We apply the continuation theorem of Mawhin to ensure that a
fourth-order nonlinear difference equation of the form
with periodic boundary conditions possesses at least one nontrivial
positive solution, where $\Delta u(k)=u(k+1)-u(k)$ is
the forward difference operator,
$a(k),b(k)$ are $T$-periodic functions and
$a(k)b(k)>0$. As applications, we will give some examples
to illustrate the application of these theorems.