Global Properties of Linear Ordinary Differential Equations

1. Introduction with historical remark.- 2. Notation, definitions and some basic facts.- 2.1 Generalities.- 2.2 Maps.- 2.3 Topology.- 2.4 Algebraic structures.- 2.5 Vector spaces.- 2.6 Linear differential equations.- 2.7 Functional equations.- 3. Global transformations.- 3.1 Definition of the global transformations.- 3.2 Smoothness of global transformations.- 3.3 Algebraic approach to global transformations.- 3.4 Fundamental problems.- 4. Analytic, algebraic and geometrical aspects of global transformation.- 4.1 Some useful formulas.- 4.2 Global transformations of special classes of linear differential equations.- 4.3 Covariant constructions of linear differential equations.- 4.4 Geometrical approach to global transformations.- 5. Criterion of global equivalence.- 5.1 Bor?vka’s criterion of global equivalence of the second order equations.- 5.2 Criterion of global equivalence of the third and higher order equations.- 6. Stationary groups.- 6.1 Notation and preliminary results.- 6.2 Preparatory results.- 6.3 Subgroups of stationary groups with increasing elements.- 6.4 Stationary groups with decreasing elements.- 6.5 Complete list of stationary groups and characterization of the corresponding equations.- 7. Canonical forms.- 7.1 Notion of canonical forms.- 7.2 The Laguerre-Forsyth and Halphen forms.- 7.3 Cartan’s moving-frame-of-reference method.- 7.4 Hereditary property.- 7.5 Global canonical forms: geometrical approach.- 7.6 Global canonical forms: analytic approach.- 7.7 List of canonical forms of the second and third order equations.- 8. Invariants.- 8.1 Notion of invariant and covariant.- 8.2 Covariants.- 8.3 Local invariants and covariants.- 8.4 Global invariants.- 8.5 Smoothness of coefficients as an invariant.- 9. Equations with solutions of prescribedproperties.- 9.1 Coordinate approach.- 9.2 Asymptotic properties of solutions of the second order equations.- 9.3 Periodic solutions of the second order equations.- 9.4 Geometrical approach.- 10. Zeros of solutions.- 10.1 Notation and definitions.- 10.2 Representation of zeros.- 10.3 Second order equations.- 10.4 Third order equations.- 10.5 Iterative nth order equations.- 10.6 Periodic solutions of nth order equations.- 11. Related results and some applications.- 11.1 Asymptotic properties and zeros of solutions of second order equations.- 11.2 Integral inequalities.- 11.3 Affine geometry of plane curves.- 11.4 Isoperimetric theorems.- 11.5 Related results and comments, possible trends of further research.- 12. Appendix: Two functional equations.- 12.1 Abel functional equation.- 12.2 Euler functional equation for homogeneous functions.- Literature cited in the book and/or for supplementary reading.- Index of names.