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A Variational Approach to Lyapunov Type Inequalities

From ODEs to PDEs

  • Antonio Cañada
  • Salvador Villegas

Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Table of contents

  1. Front Matter
    Pages i-xviii
  2. Antonio Cañada, Salvador Villegas
    Pages 1-7
  3. Antonio Cañada, Salvador Villegas
    Pages 9-45
  4. Antonio Cañada, Salvador Villegas
    Pages 47-68
  5. Antonio Cañada, Salvador Villegas
    Pages 69-93
  6. Antonio Cañada, Salvador Villegas
    Pages 95-118
  7. Back Matter
    Pages 119-120

About this book

Introduction

This book highlights the current state of Lyapunov-type inequalities through a detailed analysis. Aimed toward researchers and students working in differential equations and those interested in the applications of stability theory and resonant systems, the book begins with an overview Lyapunov’s original results and moves forward to include prevalent results obtained in the past ten years. Detailed proofs and an emphasis on basic ideas are provided for different boundary conditions for ordinary differential equations, including Neumann, Dirichlet, periodic, and antiperiodic conditions. Novel results of higher eigenvalues, systems of equations, partial differential equations as well as variational approaches are presented. To this respect, a new and unified variational point of view  is introduced for the treatment of such problems and a systematic discussion of different types of boundary conditions is featured.

Various problems make the study of Lyapunov-type inequalities of interest to those in pure and applied mathematics. Originating with the study of the stability properties of the Hill equation, other questions arose for instance in systems at resonance, crystallography, isoperimetric problems, Rayleigh type quotients and oscillation and intervals of disconjugacy and it lead to the study of Lyapunov-type inequalities for differential equations. This classical area of

mathematics is still of great interest and remains a source of inspiration.

 

Keywords

Dirichlet boundry conditions Lyapunov-type inequalities Neumann boundry conditions Periodic and antiperiodic boundry conditions Radial higher eigenvalues

Authors and affiliations

  • Antonio Cañada
    • 1
  • Salvador Villegas
    • 2
  1. 1.Department of Mathematical AnalysisUniversity of GranadaGranadaSpain
  2. 2.Department of Mathematical AnalysisUniversity of GranadaGranadaSpain

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-25289-6
  • Copyright Information The Author(s) 2015
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-25287-2
  • Online ISBN 978-3-319-25289-6
  • Series Print ISSN 2191-8198
  • Series Online ISSN 2191-8201
  • Buy this book on publisher's site