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

Sukanya Basu,Orlando Merino

doi:10.1155/2009/128602

Abstract

For nonnegative real numbers , , , , , and such that and , the difference equation , has a unique positive equilibrium. A proof is given here for the following statements: (1) For every choice of positive parameters , , , , , and , all solutions to the difference equation , converge to the positive equilibrium or to a prime period-two solution. (2) For every choice of positive parameters , , , , and , all solutions to the difference equation , converge to the positive equilibrium or to a prime period-two solution.

Highlights

Introduction and Main ResultsIn their book 1, Kulenovicand Ladas initiated a systematic study of the difference equation xn 1 α A βxn Bxn γ xn−1 Cxn−1, n1.1 for nonnegative real numbers α, β, γ, A, B, and C such that B C > 0 and α β γ > 0, and for nonnegative or positive initial conditions x−1, x0
There are a total of 42 cases that arise from 1.1 in the manner just discussed, under the hypotheses B C > 0 and α β γ > 0
The results in this subsection are from literature, and they are given here for easy reference

Summary

Introduction and Main Results

In their book 1 , Kulenovicand Ladas initiated a systematic study of the difference equation xn 1 α A βxn Bxn γ xn−1 Cxn−1. One of their main ideas in this undertaking was to make the task more manageable by considering separate cases when one or more of the parameters in 1.1 is zero. Consider the following affine change of variables which is helpful to reduce number of parameters and simplify calculations: xn γ C. The two main differences between 1.15 and 1.13 are the set of initial conditions, and the possibility of having a negative value of r in 1.15 , while only positive values of r are allowed in 1.13 For both 1.15 and 1.13 the unique equilibrium has the formula: p1 y p 1 2 4r q 1. Our main results Theorems 1.2 and 1.3 imply that when prime period-two solutions to 1.11 or 1.13 do not exist, the unique equilibrium is a global attractor. We refer the reader to 1 for terminology and definitions that concern difference equations

Results from Literature

Existence of an Invariant and Attracting Interval

14 Then a calculation shows