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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.
This book provides an extensive survey on Lyapunov-type inequalities. It summarizes and puts order into a vast literature available on the subject, and sketches recent developments in this topic. In an elegant and didactic way, this work presents the concepts underlying Lyapunov-type inequalities, covering how they developed and what kind of problems they address. This survey starts by introducing basic applications of Lyapunov’s inequalities. It then advances towards even-order, odd-order, and higher-order boundary value problems; Lyapunov and Hartman-type inequalities; systems of linear, nonlinear, and quasi-linear differential equations; recent developments in Lyapunov-type inequalities; partial differential equations; linear difference equations; and Lyapunov-type inequalities for linear, half-linear, and nonlinear dynamic equations on time scales, as well as linear Hamiltonian dynamic systems. Senior undergraduate students and graduate students of mathematics, engineering, and science will benefit most from this book, as well as researchers in the areas of ordinary differential equations, partial differential equations, difference equations, and dynamic equations. Some background in calculus, ordinary and partial differential equations, and difference equations is recommended for full enjoyment of the content.
​The eigenvalue problems for quasilinear and nonlinear operators present many differences with the linear case, and a Lyapunov inequality for quasilinear resonant systems showed the existence of eigenvalue asymptotics driven by the coupling of the equations instead of the order of the equations. For p=2, the coupling and the order of the equations are the same, so this cannot happen in linear problems. Another striking difference between linear and quasilinear second order differential operators is the existence of Lyapunov-type inequalities in R^n when p>n. Since the linear case corresponds to p=2, for the usual Laplacian there exists a Lyapunov inequality only for one-dimensional problems. For linear higher order problems, several Lyapunov-type inequalities were found by Egorov and Kondratiev and collected in On spectral theory of elliptic operators, Birkhauser Basel 1996. However, there exists an interesting interplay between the dimension of the underlying space, the order of the differential operator, the Sobolev space where the operator is defined, and the norm of the weight appearing in the inequality which is not fully developed. Also, the Lyapunov inequality for differential equations in Orlicz spaces can be used to develop an oscillation theory, bypassing the classical sturmian theory which is not known yet for those equations. For more general operators, like the p(x) laplacian, the possibility of existence of Lyapunov-type inequalities remains unexplored. ​
Theories, methods and problems in approximation theory and analytic inequalities with a focus on differential and integral inequalities are analyzed in this book. Fundamental and recent developments are presented on the inequalities of Abel, Agarwal, Beckenbach, Bessel, Cauchy–Hadamard, Chebychev, Markov, Euler’s constant, Grothendieck, Hilbert, Hardy, Carleman, Landau–Kolmogorov, Carlson, Bernstein–Mordell, Gronwall, Wirtinger, as well as inequalities of functions with their integrals and derivatives. Each inequality is discussed with proven results, examples and various applications. Graduate students and advanced research scientists in mathematical analysis will find this reference essential to their understanding of differential and integral inequalities. Engineers, economists, and physicists will find the highly applicable inequalities practical and useful to their research.
This brief examines mathematical models in nonsmooth mechanics and nonregular electrical circuits, including evolution variational inequalities, complementarity systems, differential inclusions, second-order dynamics, Lur'e systems and Moreau's sweeping process. The field of nonsmooth dynamics is of great interest to mathematicians, mechanicians, automatic controllers and engineers. The present volume acknowledges this transversality and provides a multidisciplinary view as it outlines fundamental results in nonsmooth dynamics and explains how to use them to study various problems in engineering. In particular, the author explores the question of how to redefine the notion of dynamical systems in light of modern variational and nonsmooth analysis. With the aim of bridging between the communities of applied mathematicians, engineers and researchers in control theory and nonlinear systems, this brief outlines both relevant mathematical proofs and models in unilateral mechanics and electronics.
This two-part treatment explains basic theory and details, including oscillatory solutions, intervals of stability and instability, discriminants, and coexistence. Particular attention to stability problems and coexistence of periodic solutions. 1966 edition.
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In this book the authors reduce a wide variety of problems arising in system and control theory to a handful of convex and quasiconvex optimization problems that involve linear matrix inequalities. These optimization problems can be solved using recently developed numerical algorithms that not only are polynomial-time but also work very well in practice; the reduction therefore can be considered a solution to the original problems. This book opens up an important new research area in which convex optimization is combined with system and control theory, resulting in the solution of a large number of previously unsolved problems.