# Nonlinear Analysis

## 1st Edition

**Authors:**Erich H. Rothe

**Editors:**Lamberto Cesari Rangachari Kannan Hans F. Weinberger

**eBook ISBN:**9781483262543

**Imprint:**Academic Press

**Published Date:**1st January 1978

**Page Count:**252

## Description

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.

## Table of Contents

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 <QNx, x

*> in the Uniform Topology*

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 ()

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. BBeДeHиe

2. OcHOBHъie ПOЯЯTя и BcпomraTeлbHbie ПpeДлoЖeHИЯ

3. ДOKa3aTeЛbCTBO TeopembI 4

ЛИTepaTypa

On the Stability of Bifurcating Solutions

1. Introduction

2. The Steady-State Bifurcation

References

Published Works of Erich H. Rothe

## Details

- No. of pages:
- 252

- Language:
- English

- Copyright:
- © Academic Press 1978

- Published:
- 1st January 1978

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483262543