The authors consider the nonlinear difference equation
\[ x_{n+1}=\alpha x_n + x_{n-k}f(x_{n-k}), \quad n=0, 1,\dots .1 \text{where} \alpha \in (0, 1),\hspace{5.0pt}k \in \lbrace 0, 1, \dots \rbrace \hspace{5.0pt}\text{and}\hspace{5.0pt}f\in C^1[[0, \infty),[0, \infty )] \qquad \mathrm{(0)}\] with $f^{\prime }(x)<0$. They give sufficient conditions for the unique positive equilibrium of (0.1) to be a global attractor of all positive solutions. The results here are somewhat easier to apply than those of other authors. An application to a model of blood cell production is given.
By means of Riccati transformation techniques, we establish some new oscillation criteria for second-order nonlinear difference equation which are sharp.