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E-raamat: Lectures In Nonlinear Functional Analysis: Synopsis Of Lectures Given At The Faculty Of Physics Of Lomonosov Moscow State University

(M V Lomonosov Moscow State University, Russia & Russian Institute For Scientific And Technical Information, Russia), (Lomonosov Moscow State Univ, Russia), (Lomonosov Moscow State Univ, Russia)
  • Formaat: 376 pages
  • Ilmumisaeg: 28-Dec-2021
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789811248948
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Lectures In Nonlinear Functional Analysis: Synopsis Of Lectures Given At The Faculty Of Physics Of Lomonosov Moscow State UniversitySuurem pilt
  • Formaat: 376 pages
  • Ilmumisaeg: 28-Dec-2021
  • Kirjastus: World Scientific Publishing Co Pte Ltd
  • Keel: eng
  • ISBN-13: 9789811248948
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This book is a systematic presentation of basic notions, facts, and ideas of nonlinear functional analysis and their applications to nonlinear partial differential equations. It begins from a brief introduction to linear functional analysis, including various types of convergence and functional spaces. The main part of the book is devoted to the theory of nonlinear operators. Various methods of the study of nonlinear differential equations based on the facts of nonlinear analysis are presented in detail. This book may serve as an introductory textbook for students and undergraduates specializing in modern mathematical physics.

Preface vii
Part 1 Functional Analysis: Preliminaries
1(50)
Lecture 1 Linear Operators
3(8)
1.1 Notation
3(2)
1.2 Transposed Operator and Its Norm
5(2)
1.3 Topological Embedding Operators
7(3)
1.4 Theorem on the Equality of Duality Brackets
10(1)
1.5 Bibliographical Notes
10(1)
Lecture 2 Weak and *-Weak Convergence
11(6)
2.1 Criteria of Weak and *-Weak Convergence
11(4)
2.2 Uniform Convexity of Banach Spaces
15(1)
2.3 Bibliographical Notes
16(1)
Lecture 3 Bochner Integral
17(14)
3.1 Basic Definitions
17(1)
3.2 Strong and Weak Measurability
18(8)
3.3 Integrability in the Bochner Sense
26(4)
3.4 Bibliographical Notes
30(1)
Lecture 4 Spaces of B-Valued Functions and Distributions
31(20)
4.1 Spaces Lp(0,T;B)
31(5)
4.2 Weak Derivatives and B-Valued Sobolev Spaces
36(5)
4.3 Space of B-Valued Distributions @'(a, b\ B)
41(4)
4.4 Lions-Aubin Theorem
45(5)
4.5 Bibliographical Notes
50(1)
Part 2 Nonlinear Operators: Continuity, Differentiability, and Compactness
51(68)
Lecture 5 Nonlinear Operators
53(10)
5.1 Gateaux and Frechet Derivatives of Nonlinear Operators
53(7)
5.2 Frechet Derivatives of Some Important Functionals
60(2)
5.3 Bibliographical Notes
62(1)
Lecture 6 Nemytsky Operator
63(16)
6.1 Basic Definitions
63(1)
6.2 Measure Convergence
64(3)
6.3 Continuity of Nemytsky Operators
67(5)
6.4 Boundedness of Nemytsky Operators
72(1)
6.5 Necessary and Sufficient Conditions of the Continuity of Nemytsky Operators
73(3)
6.6 Potential of Nemytsky Operator
76(2)
6.7 Bibliographical Notes
78(1)
Lecture 7 Frdchet Derivatives of Implicit Functionals in Control Theory
79(16)
7.1 Elliptic Boundary-Value Control Problems
79(4)
7.2 Parabolic Mixed Control Problems
83(6)
7.3 Coefficient Control Problem for the Burgers Equation
89(3)
7.4 Coefficient Control Problem for the Korteweg-de Vries equation
92(1)
7.5 Bibliographical Notes
93(2)
Lecture 8 Compact, Completely Continuous, and Totally Continuous Operators
95(10)
8.1 Compact Operators
95(5)
8.2 Compact Sets: A Reminder
100(1)
8.3 Completely Continuous Operators
101(2)
8.4 Bibliographical Notes
103(2)
Lecture 9 Local Invertibility Theorem
105(14)
9.1 Space Isom(X; Y)
105(2)
9.2 Auxiliary Results
107(5)
9.3 Local Invertibility Theorem
112(2)
9.4 Two Examples of Application of the Local Invertibility Theorem
114(4)
9.5 Bibliographical Notes
118(1)
Part 3 Variational Methods of the Study of Nonlinear Operator Equations
119(86)
Lecture 10 Potential Operators
121(10)
10.1 Basic Definitions
121(4)
10.2 Taylor's Formula
125(2)
10.3 Extremum Conditions for Functionals
127(2)
10.4 Bibliographical Notes
129(2)
Lecture 11 Semicontinuous and Weakly Coercive Functionals
131(8)
11.1 Introduction
131(1)
11.2 Semicontinuous Functionals
131(2)
11.3 Compact and Weakly Compact Sets
133(1)
11.4 Example
134(3)
11.5 Bibliographical Notes
