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Báo cáo hóa học: "Research Article Results and Conjectures about Order q Lyness’ Difference Equation un q un a un q−1 · · · un 1 in R∗ , with a Particular Study of the Case q 3"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Results and Conjectures about Order q Lyness’ Difference Equation un q un a un q−1 · · · un 1 in R∗ , with a Particular Study of the Case q 3 | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 134749 36 pages doi 10.1155 2009 134749 Research Article Results and Conjectures about Order q Lyness Difference Equation un qUn a un q-1 un 1 in R with a Particular Study of the Case q 3 G. Bastien1 2 and M. Rogalski2 3 1 Institut de Mathematiques de Jussieu UPMC-Paris 06 UMR-CNRS 7586 75251 Paris France 2 Equipe d Analyse fonctionnelle IMJ 16 rue Clisson 75013 Paris France 3 Laboratoire Paul Painleve USTL Lille 1 UMR-CNRS 8524 59655 Villeneuve France Correspondence should be addressed to G. Bastien bastien@upmc.fr Received 4 March 2009 Accepted 14 July 2009 Recommended by Istvan Gyori We study order q Lyness difference equation in R un qUn a un q-1 -I 1- un 1 with a 0 and the associated dynamical system Fa in R q. We study its solutions divergence permanency local stability of the equilibrium . We prove some results about the first three invariant functions and the topological nature of the corresponding invariant sets about the differential at the equilibrium about the role of 2-periodic points when q is odd about the nonexistence of some minimal periods and so forth and discuss some problems related to the search of common period to all solutions or to the second and third invariants. We look at the case q 3 with new methods using new invariants for the map F2 and state some conjectures on the associated dynamical system in R q in more general cases. Copyright 2009 G. Bastien and M. Rogalski. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction We will study the global behavior of solutions of the order q Lyness difference equation for q 3 and a 0 a un q-1 un 1 un q f u0f u1f f uq-1 0- un 1.1 The associated dynamical system in R q is given by the map Fa Fa X x1f f x - Fa j a X1 Xq-1 fX1fX2 f.fXq-1 Xq 1.2