Boundedness, periodicity, and stability of some difference equations

Abstract

We investigate the boundedness character, the periodic nature, and the stability behavior of solutions of three difference equations. The first equation is$$x\sb{n+1} = {\rm max}\left\{{A\over {x\sb{n}}},{B\over {x\sb{n-1}}}\right\},\ n = 0,1,\...$$where the parameters A and B and the initial conditions $x\sb{-1}$ and $x\sb0$ are nonzero real numbers. We obtain necessary and sufficient conditions for every solution to be eventually periodic. We also give a precise description of the period in terms of A, B, and the initial conditions.^ The second equation is$$x\sb{n+1} = {{x\sb{n}+x\sb{n-1}x\sb{n-2}}\over {x\sb{n}x\sb{n-1}+x\sb{n-2}}},\quad n = 0,1,\...$$where the initial conditions $x\sb{-2},\ x\sb{-1}$, and $x\sb0$ are positive real numbers. We present a detailed analysis of the semicycles of solutions and show that the equilibrium of the equation is globally asymptotically stable. We also extend the global stability results to a general class of difference equations of the form$$x\sb{n+1} = f(x\sb{n},\ x\sb{n-1},\...,\ x\sb{n-k}),\ n = 0,1,\...$$where the function f satisfies the strong negative feedback property.^ Finally we investigate the boundedness character of the positive solutions of the Plant-Herbivore System\left.\eqalign{x\sb{n+1} &= {\alpha x\sb{n}\over\beta x\sb{n}+e\sp{y\sb{n}}}\cr y\sb{n+1} &= \gamma(x\sb{n}+1)y\sb{n}\cr}\right\}\quad ,n = 0,1\...where $\alpha\in (1,\infty),\beta\in (0,\infty$), and $\gamma\in$ (0,1) and the initial conditions $x\sb0$ and $y\sb0$ are arbitrary positive numbers. ^

Mathematics

Recommended Citation

Amal M Al-Amleh, "Boundedness, periodicity, and stability of some difference equations" (1998). Dissertations and Master's Theses (Campus Access). Paper AAI9908234.
http://digitalcommons.uri.edu/dissertations/AAI9908234

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