Solutions of the Difference Equation xn+1=xnxn-1-1

and Applied Analysis 3 Proof. We can write terms of a period-three solution of 1.1 as x−1 a, x0 b, x1 ab − 1, x2 b ab − 1 − 1 a ⇐⇒ ab2 − b − 1 a ⇐⇒ ab2 a b 1, x3 a ab − 1 − 1 b ⇐⇒ a2b − a − 1 b ⇐⇒ a2b a b 1. 2.4 Therefore, this is indeed a solution of period three if the system below is satisfied. ab2 a b 1, a2b a b 1. 2.5 Hence we see that ab2 a2b. Thus, this is true if a 0, or if b 0, or if a b when a/ 0 and b / 0. Now it suffices to consider the following three cases. Case 1. Suppose that a 0. Then a2b a b 1 implies that 0 b 1 and hence gives us b −1. Therefore, a period-three solution exists if a 0 and b −1. Case 2. Suppose that b 0. Then a2b a b 1 implies that 0 a 1 and hence gives us a −1. Therefore, a period-three solution exists if a −1 and b 0. Case 3. Suppose that a b such that a/ 0 and b / 0. Then a2b a b 1 implies that a3 2a 1, which gives us a −1. Therefore, a period-three solution exists if a −1 and b −1. Note that a 1 ± √5 /2 are also solutions of a3 2a 1, which are the two equilibria of 1.1 . Hence, there exist exactly three periodic solutions with minimal period three of 1.1 given by x−1 −1, x0 −1, x1 0, . . . ; x−1 −1, x0 0, x1 −1, . . . ; x−1 0, x0 −1, x1 −1, . . . , 2.6 as claimed. In the sequel, we will refer to any one of these three periodic solutions of 1.1 as . . . , 0,−1,−1, 0,−1,−1, . . . . 2.7 The following theorem demonstrates, in a similar way as in 14, 15 , that 1.1 has solutions that are eventually periodic with minimal period three. 4 Abstract and Applied Analysis Theorem 2.2. All eventually periodic solutions with minimal period three are of the form x−1, x0, . . . , xN, xN 1, 0,−1,−1, 0,−1,−1, 0,−1,−1, . . . , 2.8 WhereN ≥ −1, xN 1 a ∈ R\{0}, xN 1/a, and, ifN/ −1, xn−1 xn 1 1 /xn for 0 ≤ n ≤ N. Proof. If {xn}n −1 is an eventually periodic solution with minimal period three, then by Theorem 2.1 there is an N ≥ −1 such that xN 2 0 and xN 3 −1. Hence 0 xN 2 xN 1xN − 1 and consequently xN / 0/ xN 1 and xN 1 1/xN . Let xN 1 a which implies that xN 1/a . From 1.1 , if N/ − 1, we get xn−1 xn 1 1 /xn, for 0 ≤ n ≤ N, as desired. Remark 2.3. If, in Theorem 2.2, a −1, then, among some others, we obtain the solutions in Theorem 2.1. The next theorem shows that no periodic or eventually periodic solution of 1.1 converges to a minimal period-three solution. Theorem 2.4. Let {xn}n −1 be a solution of 1.1 that is neither periodic nor eventually periodic with minimal period three. Then {xn}n −1 does not converge to the minimal period-three solution . . . , 0,−1,−1, 0,−1,−1, . . . . 2.9 Proof. Let {xn}n −1 be a solution of 1.1 that is neither periodic nor eventually periodic with minimal period three. For the sake of contradiction, assume that {xn}n −1 converges to the minimal period-three solution . . . , 0,−1,−1, 0,−1,−1, . . . . 2.10 Then we can choose N ≥ 0 such that there exist − 2 < δ1, δ2, δ3, . . . < 1 2 , −1 < 0, 1, 2, 3, . . . < 1, 2.11