Maximum Principles for the Hill's Equation

Author： Cabada   Alberto;Cid   José Ángel;López-Somoza   Lucía  

Publisher： Elsevier Science‎

Publication year： 2017

E-ISBN: 9780128041260

P-ISBN(Paperback): 9780128041178

Subject： O175.1 Ordinary Differential Equations

Keyword： 复分析、复变函数,数学

Language： ENG

Access to resources Favorite

Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.

Description

Maximum Principles for the Hill's Equation focuses on the application of these methods to nonlinear equations with singularities (e.g. Brillouin-bem focusing equation, Ermakov-Pinney,…) and for problems with parametric dependence. The authors discuss the properties of the related Green’s functions coupled with different boundary value conditions. In addition, they establish the equations’ relationship with the spectral theory developed for the homogeneous case, and discuss stability and constant sign solutions. Finally, reviews of present classical and recent results made by the authors and by other key authors are included.

  • Evaluates classical topics in the Hill’s equation that are crucial for understanding modern physical models and non-linear applications
  • Describes explicit and effective conditions on maximum and anti-maximum principles
  • Collates information from disparate sources in one self-contained volume, with extensive referencing throughout

Chapter

Front Cover

Maximum Principles for the Hill's Equation

Copyright

Contents

About the Authors

Preface

Acknowledgment

1 Introduction

1.1 Hill's Equation

1.2 Stability in the Sense of Lyapunov

1.3 Floquet's Theorem for the Hill's Equation

References

2 Homogeneous Equation

2.1 Introduction

2.2 Sturm Comparison Theory

2.3 Spectral Properties of Dirichlet Problem

2.4 Spectral Properties of Mixed and Neumann Problems

2.5 Spectral Properties of the Periodic Problem: Intervals of Stability and Instability

2.6 Relation Between Eigenvalues of Neumann, Dirichlet, Periodic, and Antiperiodic Problems

References

3 Nonhomogeneous Equation

3.1 Introduction

3.2 The Green's Function

3.3 Periodic Conditions

3.3.1 Properties of the Periodic Green's Function

3.3.2 Optimal Conditions for the Periodic MP and AMP

3.3.3 Explicit Criteria for the Periodic AMP and MP

3.3.4 More on Explicit Criteria

3.3.5 Examples

3.4 Non-Periodic Conditions

3.4.1 Neumann Problem

3.4.2 Dirichlet Problem

3.4.3 Relation Between Neumann and Dirichlet Problems

3.4.4 Mixed Problems and their Relation with Neumann and Dirichlet Ones

3.4.5 Order of Eigenvalues and Constant Sign of the Green's Function

3.4.6 Relations Between Green's Functions. Comparison Principles

3.4.7 Constant Sign for Non-Periodic Green's Functions

3.4.8 Global Order of Eigenvalues

3.4.9 Examples

3.5 General Second Order Equation

3.5.1 Periodic Problem

3.5.2 Non-Periodic Conditions

References

4 Nonlinear Equations

4.1 Introduction

4.2 Fixed Point Theorems and Degree Theory

4.2.1 Leray-Schauder Degree

4.2.2 Fixed Point Theorems

4.2.2.1 Application to Nonlinear Boundary Value Problems

4.2.3 Extremal Fixed Points

4.2.4 Monotone Operators

4.2.4.1 Existence of Solutions of Periodic Boundary Value Problems

4.2.5 Non-increasing Operators

4.2.6 Non-decreasing Operators

4.2.6.1 Multiplicity of Solutions

4.2.7 Problems with Parametric Dependence

4.2.7.1 Introduction and Preliminaries

4.2.7.2 Positive Green's Function

Auxiliary Results

The case γ*>0

The case c(t)=0

4.2.7.3 Non-negative Green's Function

Applications to Singular Equations

4.3 Lower and Upper Solutions Method

4.3.1 Well Ordered Lower and Upper Solutions

Construction of the modified problem

4.3.2 Existence of Extremal Solutions

4.3.2.1 Periodic Boundary Value Problem

4.3.3 Non-Well-Ordered Lower and Upper Solutions

4.4 Monotone Iterative Techniques

4.4.1 Well Ordered Lower and Upper Solutions

4.4.2 Reversed Ordered Lower and Upper Solutions

4.4.2.1 Final Remarks

References

A Sobolev Inequalities

References

Glossary

Index

Back Cover

The users who browse this book also browse