The Nonlinear Limit-Point/Limit-Circle Problem

by Bartuek, Miroslav; Dosla, Zuzana; Graef, John R.; Bartisek, Miroslav

• ISBN13: 9780817635626
• ISBN10: 0817635629
• Format: Paperback
• Copyright: 2003-12-01
• Publisher: Birkhauser

Summary

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 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.

Table of Contents

Preface	v
Basic Notation	xi
1 Origins of the Limit-Point/ Limit-Circle Problem	1	(12)
1.1 The Weyl Alternative	1	(2)
1.2 The Deficiency Index Problem	3	(2)
1.3 Second Order Linear Equations	5	(8)
2 Basic Definitions	13	(16)
2.1 Description of the Limit-Point/Limit-Circle Problem	13	(5)
2.2 Continuable and Singular solutions	18	(4)
2.3 Extension of the LP/LC Properties to singular Solutions	22	(7)
3 Second Order Nonlinear Equations	29	(30)
3.1 Introduction	79
3.2 The superlinear Equation	30	(13)
3.2.1 Limit-Circle Criteria	30	(4)
3.2.2 Necessary Conditions for Limit-Circle Behavior	34	(3)
3.2.3 Limit-Point Criteria	37	(3)
3.2.4 Necessary and Sufficient Conditions	40	(1)
3.2.5 The superlinear Forced Equation	41	(2)
3.3 The Sublinear Equation	43	(9)
3.3.1 Limit-Circle Criteria	43	(4)
3.3.2 Limit-Point Criteria	47	(3)
3.3.3 Necessary and Sufficient Conditions	50	(1)
3.3.4 The sublinear Forced Equation	51	(1)
3.4 Equations with r(t) less than 0 51
3.4.1 Nonlinear Limit-Point Results	52	(2)
3.4.2 Nonlinear Limit-Circle Results	54	(5)
4 Some Early Limit-Point and Limit-Circle Results	59	(14)
4.1 Wintner's Result	59	(1)
4.2 Early Results on Higher Order Linear Equations	60	(3)
4.2.1 Naimark's Results	60	(2)
4.2.2 Fedoryuk's Results	62	(1)
4.3 Nonlinear Limit-Point Results for Second Order Equations	63	(2)
4.4 Nonlinear Limit-Point Results for Higher Order Equations	65	(2)
4.5 Some New Generalizations of the Early Results	67	(6)
5 Relationship to Other Asymptotic Properties	73	(10)
5.1 Second Order Linear Equations	73	(3)
5.2 Second Order Nonlinear Equations	76	(7)
5.2.1 The Superlinear Case	76	(3)
5.2.2 The Sublinear Case	79	(4)
6 Third Order Differential Equations	83	(24)
6.1 Equations with Quasiderivatives	83	(9)
6.2 Linear Equations	92	(7)
6.3 Nonlinear Three-Term Equations	99	(8)
7 Fourth Order Differential Equations	107	(18)
7.1 Equations with Quasiderivatives	107	(9)
7.2 Sublinear Equations in Self-Adjoint Form	116	(2)
7.3 Two-Term Equations	118	(4)
7.4 Linear Equations	122	(3)
8 Nonlinear Differential Equations of n-th Order	125	(18)
8.1 Introduction	125	(1)
8.2 Basic Lemmas	126	(4)
8.3 Limit-Point Results	130	(13)
9 Relationship to spectral Theory	143	(8)
9.1 Introduction	143	(1)
9.2 Self-Adjoint Linear Fourth Order Equations	144	(4)
9.3 Two-Term Even Order Linear Equations	148	(3)
Bibliography	151	(10)
Author Index	161	(2)
Subject Index	163