In this paper we discuss the Morishima's example, which implies a kind of eventually asymptotical stability of solutions for a difference equation $x(n + 1) = f(x(n))$ for $n = 0, 1, 2, \cdots$. We define new definitions of eventual stability of periodic points in the meaning of the large in the same way as ones of Lakshmikantham et. al. and Yoshizawa. By applying the Lyapunov's second method we give eventual stability criteria in the large of the difference equation. In order to illustrate our main results on eventual stability an example of a set of 2-periodic points for eventual stability is given with an analytical estimation.