137(2)
Lecture 12 L. A. Lyusternik's Theorem on Conditional Extrema
139(14)
12.1 L. A. Lyusternik Theorem: A Particular Case
139(3)
12.2 Lyusternik Theorem: The General Case
142(5)
12.3 Example
147(4)
12.4 Bibliographical Notes
151(2)
Lecture 13 S. L Pokhozhaev's Method of Spherical Fibering
153(12)
13.1 Ovsyannikov's Equation
153(1)
13.2 Statement of the Problem
154(1)
13.3 Variational Statement of the Problem
154(10)
13.4 Bibliographical Notes
164(1)
Lecture 14 M. A. Krasnosel'sky's Theory of the Genus of a Set
165(6)
14.1 Genus of a Set
165(3)
14.2 Classes Mk
168(1)
14.3 Conditions for Functionals
168(2)
14.4 Bibliographical Notes
170(1)
Lecture 15 Auxiliary Results
171(8)
15.1 Class F of Real-Valued Functionals
171(7)
15.2 Bibliographical Notes
178(1)
Lecture 16 Applications of M. A. Krasnosel'sky's Theory of the Genus of a Set
179(8)
16.1 Auxiliary Lemma
179(1)
16.2 Critical Numbers of Functionals
180(3)
16.3 Critical Points of Functionals
183(1)
16.4 Example of a Countable Set of Solutions
184(2)
16.5 Bibliographical Notes
186(1)
Lecture 17 Mountain Pass Theorem
187(10)
17.1 Theorem on Deformations
187(7)
17.2 Mountain Pass Theorem
194(3)
Lecture 18 An Application of the Mountain Pass Theorem
197(8)
18.1 Theorem on the Existence of a Solution
197(6)
18.2 Bibliographical Notes
203(2)
Part 4 Methods of Monotonicity and Compactness
205(70)
Lecture 19 Galerkin Method and Method of Monotonicity. Elliptic Equations
207(10)
19.1 Galerkin Method and Monotonicity Method
207(8)
19.2 Bibliographical Notes
215(2)
Lecture 20 Method of Monotonic Operators. General Results
217(8)
20.1 Fundamentals of the Theory of Monotonic Operators
217(4)
20.2 Browder-Minty Existence Theorem
221(3)
20.3 Bibliographical Notes
224(1)
Lecture 21 Properties of the p-Laplacian
225(8)
21.1 Important Auxiliary Inequalities
225(2)
21.2 Boundedness and Continuity of the p-Laplacian
227(4)
21.3 Bibliographical Notes
231(2)
Lecture 22 Galerkin Method and Method of Compactness
Parabolic Equations, I
233(1)
22.1 Parabolic Equation with the p-Laplacian
233(13)
22.2 Bibliographical Notes
246(1)
Lecture 23 Galerkin Method and Method of Compactness
Parabolic Equations, II
247(1)
23.1 Parabolic Equation with the p-Laplacian
247(10)
23.2 Bibliographical Notes
257(2)
Lecture 24 Galerkin Method and Method of Compactness
Hyperbolic Equations
259(1)
24.1 Nonlinear Hyperbolic Equations
259(15)
24.2 Bibliographical Notes
274(1)
Part 5 Methods Based on the Maximum Principle
275(22)
Lecture 25 Method of Upper and Lower Solutions for Elliptic Equations
277(6)
25.1 Upper and Lower Solutions: Definitions
277(1)
25.2 Main Theorem
277(5)
25.3 Bibliographical Notes
282(1)
Lecture 26 Method of Upper and Lower Solutions for Parabolic Equations
283(8)
26.1 Interpolation Inequality
283(2)
26.2 Definitions of Upper and Lower Solutions
285(2)
26.3 Iterative Scheme
287(1)
26.4 Main Theorem
288(2)
26.5 Bibliographical Notes
290(1)
Lecture 27 Method of Upper and Lower Weak Solutions
291(6)
Part 6 Schauder Principle and Contraction Mapping Theorem
297(60)
Lecture 28 Topological Schauder Principle
299(12)
28.1 Contraction Mapping Theorem
299(2)
28.2 Schauder Fixed-Point Theorem
301(7)
28.3 Quasilinear Equation with p-Laplacian
308(2)
28.4 Bibliographical Notes
310(1)
Lecture 29 Picard Theorem: Simplest Case
311(10)
29.1 Autonomous Equation with Globally Lipschitzian Right-Hand Side
311(3)
29.2 Example
314(5)
29.3 Problems
319(2)
Lecture 30 Theorem on Nonextendable Solutions of Cauchy Problems
321(10)
Lecture 31 Benjamin-Bona-Mahony-Burgers Equation
331(8)
31.1 Statement of the Problem and Its Equivalent Reformulations
331(2)
31.2 Integral Equation in the Space C1([ 0,T0); Z1)
333(1)
31.3 Improving Smoothness to C1([ 0, T0); Z2)
333(1)
31.4 Further Strengthening Results
334(2)
31.5 Blow-Up of Solutions
336(2)
31.6 Main Result
338(1)
31.7 Problems
338(1)
Lecture 32 Example of Global Solvability
339(6)
32.1 Application of the Picard Theorem
339(2)
32.2 Global Solvability
341(2)
32.3 Problems
343(2)
Lecture 33 Various Generalizations and Limits of Applicability
345(12)
33.1 Nonextendable Solutions of Volterra Integral Equations
345(7)
33.2 Example of Nonextendable Solution, which Does not Have a Limit
352(2)
33.3 Peano Theorem
354(1)
33.4 Problems
355(2)
Bibliography 357(4)
Index 361
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