The authors obtain some sufficient conditions such that the following discrete Neumann boundary value problem will admit at least one solution: \[ \nabla \Delta y(k)=f(k,y(k),\Delta y(k)),\quad k=1,\dots,n-1;\quad \Delta y(0)=0=\Delta y(n), \] where \(f\) is a continuous scalar-valued function, \(n\geq 2\) and \[ \Delta y(k):=\begin{cases} y(k+1)-y(k),&\text{for }k=0,\dots,n-1\\ 0,&\text{for }k=n;\end{cases} \] \[ \Delta y(k):=\begin{cases} y(k)-y(k-1),&\text{for }k=1,\dots,n\\ 0,&\text{for }k=0;\end{cases} \] \[ \nabla \Delta y(k):=\begin{cases} y(k+1)-2y(k)+y(k-1),&\text{for }k=1,\dots,n-1\\ 0,&\text{for }k=n\text{ or }k=0.\end{cases} \]