The authors present sufficient conditions for the uniform stability and uniformly asymptotic stability of the zero solution of the difference equation \[ x(n+ 1)- \lambda x(n)+ f(n,x_n)= 0,\quad n\in\mathbb{N},\tag{1} \] where \(\lambda\in[0,1]\) and \(f\) is a functional depending on \(x_n\) defined by \(x_n(m):= x(n+ m)\) for \(-k\leq m\leq 0\), \(k\in\mathbb{N}\). The equation (1) includes as the special case the linear equation \[ x(n+ 1)- x(n)+ P_nx(n- k)= 0,\quad n\in\mathbb{N}. \] If \[ \sum^n_{r= n-k} \lambda^{n-r}p_r\leq 1+{k+2\over 2(k+1)} \lambda^{k+1},\quad n\in\mathbb{N}(k),\tag{2} \] where \(\{p_r\}\) is a sequence of nonnegative real numbers and \(f\) satisfies an additional condition (\(f\) is bounded by \(p_r\)), then the zero solution of (1) is uniformly stable for \(\lambda\in [0,1]\). If \(\lambda\in [0,1)\), then under the condition (2), the zero solution of (1) is uniformly asymptotically stable. The authors use a direct method to prove their presented results instead of Lyapunov function techniques.