# RATIONAL DIFFERENCE EQUATIONS WITH POSITIVE EQUILIBRIUM POINT

• Dubickas, Arturas
• 투고 : 2009.01.12
• 발행 : 2010.05.31
• 64 4

#### 초록

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 $B=\sum{{m\atop{i=1}}\;b_i$.

#### 키워드

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

#### 참고문헌

1. E. Camouzis, Global analysis of solutions of ${x_{n+1}}=\frac{{\beta}x_n+{\delta}x_{n-2}}{A+Bx_{n}+Cx_{n-1}}$, J. Math. Anal. Appl. 316 (2006), no. 2, 616-627. https://doi.org/10.1016/j.jmaa.2005.05.008
2. E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Advances in Discrete Mathematics and Applications, 5. Chapman & Hall/CRC, Boca Raton, FL, 2008.
3. M. R. S. Kulenovic and G. Ladas, Dynamics of Second Order Rational Difference Equations, With open problems and conjectures. Chapman & Hall/CRC, Boca Raton, FL, 2002.
4. J. Park, A global behavior of the positive solutions of ${x_{n+1}}=\frac{{\beta}x_n+x_{n-2}}{A+Bx_{n}+x_{n-2}}$, Commun. Korean Math. Soc. 23 (2008), no. 1, 61-65. https://doi.org/10.4134/CKMS.2008.23.1.061

#### 피인용 문헌

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