A Variational Approach to Lyapunov Type Inequalities: From Odes to Pdes

ISBN-13: 9783319252872 / Angielski / Miękka / 2015 / 120 str.

Antonio Canada; Salvador Villegas

Opis

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

Recenzje

"The monograph is self-contained and well written. In each chapter, some main references, most of which are up-to date, are given. The monograph will be a useful reference for mathematicians and mathematical physicians working on differential equations, eigenvalue problems, the variational method, optimal control, etc." (Meirong Zhang, Mathematical Reviews, March, 2017)

"In this brief monograph, the authors present their recent results on Lyapunov-type inequalities and some applications to the stability of linear periodic equations, the sign of the eigenvalues of eigenvalue problems and nonlinear resonant problems. The book contains five chapters, each of them with a list of references, and an index. ... The book will be useful to graduate students and researchers interested in Lyapunov-type inequalities and stability problems for differential equations." (Rodica Luca, zbMATH 1360.34001, 2017)

Spis treści

1. Introduction.- 2. A variational characterization of the best Lyapunov constants.- 3. Higher eigenvalues.- 4. Partial differential equations.- 5. Systems of equations.- Index.