Rothe / Cesari / Kannan Nonlinear Analysis

1. Auflage 2014
ISBN: 978-1-4832-6254-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 252 Seiten, Web PDF

Nonlinear Analysis: A Collection of Papers in Honor of Erich H. Rothe is a collection of papers in honor of Erich H. Rothe, a mathematician who has made significant contributions to various aspects of nonlinear functional analysis. Topics covered range from periodic solutions of semilinear parabolic equations to nonlinear problems across a point of resonance for non-self-adjoint systems. Nonlinear boundary value problems for ordinary differential equations are also considered. Comprised of 14 chapters, this volume first discusses the use of fixed-point theorems in ordered Banach spaces to prove existence and multiplicity result for periodic solutions of semilinear parabolic differential equations of the second order. The reader is then introduced to linear maximal monotone operators and singular nonlinear integral equations of Hammerstein type. Subsequent chapters focus on the branching of periodic solutions of non-autonomous systems; restricted generic bifurcation; Tikhonov regularization and nonlinear problems at resonance; and minimax theorems and their applications to nonlinear partial differential equations. This monograph will be of interest to students and practitioners in the field of mathematics.

Autoren/Hrsg.

Rothe, Erich H.

Cesari, Lamberto

Kannan, Rangachari

Weinberger, Hans F.

Inhaltsverzeichnis

1;Front Cover;1
2;Nonlinear Analysis: A Collection of Papers in Honor of Erich H. Rothe;4
3;Copyright Page;5
4;Table of Contents;8
5;LIST OF CONTRIBUTORS;12
6;PREFACE;14
7;Chapter 1. Periodic Solutions of Semilinear Parabolic Equations;16
7.1;Introduction;16
7.2;1. Definitions and Main Results;17
7.3;2. Preliminaries on Linear Evolution Equations;21
7.4;3. Semilinear Evolution Equations;27
7.5;4. Semilinear Parabolic Equations;30
7.6;5. Proof of the Main Results;35
7.7;References;43
8;Chapter 2. Linear Maximal Monotone Operatorsand Singular Nonlinear Integral Equations of Hammerstein Type;46
8.1;Introduction;46
8.2;1. Proof of the Main Theorem;48
8.3;2. Application to Hammerstein Equations;54
8.4;References;57
9;Chapter 3. Nonlinear Problems across a Point of Resonance for Nonselfadjoint Systems;58
9.1;1. Introduction;58
9.2;2. Notations;60
9.3;3. Some Abstract Existence Theorems;62
9.4;4. A Study on in the Uniform Topology;65
9.5;5. Existence Theorems for the Scalar Case by Uniform Topology;73
9.6;6. Extensions to the Case of P and Q Orthogonal;77
9.7;References;80
10;Chapter 4. Branching of Periodic Solutions of Nonautonomous Systems;84
10.1;Introduction;84
10.2;1. Existence of Periodic Solutions;85
10.3;2. Stability of Periodic Solutions;93
10.4;References;96
11;Chapter 5. Restricted Generic Bifurcation;98
11.1;Introduction;98
11.2;1. Motivation and Statement of the Problems;98
11.3;2. Restricted Generic Bifurcation, p = 1;104
11.4;3. Restricted Generic Bifurcation, p = 2;110
11.5;References;112
12;Chapter 6. On a Second-Order Nonlinear Elliptic Boundary Value Problem;114
12.1;1. Introduction and Statement of the Result;114
12.2;2. Proof of the Theorem;116
12.3;References;122
13;Chapter 7. Tikhonov Regularization and Nonlinear Problems at Resonance—Deterministic and Random;124
13.1;1. Introduction;124
13.2;2. Linear Case;126
13.3;3. Alternative Method;129
13.4;4. Perturbation Method;130
13.5;5. The Proximal Point Algorithm;133
13.6;6. Existence of Random Solutions;134
13.7;7. Tikhonov Regularization and Random Problems;136
13.8;References;138
14;Chapter 8. The Eigenvalue Problem for Variational Inequalities and a New Version of the Ljusternik–Schnirelmann Theory;140
14.1;Introduction: Motivation;140
14.2;1. A Simple Example;143
14.3;2. General Theory. The Penalty Method and a New Version of the Ljusternik-Schnirelmann Theory;146
14.4;3. Special Case of a Halfspace;155
14.5;4. An Open Problem;157
14.6;References;158
15;Chapter 9. Nonlinear Boundary Value Problems for Ordinary Differential Equations: From Schauder Theorem to Stable Homotopy;160
15.1;1. Introduction;160
15.2;2. Formulation of the Problem;161
15.3;3. The Case Where L Has an Inverse;162
15.4;4. The Case of Linear Boundary Conditions Such That L Has No Inverse;165
15.5;5. The Case of Nonlinear Boundary Conditions Such That Ind L = 0;168
15.6;6. The Case of Nonlinear Boundary Conditions Such That Ind L > 0;171
15.7;References;174
16;Chapter 10. Some Minimax Theorems and Applications to Nonlinear Partial Differential Equations;176
16.1;Introduction;176
16.2;1. The Abstract Theorems;177
16.3;2. Applications to Elliptic Partial Differential Equations;182
16.4;3. An Application to a Hyperbolic Partial Differential Equation;188
16.5;References;192
17;Chapter 11. Branching and Stability for Nonlinear Gradient Operators;194
17.1;1. Introduction;194
17.2;2. Preliminaries;196
17.3;3. The Branching Theorems for Equation;198
17.4;4. The Stability Results;202
17.5;References;206
18;Chapter 12. Recent Progress in Bifurcation Theory;208
18.1;1. Bifurcation Theory;208
18.2;2. Bifurcation at Multiple Eigenvalues;211
18.3;3. Bifurcation in the Presence of a Symmetry Group;216
18.4;4. Bifurcation of Doubly Periodic Solutions;218
18.5;References;222
19;Chapter 13. On the Subgradient of Convex Functionals;226
19.1;1. Introduction;234
19.2;2. The Steady-State Bifurcation;237
19.3;3. The Hopf Bifurcation;241
19.4;References;247
20;14. On the Stability of Bifurcating Solutions;234
20.1;1. Bee.eH.e;227
20.2;2. Oc.o..bie .o..... . Bcno.ora.e.b.ble .pe..o..e...;227
20.3;3. .o.a.a.e.bc..o Teope.bi 4;231
21;PUBLISHED WORKS OF ERICH H. ROTHE;250