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BL
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| Record Number
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872345
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| Main Entry
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Cabada, Alberto.
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| Title & Author
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Maximum principles for the Hill's equation /\ Alberto Cabada, José Ángel Cid, Lucía López-Somoza.
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| Edition Statement
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1st ed.
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| Publication Statement
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London :: Academic Press,, ©2018.
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| Page. NO
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1 online resource
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| ISBN
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012804117X
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: 0128041269
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: 9780128041178
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: 9780128041260
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012804117X
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9780128041178
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| Contents
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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
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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
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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
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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
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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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| Abstract
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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.
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| Subject
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Hill's equation.
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| Subject
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Hill's equation.
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| Subject
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MATHEMATICS-- Calculus.
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| Subject
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MATHEMATICS-- Mathematical Analysis.
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| Dewey Classification
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515/.352
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| LC Classification
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QA372
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| Added Entry
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Cid, José Ángel.
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López-Somoza, Lucía.
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