Let \[ x_{n+1}=f(x_n,x_{n-1},\dots,x_{n-k}),\;n\in \mathbb{N}=\{0,1,2,\dots\}\tag{\(*\)} \] be a scalar difference equation, where \(k\) is a positive integer and \(f:\mathbb{R}^{k+1}\to \mathbb{R}\) is a continuous function. Using the concept of discrete exponential ordering, the authors are able to show under some conditions that the boundedness of all solutions as well as the local and global stability of (\(*\)) hold if and only if they hold for the equation \(x_{n+1}=h(x_n)\), where \(h(x)=f(x,x,\dots,x)\). The authors apply their results to the equations: \[ x_{n+1}=bx_n+g(x_n-x_{n-1}), \quad 0<b<1,\tag{A} \] where \(f:\mathbb R\to\mathbb R\) is a nondecreasing function and \[ x_{n+1}=ax_n+H(x_{n-k}), \quad 0<a<1, \tag{B} \] where \(H:(0,\infty )\to (0,\infty )\) is differentiable.
For equation (B), the following interesting result is obtained.
Theorem: Assume that \(H^{\prime}(x)\geq -a^{k+1}\frac{k^k}{(k+1)^{k+1}}\) for \(x>0\). If equation (B) has a unique equilibrium \(x^{*},\) then \(x^{*}\) is globally asymptotically stable if and only if \(H(x)<(1-a)x\) for \(x\in (x^{*},\infty)\) and \(H(x)>(1-a)x\) for \(x\in (0,x^{*})\).