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RATIONAL DIFFERENCE EQUATIONS WITH POSITIVE EQUILIBRIUM POINT

  • Dubickas, Arturas (Department of Mathematics and Informatics Vilnius University)
  • Received : 2009.01.12
  • Published : 2010.05.31

Abstract

In this note we study positive solutions of the mth order rational difference equation $x_n=(a_0+\sum{{m\atop{i=1}}a_ix_{n-i}/(b_0+\sum{{m\atop{i=1}}b_ix_{n-i}$, where n = m,m+1,m+2, $\ldots$ and $x_0,\ldots,x_{m-1}$ > 0. We describe a sufficient condition on nonnegative real numbers $a_0,a_1,\ldots,a_m,b_0,b_1,\ldots,b_m$ under which every solution $x_n$ of the above equation tends to the limit $(A-b_0+\sqrt{(A-b_0)^2+4_{a_0}B}$/2B as $n{\rightarrow}{\infty}$, where $A=\sum{{m\atop{i=1}}\;a_i$ and $B=\sum{{m\atop{i=1}}\;b_i$.

Keywords

difference equations;equilibrium point;convergence of sequences;upper and lower limits

References

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Cited by

  1. On the Difference equation xn+1=axn−l+bxn−k+cxn−sdxn−s−e vol.40, pp.3, 2017, https://doi.org/10.1002/mma.3980