9780521843973

An Introduction to Nonlinear Analysis

by
  • ISBN13:

    9780521843973

  • ISBN10:

    0521843979

  • Format: Hardcover
  • Copyright: 2005-01-10
  • Publisher: Cambridge University Press
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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

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