementing earlier results are obtained. Section 7.2 deals with time singularities, Section 7.3 with space singularities, and Section 7.4 with mixed singularities, that is, both time and space ones. In Chapter 8, the authors consider the existence of periodic solutions. They start with the method of lower and upper solutions and with its relationship to the Leray-Schauder degree in Section 8.1. Section 8.2 is devoted to problems with a nonlinearity having an attractive singularity in its first space variable. Sections 8.3 and 8.4 deal with problems on strong and weak repulsive space singularities, respectively. An existence theorem for periodic problems with time singularities is given in the last section of Chapter 8. In Chapter 9, the authors study two singular mixed boundary value problems. The latter arises in the theory of shallow membrane caps and the authors discuss its solvability in dependence of parameters which appear in the differential equation. In Chapter 10, they treat problems which may have singularities in space variables. The boundary conditions under discussion are generally nonlinear and nonlocal. There are presented general principles for solvability of regular and singular nonlocal problems. Chapter 11 is devoted to a class of problems having singularities in the space variables. The implementation of a parameter into the equation enables the authors to prove solvability of problems with three independent (generally nonlocal) boundary conditions. The authors deliver an existence principle and its specialization to the problem with given maximal values for positive solutions. Appendices give an overview of some basic classical theorems and assertions which are used in Chapters 1–11.

The monograph deals with boundary value problems which are considered in the frame of Carathéodory theory. If nonlinearities in differential equations fulfil the Carathéodory conditions, the boundary value problems are called regular, while, if the Carathéodory conditions are not fulfilled on the whole region, the problems are called singular. Two types of singularities are distinguished – time and space ones. For singular boundary value problems, the notions of a solution and of a \(w\)-solution are introduced. Solutions of \(n\)th-order differential equations are understood as functions having absolutely continuous derivatives up to order \(n - 1\) on the whole basic compact interval. On the other hand, \(w\)-solutions have these derivatives only locally absolutely continuous on a noncompact subset of the basic interval. The main attention is paid to the existence of solutions of singular problems. The proofs are mostly based on regularization and the sequential technique. The impact of the theoretical results is demonstrated by illustrative examples. Essentially, the book is divided into two parts and four appendices.

Part I consists of 6 chapters and is devoted to scalar higher-order singular boundary value problems. In Chapter 1, time and space singularities are defined, three existence principles for problems with time singularities and two for problems with space singularities are formulated and proved. Chapter 2 presents existence results for focal problems with a time singularity and for focal problems having space singularities in all variables. Chapters 3–6 investigate other higher-order boundary value problems having only space singularities which appear most frequently in literature. They provide existence results for \((n,p)\)-problems, conjugate problems, Sturm-Liouville problems, and Lidstone problems.

Part II consists of Chapters 7–11 and deals with scalar second-order singular boundary value problems with one-dimensional \(\varphi\)-Laplacian. The exposition is focused mainly on Dirichlet and periodic problems which are considered in Chapters 7 and 8, respectively. Section 7.1 is fundamental for the further investigation. The operator representation of the regular Dirichlet problem with \(\varphi\)-Laplacian is derived here and the techniques of a priori estimates and of lower and upper functions are developed. In Sections 7.2–7.4, three existence principles are presented. These principles together with the principles of Chapter 1 are then specialized to important particular cases and existence theorems and criteria extending as well as supplementing earlier results are obtained. Section 7.2 deals with time singularities, Section 7.3 with space singularities, and Section 7.4 with mixed singularities, that is, both time and space ones. In Chapter 8, the authors consider the existence of periodic solutions. They start with the method of lower and upper solutions and with its relationship to the Leray-Schauder degree in Section 8.1. Section 8.2 is devoted to problems with a nonlinearity having an attractive singularity in its first space variable. Sections 8.3 and 8.4 deal with problems on strong and weak repulsive space singularities, respectively. An existence theorem for periodic problems with time singularities is given in the last section of Chapter 8. In Chapter 9, the authors study two singular mixed boundary value problems. The latter arises in the theory of shallow membrane caps and the authors discuss its solvability in dependence of parameters which appear in the differential equation. In Chapter 10, they treat problems which may have singularities in space variables. The boundary conditions under discussion are generally nonlinear and nonlocal. There are presented general principles for solvability of regular and singular nonlocal problems. Chapter 11 is devoted to a class of problems having singularities in the space variables. The implementation of a parameter into the equation enables the authors to prove solvability of problems with three independent (generally nonlocal) boundary conditions. The authors deliver an existence principle and its specialization to the problem with given maximal values for positive solutions. Appendices give an overview of some basic classical theorems and assertions which are used in Chapters 1–11.

Reviewer: Denis Bonheure (Bruxelles)