The Periodic Character of the Difference Equation xn+1=f(xn-l+1,xn-2k+1)
نویسندگان
چکیده
In this paper, we consider the nonlinear difference equation xn 1 f xn−l 1, xn−2k 1 , n 0, 1, . . . , where k, l ∈ {1, 2, . . . } with 2k / l and gcd 2k, l 1 and the initial values x−α, x−α 1, . . . , x0 ∈ 0, ∞ with α max{l − 1, 2k − 1}. We give sufficient conditions under which every positive solution of this equation converges to a not necessarily prime 2-periodic solution, which extends and includes corresponding results obtained in the recent literature.
منابع مشابه
Long-Term Behavior of Solutions of the Difference Equation xn+1=xn-1xn-2-1
and Applied Analysis 3 Proof. Assume that xN xN 2k and xN 1 xN 2k 1, for every k ∈ N0, and some N ≥ −2, with xN / xN 1. Then, we have xN 4 xN 2xN 1 − 1 xNxN 1 − 1 xN 3 xN 1. 2.4 From this and since xN 4 xN , we obtain a contradiction, finishing the proof of the result. Theorem 2.3. There are no periodic or eventually periodic solutions of 1.1 with prime period three. Proof. If xN xN 3k, xN 1 xN...
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