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Zbl 1119.39007
Kipnis, M.M.; Komissarova, D.A.
A note on explicit stability conditions for autonomous higher order difference equations. (English)
[J] J. Difference Equ. Appl. 13, No. 5, 457-461 (2007). ISSN 1023-6198

Using an elementary approach based on the associated characteristic equation, the authors prove that the linear difference equation of order $K\geq 1$ with constant coefficients $$ x_n-x_{n-1}+\sum_{i=1}^Ka_i x_{n-i}=0\tag{1} $$ is asymptotically stable if $a_i\geq 0$ for all $i=1,2,\dots , K$, and $$0<\sum_{i=1}^K\frac{a_i}{2\sin(\pi/(4i-2)}<1.$$ This result generalizes a well-known stability criterion of {\it Simon A. Levin} and {\it Robert M. May} [Theor. Population Biology 9, 178--187 (1976; Zbl 0338.92021)] for equation $$ x_{n}-x_{n-1}+ax_{n-K}=0,\, a>0. $$ As a corollary, the authors show that Eq. (1) is asymptotically stable if $a_i\geq 0$ for all $i=1,2,\dots , K$, and $$ 0<\sum_{i=1}^K ia_i\leq\frac{\pi}{2}. $$ Moreover, the constant $\pi/2$ cannot be replaced by a greater one.
[Eduardo Liz (Vigo)]

MSC 2000:: *39A11 Stability of difference equations

Keywords: linear difference equations; characteristic equation; asymptotic stability

Citations: Zbl 0338.92021

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