The Nonlinear Limit-Point/Limit-Circle Problem (eBook, PDF)

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This self-contained monograph traces the evolution of the limit-point/limit-circle problem from its 1910 inception, in a paper by Hermann Weyl, to its modern-day extensions to the asymptotic analysis of nonlinear differential equations. The authors distill the classical theorems in the linear case and carefully map the progress from linear 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 ...

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From the reviews:

"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

"The limit-point/limit-cicle classification for Sturm-Liouville differential equations on the interval [0,8) has been one of the most influential topic in ordinary differential equations over the last century, the majority of these results being on linear differential equations." -- ZENTRALBLATT MATH

"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 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

"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