Global asymptotic stability for two kinds of higher order recursive sequences

In this paper, the method named subsequence analysis is introduced to study the global asymptotic stability of the following two families of difference equations: x(n) = x(n-k)x(n-l) + a/x(n-k) + x(n-l), n = 1, 2, ..., and x(n) = x(n-k) + x(n-l)/1 + ax(n-k)x(n-l), n = 1, 2, ..., where k and l are integers, 1 <= k <= l, a > 0, x(0), x(-1), ..., x(-(l-1)) is an element of (0,8). This method is novel and can be used to simplify the proof of similar kinds of stability problems with complicated semi-cycle structure.