Stability of Hyperbolic Equilibrium Solution of Second Order Nonlinear Rational Difference Equation

Abstract

Our goal in this paper is to investigate the global asymptotic stability of the hyperbolic equilibrium solution of the second order rational difference equationxn+1=α+βxn+γxn-1/A+Bxn+Cxn-1,n=0,1,2,…, where the parametersA≥0andB,C,α,β,γare positive real numbers and the initial conditionsx-1, x0are nonnegative real numbers such thatA+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).

Stability of Nonhyperbolic Equilibrium Solution of Second Order Nonlinear Rational Difference Equation

“…Our approach handles the aforementioned case as well as other cases. Furthermore, the results in this paper, together with the established results in [1,2,4], give a complete picture of the nature of solutions of the second order rational difference equation of form (1).…”

“…In part 1 of this investigation [1], we have established the global stability of the hyperbolic equilibrium solution of the second order rational difference equation:…”

“…Many rational difference equations were studied extensively in [3]. A systematic study of the second order rational difference equation of form (1), where the parameters , , , , , and the initial conditions −1 , 0 are nonnegative real numbers, was considered in the monograph of Kulenovic and Ladas [4]. They presented the known results up to 2002.…”

“…They presented the known results up to 2002. Next, Kulenovic and Ladas [4] derived several ones on the boundedness, the global stability, and the periodicity of solutions of all rational difference equations of form (1). Furthermore, they posed several open problems and conjectures related to this equation and its functional generalization.…”

“…Camouzis and Ladas in [12] summarize the progress up to 2008. Recently, the work done by many researchers such as [13][14][15][16][17][18][19][20][21] have solved many open problems and conjectures proposed in [4,[10][11][12] related to (1) and have led to the development of some general theory about difference equation. However, as confirmed by Professor Kulenovic (personal communication, August, 24, 2014), the case = 0 remains open.…”