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标题：On the difference equation x n + 1 = a x n − l + b x n − k + f ( x n − l , x n − k ) $x_{n+1}=ax_{n-l}+bx_{n-k}+f ( x_{n-l},x_{n-k} )$
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作者：Mahmoud A. E. Abdelrahman ; George E. Chatzarakis ; Tongxing Li 等
期刊名称：Advances in Difference Equations
印刷版ISSN：1687-1839
电子版ISSN：1687-1847
出版年度：2018
卷号：2018
期号：1
页码：431
DOI：10.1186/s13662-018-1880-8
语种：English
出版社：Hindawi Publishing Corporation
摘要：In this paper, we study the asymptotic behavior of the solutions of a new class of difference equations x n + 1 = a x n − l + b x n − k + f ( x n − l , x n − k ) , $$x_{n+1}=ax_{n-l}+bx_{n-k}+f ( x_{n-l},x_{n-k} ), $$ where l and k are nonnegative integers, a and b are nonnegative real numbers, the initial values x − s , x − s + 1 , … , x 0 $x_{-s}, x_{-s+1},\ldots, x_($ are positive real numbers, s = max { l , k } $s=\max\{l,k\}$ , and f ( u , v ) : ( 0 , ∞ ) 2 → ( 0 , ∞ ) $f (u,v ): ( 0,\infty ) ^,\rightarrow ( 0,\infty ) $ is a continuous and homogeneous real function of degree zero. We consider the stability, boundedness, and periodicity of the solutions of this equation which is the most general form of linear difference equations. Thus, the results in this paper apply to several other equations that are special cases of the studied equation. Moreover, we present a new method to study periodic solutions of period two.
关键词：Difference equation ; Equilibrium point ; Local stability ; Periodic solution