This paper studies the existence of at least three distinct solutions of the discrete boundary value problem \[ -\Delta\left(\phi_p(\Delta u(k-1))\right)+q_k\phi_p(u(k))=\lambda f(k,u(k))+\mu g(k,u(k))+h(u(k)),\;\;k\in[1,T], \]
\[ u(0)=u(T+1)=0, \] where \(1<p<+\infty\), \(\lambda>0\), \(\mu\geq0\), \(\phi_p(s)=|s|^{p-2}s\), \(T\geq 2\) is a fixed integer.
The first section is devoted to literature reviews. In the second section the authors give the preliminaries which include some definitions and known results.
The main results are in the third section. In Theorem 3.1, the authors achieve the sufficient conditions that will guarantee that the boundary value problem above has at least three distinct non-negative solutions. Moreover, they obtain a few additional theorems including a consequence of the main theorem and a special case of the consequence. They provide an example to demonstrate the results. At last, they point out one more consequence of the special case by the way of example.