tailieunhanh - Báo cáo hóa học: "Global behavior of the solutions of some difference equations"

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: Global behavior of the solutions of some difference equations | Elabbasy et al. Advances in Difference Equations 2011 2011 28 http content 2011 1 28 o Advances in Difference Equations a SpringerOpen Journal RESEARCH Open Access Global behavior of the solutions of some difference equations Elmetwally M Elabbasy1 Hamdy A El-Metwally2 4 and Elsayed M Elsayed3 4 Correspondence emelabbasy@ 1Mathematics Department Faculty of Science Mansoura University Mansoura 35516 Egypt Full list of author information is available at the end of the article Abstract In this article we study the difference equation aXn-lXn-k Xn 1 ----- n 0 1 . bXn-p CXn-q where the initial conditions x-r x-r 1 x-r 2 . x0 are arbitrary positive real numbers r max l k p q is nonnegative integer and a b c are positive constants Also we study some special cases of this equation. Keywords Stability Solutions of the difference equations 1 Introduction The purpose of this article is to investigate the global attractivity of the equilibrium point and the asymptotic behavior of the solutions of the following difference equation aXn-lXn-k Xn 1 . n 0 1 . 1 bXn-p - CXn-q where the initial conditions x-r x-r 1 x-r 2 . x0 are arbitrary positive real numbers r max l k p q is nonnegative integer and a b c are positive constants Moreover we obtain the form of the solution of some special cases of Equation 1 and some numerical simulations to the equation are given to illustrate our results. Let us introduce some basic definitions and some theorems that we need in the sequel. Let I be some interval of real numbers and let f Ik 1 I be a continuously differentiable function. Then for every set of initial conditions x-k x-k 1 . x0 e I the difference equation Xn 1 f Xn Xn-1 . Xn-k n 0 1 . 2 has a unique solution Xnint-k 1 . A point X e I is called an equilibrium point of Equation 2 if X f X X . X . Springer 2011 Elabbasy et al licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons .