Global Behavior of Solutions to Two Classes of Second-Order Rational Difference Equations

• Main Authors: Basu Sukanya, Merino Orlando
• Format: Article
• Language: English
• Published: SpringerOpen 2009-01-01
• Series: Advances in Difference Equations
• Online Access: http://www.advancesindifferenceequations.com/content/2009/128602

<p>Abstract</p> <p>For nonnegative real numbers <inline-formula> <graphic file="1687-1847-2009-128602-i1.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1847-2009-128602-i2.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1847-2009-128602-i3.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1847-2009-128602-i4.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1847-2009-128602-i5.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1847-2009-128602-i6.gif"/></inline-formula> and <inline-formula> <graphic file="1687-1847-2009-128602-i7.gif"/></inline-formula> such that <inline-formula> <graphic file="1687-1847-2009-128602-i8.gif"/></inline-formula> and <inline-formula> <graphic file="1687-1847-2009-128602-i9.gif"/></inline-formula>, the difference equation <inline-formula> <graphic file="1687-1847-2009-128602-i10.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1847-2009-128602-i11.gif"/></inline-formula> has a unique positive equilibrium. A proof is given here for the following statements: (1) <it>For every choice of positive parameters</it><inline-formula> <graphic file="1687-1847-2009-128602-i12.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1847-2009-128602-i13.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1847-2009-128602-i14.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1847-2009-128602-i15.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1847-2009-128602-i16.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1847-2009-128602-i17.gif"/></inline-formula> and <inline-formula> <graphic file="1687-1847-2009-128602-i18.gif"/></inline-formula>, <it>all solutions to the difference equation</it><inline-formula> <graphic file="1687-1847-2009-128602-i19.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1847-2009-128602-i20.gif"/></inline-formula><inline-formula> <graphic file="1687-1847-2009-128602-i21.gif"/></inline-formula><it>converge to the positive equilibrium or to a prime period-two solution</it>. (2) <it>For every choice of positive parameters</it><inline-formula> <graphic file="1687-1847-2009-128602-i22.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1847-2009-128602-i23.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1847-2009-128602-i24.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1847-2009-128602-i25.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1847-2009-128602-i26.gif"/></inline-formula> and <inline-formula> <graphic file="1687-1847-2009-128602-i27.gif"/></inline-formula>, <it>all solutions to the difference equation</it><inline-formula> <graphic file="1687-1847-2009-128602-i28.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1847-2009-128602-i29.gif"/></inline-formula><inline-formula> <graphic file="1687-1847-2009-128602-i30.gif"/></inline-formula><it>converge to the positive equilibrium or to a prime period-two solution</it>.</p>