Qualitative methods in mathematical analysis / L.E. El'sgol'c ; A. A. Brown, J. M. DanskinType de document : MonographieCollection : Translations of mathematical monographs, 12Langue : anglais.Pays : Etats Unis.Éditeur : Providence : American Mathematical Society, 1964Description : 1 vol. (VII-250 p.) ; 24 cmISBN : 9780821815625.ISSN : 0065-9282.Bibliographie : Bibliogr. p. 231-250.Sujet MSC : 49-02, Calculus of variations and optimal control; optimization, Research exposition (monographs, survey articles)
34Cxx, Ordinary differential equations, Qualitative theory
58E05, Global analysis, analysis on manifolds -- Variational problems in infinite-dimensional spaces, Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel'man) theory, etc.)
34K05, Ordinary differential equations -- Functional-differential and differential-difference equations, General theoryEn-ligne : Zentralblatt | MathSciNet
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Bibliogr. p. 231-250
This is a good translation (with only occasional misprints) of a useful Russian book [GITTL, Moscow, 1955; MR0076109 (17,847c)] devoted primarily to the qualitative theory of ordinary differential equations and delay-differential equations (differential equations with retarded arguments).
In Chapter I the author discusses the theory of critical points of a C2 function on a C3 manifold, and in Chapter II he applies this to functions of several complex variables. Chapter III treats some fixed-point theorems and their applications to differential equations. These chapters assume some knowledge of topology.
The last three chapters, which are almost independent of the first three, are concerned with differential equations with and without delays and variational problems with delays. About half of this material, or one-third of the book, is devoted to problems with delays, and it provides a readable and broad introduction to this currently popular field of study.
Topics discussed for both ordinary and delay-differential equations include existence theory, methods of approximate solution, dependence of solutions on a small coefficient of the derivative, oscillation theorems, and stability theory. Some results are proved in full, but many are stated without proof, and references are given to papers containing the proofs. Most of the material given is still up to date with the exception of that concerning Ljapunov's second method for delay-differential equations (see #290 below). (MathSciNet)