The Nonlinear Limit-Point/Limit-Circle Problem

by Miroslav Bartusek, Zuzana Dosla, John R. Graef

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.

176 pages, Paperback

First published April 13, 2011