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Author Cabada, Alberto.

Title Maximum principles for the Hill's equation / Alberto Cabada, José Ángel Cid, Lucía López-Somoza.

Imprint London : Academic Press, ©2018.

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Location Call No. OPAC Message Status
 Axe Elsevier ScienceDirect Ebook  Electronic Book    ---  Available
Edition 1st ed.
Description 1 online resource
text txt rdacontent
computer c rdamedia
online resource cr rdacarrier
Contents 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 γ*>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
Summary 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.
Subject Hill's equation.
Équation de Hill.
MATHEMATICS -- Calculus.
MATHEMATICS -- Mathematical Analysis.
Hill's equation
Added Author Cid, José Ángel.
López-Somoza, Lucía.
Other Form: Print version: 9780128041178 012804117X (OCoLC)979562421
ISBN 9780128041260 (electronic bk.)
0128041269 (electronic bk.)
012804117X
9780128041178
9780128041178
012804117X
Standard No. 9780128041178
AU@ 000061362393
AU@ 000068128410
UKMGB 018557300

 
    
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