On the rational difference equation x_{n}=1+frac{(1-x_{n-k})(1-x_{n-l})(1-x_{n-m})}{x_{n-k}+x_{n-l}+x_{n-m}} |
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Authors: | Maoxin Liao Xianhua Tang Changjin Xu |
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Affiliation: | 1. School of Mathematics and Physics, University of South China, Hengyang, Hunan, 421001, P.R. China 2. School of Mathematical Sciences and Computing Technology, Central South University, Changsha, Hunan, 410083, P.R. China
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Abstract: | In this note we consider the following higher order rational difference equations $$x_{n}=1+frac{(1-x_{n-k})(1-x_{n-l})(1-x_{n-m})}{x_{n-k}+x_{n-l}+x_{n-m}},quad n=0,1,ldots,$$ where 1≤k<l<m, and the initial values x ?m ,x ?m+1,…,x ?1 are positive numbers. We give some sufficient conditions for the persistence of positive solutions for the above equation, and prove that the positive equilibrium point of this equation is globally asymptotically stable. |
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