This paper is devoted to study the discrete Sturm-Liouville problem $$ \left\{\begin{array}{ll} -\Delta(p(k)\Delta u(k-1))+q(k)u(k)=\lambda m(k)u(k)+f_1(k,u(k),\lambda)+f_2(k,u(k),\lambda),\ \ k\in[1,T]_Z,\\[2ex] a_0u(0)+b_0\Delta u(0)=0,\ a_1u(T)+b_1\Delta u(T)=0, \end{array}\right. $$ where $\lambda\in\mathbb{R}$ is a parameter, $f_1, f_2\in C([1,T]_Z\times\mathbb{R}^2, \mathbb{R})$, $f_1$ is not differentiable at the origin and infinity. Under some suitable assumptions on nonlinear terms, we prove the existence of unbounded continua of positive and negative solutions of this problem which bifurcate from intervals of the line of trivial solutions or from infinity, respectively.

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.

We apply the continuation theorem of Mawhin to ensure that a second-order nonlinear difference equation of the form $$\Delta^2 u(k-1) +a(k)u^{n}(k)-b(k)u^{n+1}(k)+c(k)u^{n+2}(k)=0$$ with periodic boundary conditions possesses at least two nontrivial positive solutions, where $n>0$ is a finite positive integer, $\Delta u(k)=u(k+1)-u(k)$ is the forward difference operator and $a(k),b(k),c(k)$ are $T$-periodic functions on $\mathbb{R}$.