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.
Inhalt
List of Contributors
Preface
Periodic Solutions of Semilinear Parabolic Equations
Introduction 1
1. Definitions and Main Results
2. Preliminaries on Linear Evolution Equations
3. Semilinear Evolution Equations
4. Semilinear Parabolic Equations
5. Proof of the Main Results
References
Linear Maximal Monotone Operators and Singular Nonlinear Integral Equations of Hammerstein Type
Introduction
1. Proof of the Main Theorem
2. Application to Hammerstein Equations
References
Nonlinear Problems Across a Point of Resonance for Non-Self-Adjoint Systems
1. Introduction
2. Notations
3. Some Abstract Existence Theorems
4. A Study on
5. Existence Theorems for the Scalar Case by Uniform Topology
6. Extensions to the Case of P and Q Orthogonal
References
Branching of Periodic Solutions of Nonautonomous Systems
Introduction
1. Existence of Periodic Solutions
2. Stability of Periodic Solutions
References
Restricted Generic Bifurcation
Introduction
1. Motivation and Statement of the Problems
2. Restricted Generic Bifurcation, p = 1
3. Restricted Generic Bifurcation, p = 2
References
On a Second-Order Nonlinear Elliptic Boundary Value Problem
1. Introduction and Statement of the Result
2. Proof of the Theorem
References
Tikhonov Regularization and Nonlinear Problems at Resonance-Deterministic and Random
1. Introduction
2. Linear Case
3. Alternative Method
4. Perturbation Method
5. The Proximal Point Algorithm
6. Existence of Random Solutions
7. Tikhonov Regularization and Random Problems
References
The Eigenvalue Problem for Variational Inequalities and a New Version of the Ljustemik-Schnirelmann Theory
Introduction: Motivation
1. A Simple Example
2. General Theory. The Penalty Method and a New Version
of the Ljustemik-Schnirelmann Theory
3. Special Case of a Halfspace
4. An Open Problem
References
Nonlinear Boundary Value Problems for Ordinary Differential Equations: From Schauder Theorem to Stable Homotopy
1. Introduction
2. Formulation of the Problem
3. The Case Where L Has an Inverse
4. The Case of Linear Boundary Conditions Such That L Has No Inverse
5. The Case of Nonlinear Boundary Conditions Such That Ind L = 0
References
Some Minimax Theorems and Applications to Nonlinear Partial Differential Equations
Introduction
1. The Abstract Theorems
2. Applications to Elliptic Partial Differential Equations
3. An Application to a Hyperbolic Partial
Differential Equation
References
Branching and Stability for Nonlinear Gradient Operators
1. Introduction
2. Preliminaries
3. The Branching Theorems for Equation (*)
4. The Stability Results
References
Recent Progress in Bifurcation Theory
1. Bifurcation Theory
2. Bifurcation at Multiple Eigenvalues
3. Bifurcation in the Presence of a Symmetry Group
4. Bifurcation of Doubly Periodic Solutions
References
On the Subgradient of Convex Functionals
1. BBe eH e
2. OcHOBH ie O T Bc omraTe bHbie pe o eH
3. OKa3aTe bCTBO TeopembI 4
TepaTypa
On the Stability of Bifurcating Solutions
1. Introduction
2. The Steady-State Bifurcation
References
Published Works of Erich H. Rothe