The global attractivity of a difference equation (Q2748238)

The present paper is devoted to the global attractivity of the equilibrium \(\widetilde x(=x_n)\equiv 1\) of the nonlinear difference equation NEWLINE\[NEWLINEx_{n+1}= \exp\bigl \{\alpha(1-x_n)/(1- \beta x_n)\bigr\},\;n\in \mathbb{N}_0 \text{ with }x_0\in \left(0,{1 \over\beta} \right), \tag{1}NEWLINE\]NEWLINE where \(\alpha >0\), \(0< \beta<1\) are real constants. It is shown that, if the condition \(\alpha\leq 1-\beta\) holds then all solution \(\{x_n\}\) \((n\in\mathbb{N})\) of (1) satisfy the relation \(x_n\in(0, {1\over \beta})\) and the equilibrium \(\widetilde x(=x_n)\equiv 1\) of (1) is asymptotically stable.NEWLINENEWLINENEWLINEThe authors prove also that the following conditions are sufficient for ensuring the attractivity of equilibrium \(\widetilde x\equiv 1\) of equation (1): NEWLINE\[NEWLINE{2\beta\over 1-\beta}<1 \text{ and }{2\beta \over 1-\beta}\leq\alpha =\min(\ln 2,1-\beta); \text{ or}\tag{i}NEWLINE\]NEWLINE NEWLINE\[NEWLINE{2\beta \over 1-\beta}\geq 1\text{ and }\alpha\leq 1-\beta.\tag{ii}NEWLINE\]NEWLINE The auxiliary function \(H(x)=x_0+ f(x_0)-x-f(x)\) with \(f(x):=\exp \{\alpha(1-x_n)/ (1-\beta x_n)\}\) and its first two derivatives are used in the proof.