Global Attractiveness And Asymptotic Stability Of A Kind Of High Order Difference Equation

Difference equation is a sort of powerful means to investigate the rule of the discrete natural phenomena. In recent 10 years, therefore, the qualitative analysis of difference equation has always been the hot spot subject studied by the researchers home and abroad. Especially, an increasingly more number of results on rational difference equation, in view that which has a simpler form, are published.In this dissertation, we introduce a new method, which is called "subsequence analysis" , to investigate the global asymptotical stability of a kind of rational difference equations with high order. At first, we study a general difference equation:where l ≥ 1 is an integer, f ∈ C((0, ∞)^l, (0, ∞)), x_i-1 ∈ (0, ∞) for i = 1,2,…,l.Under certain assumptions, we obtain a set of sufficient conditions for the global attractiveness of equation (E). Then, when the above result is applied to the following difference equation:the global asymptotical stability is obtained if some conditions are satisfied, where k ∈ {1,2…},m,p,q ∈ {1,2, … , k} and x₀,x-1,… ,x-(k-1) are positive real initial values.As we can see that the difference equations studied in the cited references are all special cases of equation (E₁₀). Hence, our result includes and extends many corresponding results. Moreover, it is confirmed that the idea of "subsequence analysis" is quite novel and potent to prove the global asymptotical stability of this kind of rational difference equations with high order.