Existence of Periodic Solutions for a Class of Fourth-order Difference
Equation

We apply the continuation theorem of Mawhin to ensure that a
fourth-order nonlinear difference equation of the form
$$\Delta^4 u(k-2)
-a(k)u^{\alpha}(k)+b(k)u^{\beta}(k)=0,$$
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,
$\alpha,\beta\in\mathbb{N}^+$
and $\alpha\neq\beta$.
$a(k),b(k)$ are $T$-periodic functions and
$a(k)b(k)>0$. As applications, we will give some examples
to illustrate the application of these theorems.