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

Abstract

We apply the continuation theorem of Mawhin to ensure that a fourth-order nonlinear difference equation of the form ${\Delta }^{4}u\left(k-2\right)-a\left(k\right)u^{\alpha }\left(k\right)+b\left(k\right)u^{\beta }\left(k\right)=0$ with periodic boundary conditions possesses at least one nontrivial positive solution, where $\Delta u\left(k\right)=u\left(k+1\right)-u\left(k\right)$ is the forward difference operator, $\alpha >0,\beta >0$ and $\alpha \ne \beta$ , $a\left(k\right),b\left(k\right)$ are $T$ -periodic functions and $a\left(k\right)b\left(k\right)>0$ . As applications, we give some examples to illustrate the application of these theorems.