Maximum principles for the Hill's equation /

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...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Cabada, Alberto
Otros Autores: Cid, Jos�e �Angel, L�opez-Somoza, Luc�ia
Formato: Electrónico eBook
Idioma:Inglés
Publicado: London : Academic Press, �2018.
Edición:1st ed.
Temas:
Hill's equation.
�Equation de Hill.
MATHEMATICS > Calculus.
MATHEMATICS > Mathematical Analysis.
Hill's equation
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 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 Ones3.4.5 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 Theorems4.2.2.1 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 �I³*>0
  • The case c(t)=0
  • 4.2.7.3 Non-negative Green's Function
  • Applications to Singular Equations4.3 Lower and Upper Solutions Method
  • 4.3.1 Well Ordered Lower and Upper Solutions
  • Construction of the modi ed 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

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