RATIONAL DIFFERENCE EQUATIONS WITH POSITIVE EQUILIBRIUM POINT

Title & Authors
RATIONAL DIFFERENCE EQUATIONS WITH POSITIVE EQUILIBRIUM POINT
Dubickas, Arturas;

Abstract
In this note we study positive solutions of the mth order rational difference equation $\small{x_n=(a_0+\sum{{m\atop{i=1}}a_ix_{n-i}/(b_0+\sum{{m\atop{i=1}}b_ix_{n-i}}$, where n = m,m+1,m+2, $\small{\ldots}$ and $\small{x_0,\ldots,x_{m-1}}$ > 0. We describe a sufficient condition on nonnegative real numbers $\small{a_0,a_1,\ldots,a_m,b_0,b_1,\ldots,b_m}$ under which every solution $\small{x_n}$ of the above equation tends to the limit $\small{(A-b_0+\sqrt{(A-b_0)^2+4_{a_0}B}}$/2B as $\small{n{\rightarrow}{\infty}}$, where $\small{A=\sum{{m\atop{i=1}}\;a_i}$ and $\small{B=\sum{{m\atop{i=1}}\;b_i}$.
Keywords
difference equations;equilibrium point;convergence of sequences;upper and lower limits;
Language
English
Cited by
1.
On the Difference equation xn+1=axn−l+bxn−k+cxn−sdxn−s−e, Mathematical Methods in the Applied Sciences, 2017, 40, 3, 535
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