Jean Mawhin

FACHGEB The last decade has seen a tremendous development in critical point theory in infinite dimensional spaces and its application to nonlinear boundary value problems. In particular, striking results were obtained in the classical problem of periodic solutions of Hamiltonian systems. This book provides a systematic presentation of the most basic tools of critical point theory: minimization, convex functions and Fenchel transform, dual least action principle, Ekeland variational principle, minimax methods, Lusternik- Schirelmann theory for Z2 and S1 symmetries, Morse theory for possibly degenerate critical points and non-degenerate critical manifolds. Each technique is illustrated by applications to the discussion of the existence, multiplicity, and bifurcation of the periodic solutions of Hamiltonian systems. Among the treated questions are the periodic solutions with fixed period or fixed energy of autonomous systems, the existence of subharmonics in the non-autonomous case, the asymptotically linear Hamiltonian systems, free and forced superlinear problems. Application of those results to the equations of mechanical pendulum, to Josephson systems of solid state physics and to questions from celestial mechanics are given. The aim of the book is to introduce a reader familiar to more classical techniques of ordinary differential equations to the powerful approach of modern critical point theory. The style of the exposition has been adapted to this goal. The new topological tools are introduced in a progressive but detailed way and immediately applied to differential equation problems. The abstract tools can also be applied to partial differential equations and the reader will also find the basic references in this direction in the bibliography of more than 500 items which concludes the book. ERSCHEIN

Critical Point Theory and Hamiltonian Systems

Freddy Dumortier

Henk Broer

Andre Vanderbauwhede

Sjoerd Verduyn Lunel

' This comprehensive volume contains the state of the art on ODE's and PDE's of different nature, functional differential equations, delay equations, and others, mostly from the dynamical systems point of view. A broad range of topics are treated through contributions by leading experts of their fields, presenting the most recent developments. A large variety of techniques are being used, stressing geometric, topological, ergodic and numerical aspects. The scope of the book is wide, ranging from pure mathematics to various applied fields. Examples of the latter are provided by subjects from earth and life sciences, classical mechanics and quantum-mechanics, among others. The proceedings have been selected for coverage in: • Index to Scientific & Technical Proceedings® (ISTP® / ISI Proceedings) • Index to Scientific & Technical Proceedings (ISTP CDROM version / ISI Proceedings) • CC Proceedings — Engineering & Physical Sciences Contents: Computational Aspects of Differential Equations and ApplicationsWater WavesTopological and Variational MethodsQualitative Theory of Nonlinear Parabolic and Elliptic EquationsAround Hilbert's 16th ProblemNavier–Stokes Equations and Reaction Diffusion EquationsHyperbolic Dynamics and BeyondSymmetry and MechanicsShock Waves and Conservation LawsNonlinear Elliptic Partial Differential EquationsAlgebraic Aspects and Optimisation in Dynamical SystemsCase Studies in Theoretical Interpretation of Numerical ExperimentsInfinite-Dimensional DynamicsQuasiperiodicityDelay EquationsWave Stability and Pattern FormationNonautonomous DynamicsNormal Forms and Invariant ManifoldsSingular PerturbationsDifferential Geometric Foliations and FlowsHomoclinic and Heteroclinic DynamicsMathematical Aspects of Celestical Mechanics Readership: Graduate students and researchers in mathematics, especially in ODE and PDE areas. Keywords:Differential Equations;Dynamical Systems;ODE;PDE;Delay Equations;Water Waves;Hilbert''s 16th Problem'

EQUADIFF 2003

George Dinca

This monograph explores the concept of the Brouwer degree and its continuing impact on the development of important areas of nonlinear analysis. The authors define the degree using an analytical approach proposed by Heinz in 1959 and further developed by Mawhin in 2004, linking it to the Kronecker index and employing the language of differential forms. The chapters are organized so that they can be approached in various ways depending on the interests of the reader. Unifying this structure is the central role the Brouwer degree plays in nonlinear analysis, which is illustrated with existence, surjectivity, and fixed point theorems for nonlinear mappings. Special attention is paid to the computation of the degree, as well as to the wide array of applications, such as linking, differential and partial differential equations, difference equations, variational and hemivariational inequalities, game theory, and mechanics. Each chapter features bibliographic and historical notes, and the final chapter examines the full history. Brouwer Degree will serve as an authoritative reference on the topic and will be of interest to professional mathematicians, researchers, and graduate students.

The Core of Nonlinear Analysis

Brouwer Degree

Anna Capietto

Peter Kloeden

Sylvia Novo

Miguel Ortega

This volume contains the notes from five lecture courses devoted to nonautonomous differential systems, in which appropriate topological and dynamical techniques were described and applied to a variety of problems. The courses took place during the C.I.M.E. Session "Stability and Bifurcation Problems for Non-Autonomous Differential Equations," held in Cetraro, Italy, June 19-25 2011. Anna Capietto and Jean Mawhin lectured on nonlinear boundary value problems; they applied the Maslov index and degree-theoretic methods in this context. Rafael Ortega discussed the theory of twist maps with nonperiodic phase and presented applications. Peter Kloeden and Sylvia Novo showed how dynamical methods can be used to study the stability/bifurcation properties of bounded solutions and of attracting sets for nonautonomous differential and functional-differential equations. The volume will be of interest to all researchers working in these and related fields.

Cetraro, Italy 2011, Editors: Russell Johnson, Maria Patrizia Pera

Stability and Bifurcation Theory for Non-Autonomous Differential Equations

Jean Mawhin

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