Abstract
Our goal in this paper is to investigate the global asymptotic stability of the hyperbolic equilibrium solution of the second order rational difference equation xn+1=α+βxn+γxn-1/A+Bxn+Cxn-1, n=0,1,2,…, where the parameters A≥0 and B, C, α, β, γ are positive real numbers and the initial conditions x-1, x0 are nonnegative real numbers such that A+Bx0+Cx-1>0. In particular, we solve Conjecture 5.201.1 proposed by Camouzis and Ladas in their book (2008) which appeared previously in Conjecture 11.4.2 in Kulenović and Ladas monograph (2002).
Highlights
IntroductionKulenovicand Ladas [15] derived several ones on the boundedness, the global stability, and the periodicity of solutions of all rational difference equations of the form (2)
Rational difference equations, bilinear ones, that is, xn+1 = α β + +∑ki=0 ∑lj=0 αixn−i βjxn−j ;n = 0, 1, 2, 3, . . . (1)attracted the attention of many researchers recently
We have established the global stability of the hyperbolic equilibrium solutions of the second order rational difference equation xn+1
Summary
Kulenovicand Ladas [15] derived several ones on the boundedness, the global stability, and the periodicity of solutions of all rational difference equations of the form (2). They posed several open problems and conjectures related to this equation and its functional generalization. The work done by many researchers such as [31,32,33,34,35,36,37,38,39] have solved many open problems and conjectures proposed in [14, 15, 29, 30] related to (2) and have led to the development of some general theory about difference equation.
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