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On boundedness of solutions of the difference equation x_{n+1}=p+frac{x_{n-1}}{x_{n}} for p<1 |
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| Authors: | Taixiang Sun Xin Wu Qiuli He Hongjian Xi |
| Affiliation: | 1. College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi, 530004, P.R. China 2. College of Electrical Engineering, Guangxi University, Nanning, Guangxi, 530004, P.R. China 3. Department of Mathematics, Guangxi College of Finance and Economics, Nanning, Guangxi, 530003, P.R. China |
| Abstract: | In this paper, we study the difference equation $$x_{n+1}=p+frac{x_{n-1}}{x_n}, quad n=0,1,ldots, $$ where initial values x ?1,x 0∈(0,+∞) and 0<p<1, and obtain the set of all initial values (x ?1,x 0)∈(0,+∞)×(0,+∞) such that the positive solutions ${x_{n}}_{n=-1}^{infty}$ are bounded. This answers the Open problem 4.8.11 proposed by Kulenovic and Ladas (Dynamics of Second Order Rational Difference Equations, with Open Problems and Conjectures, 2002). |
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