Non-autonomous homogeneous rational difference equations of degree one: convergence and monotone solutions for second and third order case

作者：Mazrooei Sebdani Reza^*

来源：Mathematical Methods in the Applied Sciences, 2014, 37(4): 518-523.

DOI：10.1002/mma.2810

• homogeneous
• rational
• non-autonomous
• quadratic
• monotonic
• non-monotonic
• period two

Consider the non-autonomous equations: %26lt;br%26gt;Xn+1 = X-n (A(n)X(n) + BnXn-1/alpha X-n(n) + beta X-n(n-1)), alpha X-n(n) + beta X-n(n-1) %26gt; 0, %26lt;br%26gt;Xn+1 = Xn-1 (BnXn + DnXn-2/alpha X-n(n) + gamma X-n(n-2)), alpha X-n(n) + gamma X-n(n-2) %26gt; 0, %26lt;br%26gt;Xn+1 = X-n (BnXn-1 + CnXn-2/beta X-n(n) + gamma X-n(n-2)), beta X-n(n) + gamma X-n(n-2) %26gt; 0, %26lt;br%26gt;where %26lt;br%26gt;A(n) %26gt;= 0, B-n %26gt;= 0, C-n %26gt;= 0, alpha(n) %26gt;= 0, beta(n) %26gt;= 0, gamma(n) %26gt;= 0, n = 0,1,2, ... , %26lt;br%26gt;and also %26lt;br%26gt;lim(n -%26gt;infinity)A(n) = A %26gt;= 0, lim(n -%26gt;infinity) B-n = B-n %26gt;= 0, lim(n -%26gt;infinity)C(n) = C-n %26gt;= 0, lim(n -%26gt;infinity)alpha(n) = alpha(n) %26gt;= 0, lim(n -%26gt;infinity)beta(n) = beta(n) %26gt;= 0, lim(n -%26gt;infinity)gamma(n) = gamma(n) %26gt;= 0. %26lt;br%26gt;These are some non-autonomous homogeneous rational difference equations of degree one. A reduction in order is considered. Convergence andmonoton character of positive solutions are studied.

• 出版日期2014-3-15