The Nonlinear Limit-Point/Limit-Circle Problem

First posed by Hermann Weyl in 1910, the limit–point/limit–circle problem has inspired, over the last century, several new developments in the asymptotic analysis of nonlinear differential equations. This self-contained monograph traces the evolution of this problem from its inception to its modern-day extensions to the study of deficiency indices and analogous properties for nonlinear equations.

The book opens with a discussion of the problem in the linear case, as Weyl originally stated it, and then proceeds to a generalization for nonlinear higher-order equations. En route, the authors distill the classical theorems for second and higher-order linear equations, and carefully map the progression to nonlinear limit–point results. The relationship between the limit–point/limit–circle properties and the boundedness, oscillation, and convergence of solutions is explored, and in the final chapter, the connection between limit–point/limit–circle problems and spectral theory is examined in detail.

With over 120 references, many open problems, and illustrative examples, this work will be valuable to graduate students and researchers in differential equations, functional analysis, operator theory, and related fields.

“With over 120 references, many open problems, and illustrative examples, this small gem of a book will be eminently valuable to graduate students and researchers in differential equations, functional analysis, operator theory, and related fields. They all will find that the book provides them with an enjoyable coverage of some new developments in the asymptotic analysis of nonlinear differential equations with particular attention paid to the limit-point/limit-circle problem.  It will open the door to further reading and to greater skill in handling further developments in and extensions of the problem.” ---CURRENT ENGINEERING PRACTICE

“The limit-point/limit-circle classification for Sturm-Liouville differential equations on the interval [0, infinity] has been one of the most influential topics in ordinary differential equations over the last century, the majority of these results being on linear differential equations.  This is the first monograph which includes nonlinear differential equations. Apart from dealing with nonlinear problems, a substantial part is devoted to an overview on the linear case, with an extensive list of references for further reading ... Conditions for continuability of all solutions are given, as well as necessary conditions and sufficient conditions for limit-circle type. Also, boundedness and (non)oscillation of solutions are investigated.” ---ZENTRALBLATT MATH