The Nonlinear Limit-Point/Limit-Circle Problem / Edition 1

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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 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.

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Editorial Reviews

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“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

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Product Details

  • ISBN-13: 9780817635626
  • Publisher: Birkhauser Verlag
  • Publication date: 12/17/2003
  • Edition description: 2004
  • Edition number: 1
  • Pages: 162
  • Product dimensions: 9.21 (w) x 6.14 (h) x 0.38 (d)

Table of Contents

1 Origins of the Limit-Point/Limit-Circle Problem.- 1.1 The Weyl Alternative.- 1.2 The Deficiency Index Problem.- 1.3 Second Order Linear Equations.- 2 Basic Definitions.- 2.1 Description of the Limit-Point/Limit-Circle Problem.- 2.2 Continuable and Singular Solutions.- 2.3 Extension of the LP/LC Properties to Singular Solutions.- 3 Second Order Nonlinear Equations.- 3.1 Introduction.- 3.2 The Superlinear Equation.- 3.2.1 Limit-Circle Criteria.- 3.2.2 Necessary Conditions for Limit-Circle behavior.- 3.2.3 Limit-Point Criteria.- 3.2.4 Necessary and Sufficient Conditions.- 3.2.5 The Superlinear Forced Equation.- 3.3 The Sublinear Equation.- 3.3.1 Limit-Circle Criteria.- 3.3.2 Limit-Point Criteria.- 3.3.3 Necessary and Sufficient Conditions.- 3.3.4 The Sublinear Forced Equation.- 3.4 Equations with r(t)— 0.- 3.4.1 Nonlinear Limit-Point Results.- 3.4.2 Nonlinear Limit-Circle Results.- 4 Some Early Limit-Point and Limit-Circle Results.- 4.1 Wintner’s Result.- 4.2 Early Results on Higher Order Linear Equations.- 4.2.1 Naimark’s Results.- 4.2.2 Fedoryuk’s Results.- 4.3 Nonlinear Limit-Point Results for Second Order Equations.- 4.4 Nonlinear Limit-Point Results for Higher Order Equations.- 4.5 Some New Generalizations of the Early Results.- 5 Relationship to Other Asymptotic Properties.- 5.1 Second Order Linear Equations.- 5.2 Second Order Nonlinear Equations.- 5.2.1 The Superlinear Case.- 5.2.2 The Sublinear Case.- 6 Third Order Differential Equations.- 6.1 Equations with Quasiderivatives.- 6.2 Linear Equations.- 6.3 Nonlinear Three-Term Equations.- 7 Fourth Order Differential Equations.- 7.1 Equations with Quasiderivatives.- 7.2 Sublinear Equations in Self-Adjoint Form.- 7.3 Two-Term Equations.- 7.4 Linear Equations.- 8 Nonlinear Differential Equations of n-th Order.- 8.1 Introduction.- 8.2 Basic Lemmas.- 8.3 Limit-Point Results.- 9 Relationship to Spectral Theory.- 9.1 Introduction.- 9.2 Self-Adjoint Linear Fourth Order Equations.- 9.3 Two-Term Even Order Linear Equations.- Author Index.

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