An Introduction to Nonlinear Analysis

by Martin Schechter

• ISBN13: 9780521843973
• ISBN10: 0521843979
• Format: Hardcover
• Copyright: 2005-01-10
• Publisher: Cambridge University Press

Summary

The techniques that can be used to solve non-linear problems are far different than those that are used to solve linear problems. Many courses in analysis and applied mathematics attack linear cases simply because they are easier to solve and do not require a large theoretical background in order to approach them. Professor Schechter's book is devoted to non-linear methods using the least background material possible and the simplest linear techniques. An understanding of the tools for solving non-linear problems is developed whilst demonstrating their application to problems in one dimension and then leading to higher dimensions. The reader is guided using simple exposition and proof, assuming a minimal set of pre-requisites. For completion, a set of appendices covering essential basics in functional analysis and metric spaces is included, making this ideal as an accompanying text on an upper-undergraduate or graduate course, or even for self-study.

Table of Contents

Preface	xiii
Extrema	1	(44)
Introduction	1	(1)
A one dimensional problem	1	(9)
The Hilbert space H	10	(7)
Fourier series	17	(3)
Finding a functional	20	(3)
Finding a minimum, I	23	(5)
Finding a minimum, II	28	(2)
A slight improvement	30	(2)
Finding a minimum, III	32	(1)
The linear problem	33	(2)
Nontrivial solutions	35	(1)
Approximate extrema	36	(4)
The Palais--Smale condition	40	(2)
Exercises	42	(3)
Critical points	45	(42)
A simple problem	45	(1)
A critical point	46	(1)
Finding a Palais--Smale sequence	47	(5)
Pseudo-gradients	52	(3)
A sandwich theorem	55	(5)
A saddle point	60	(4)
The chain rule	64	(1)
The Banach fixed point theorem	65	(1)
Picard's theorem	66	(2)
Continuous dependence of solutions	68	(1)
Continuation of solutions	69	(2)
Extending solutions	71	(1)
Resonance	72	(3)
The question of nontriviality	75	(1)
The mountain pass method	76	(3)
Other intervals for asymptotic limits	79	(3)
Super-linear problems	82	(1)
A general mountain pass theorem	83	(2)
The Palais--Smale condition	85	(1)
Exercises	85	(2)
Boundary value problems	87	(36)
Introduction	87	(1)
The Dirichlet problem	87	(1)
Mollifiers	88	(2)
Test functions	90	(2)
Differentiability	92	(7)
The functional	99	(2)
Finding a minimum	101	(6)
Finding saddle points	107	(3)
Other intervals	110	(4)
Super-linear problems	114	(2)
More mountains	116	(3)
Satisfying the Palais-Smale condition	119	(1)
The linear problem	120	(1)
Exercises	121	(2)
Saddle points	123	(22)
Game theory	123	(1)
Saddle points	123	(2)
Convexity and lower semi-continuity	125	(3)
Existence of saddle points	128	(4)
Criteria for convexity	132	(1)
Partial derivatives	133	(4)
Nonexpansive operators	137	(2)
The implicit function theorem	139	(4)
Exercises	143	(2)
Calculus of variations	145	(26)
Introduction	145	(1)
The force of gravity	145	(3)
Hamilton's principle	148	(3)
The Euler equations	151	(4)
The Gateaux derivative	155	(1)
Independent variables	156	(2)
A useful lemma	158	(1)
Sufficient conditions	159	(6)
Examples	165	(2)
Exercises	167	(4)
Degree theory	171	(36)
The Brouwer degree	171	(4)
The Hilbert cube	175	(8)
The sandwich theorem	183	(1)
Sard's theorem	184	(3)
The degree for differentiable functions	187	(6)
The degree for continuous functions	193	(4)
The Leray-Schauder degree	197	(3)
Properties of the Leray--Schauder degree	200	(1)
Peano's theorem	201	(2)
An application	203	(2)
Exercises	205	(2)
Conditional extrema	207	(30)
Constraints	207	(6)
Lagrange multipliers	213	(2)
Bang--bang control	215	(2)
Rocket in orbit	217	(3)
A generalized derivative	220	(1)
The definition	221	(1)
The theorem	222	(4)
The proof	226	(3)
Finite subsidiary conditions	229	(6)
Exercises	235	(2)
Mini-max methods	237	(8)
Mini-max	237	(3)
An application	240	(3)
Exercises	243	(2)
Jumping nonlinearities	245	(8)
The Dancer--Fucik spectrum	245	(3)
An application	248	(3)
Exercises	251	(2)
Higher dimensions	253	(60)
Orientation	253	(1)
Periodic functions	253	(1)
The Hilbert spaces Ht	254	(4)
Compact embeddings	258	(1)
Inequalities	258	(4)
Linear problems	262	(3)
Nonlinear problems	265	(6)
Obtaining a minimum	271	(3)
Another condition	274	(3)
Nontrivial solutions	277	(1)
Another disappointment	278	(1)
The next eigenvalue	278	(4)
A Lipschitz condition	282	(1)
Splitting subspaces	283	(2)
The question of nontriviality	285	(2)
The mountains revisited	287	(2)
Other intervals between eigenvalues	289	(4)
An example	293	(1)
Satisfying the PS condition	294	(3)
More super-linear problems	297	(1)
Sobolev's inequalities	297	(6)
The case q = ∞	303	(2)
Sobolev spaces	305	(3)
Exercises	308	(5)
Appendix A Concepts from functional analysis	313	(18)
A.1 Some basic definitions	313	(1)
A.2 Subspaces	314	(1)
A.3 Hilbert spaces	314	(2)
A.4 Bounded linear functionals	316	(1)
A.5 The dual space	317	(2)
A.6 Operators	319	(2)
A.7 Adjoints	321	(1)
A.8 Closed operators	322	(1)
A.9 Self-adjoint operators	323	(2)
A.10 Subsets	325	(1)
A.11 Finite dimensional subspaces	326	(1)
A.12 Weak convergence	327	(1)
A.13 Reflexive spaces	328	(1)
A.14 Operators with closed ranges	329	(2)
Appendix B Measure and integration	331	(10)
B.1 Measure zero	331	(1)
B.2 Step functions	331	(1)
B.3 Integrable functions	332	(3)
B.4 Measurable functions	335	(1)
B.5 The spaces Lp	335	(1)
B.6 Measurable sets	336	(2)
B.7 Caratheodory functions	338	(3)
Appendix C Metric spaces	341	(4)
C.1 Properties	341	(2)
C.2 Para-compact spaces	343	(2)
Appendix D Pseudo-gradients	345	(8)
D.1 The benefits	345	(1)
D.2 The construction	346	(7)
Bibliography	353	(2)
Index	355