# Stability and periodicity of some difference equations and applications

#### Abstract

We investigate the boundedness and persistence and asymptotic behavior of the two nonautonomous difference equations\eqalignno{x\sb{n+1}&={\alpha\sb{n}x\sb{n}+b\sb{n}\over x\sb{n-1}},\quad n=0,1,\...\cr \noalign{\hbox{\rm and}}y\sb{n+1}&={\max\{a\sb{n}y\sb{n},b\sb{n}\}\over y\sb{n-1}},\quad n=0,1,\...\cr}with nonnegative coefficients $\{a\sb{n}\}\sbsp{n=0}{\infty}$ and $\{b\sb{n}\}\sbsp{n=0}{\infty}$ and positive initial conditions.^ We also investigate the periodicity nature of equations which are natural extensions of the above, namely,$$x\sb{n+1}={a\sbsp{n}{k}+\sum\sbsp{i=0}{k-1}a\sbsp{n}{i}x\sb{n -i}\over x\sb{n-k}},\quad n=0,1,\...$$with nonnegative coefficients and positive initial conditions.^ In addition, we look at the global properties of the following two equations:$$x\sb{n+1}=\cases{{\alpha x\sb{n}+\beta x\sb{n-1}\over2},\ &{\rm if}\ {\it x\/}\sb{n} + {\it x\/}\sb{n-1}\ {\rm is\ even},\cr &\qquad\qquad\qquad\qquad\qquad n\ =\ 0,1,\...,\cr \gamma x\sb{n} + \delta x\sb{n-1},\quad &{\rm if}\ {\it x\/}\sb{n} + {\it x\/}\sb{n-1}\ {\rm is\ odd},\cr}$$with $\alpha,\beta,\gamma,\delta\in\{{-}1,1\}$ and $x\sb{-1}, x\sb0\in\doubz;$ and$$y\sb{n+1}=(\alpha y\sb{n}+\beta y\sb{n-1})e\sp{-y\sb{n}},\ n=0,1,\...$$with $\alpha\in\lbrack0,1),\ \beta\in(0,\infty),$ and $y\sb{-1},\ y\sb0\in\lbrack0,\infty).$ The first has its origins in number theory, and the second has applications to mathematical biology. ^

Mathematics

#### Recommended Citation

Candace Marie Kent, "Stability and periodicity of some difference equations and applications" (1998). Dissertations and Master's Theses (Campus Access). Paper AAI9902568.
http://digitalcommons.uri.edu/dissertations/AAI9902568

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