POSITIVE SOLUTIONS OF DISCRETE BOUNDARY VALUE PROBLEM FOR A SECOND-ORDER NONLINEAR DIFFERENCE EQUATION WITH SINGULAR $\phi $-LAPLACIAN

We establish the nonexistence, existence and multiplicity of positive solutions of the following discrete boundary value problem for a second-order nonlinear difference equation with singular $ \phi $-Laplacian

$ \begin{equation*} \begin{cases} \begin{split} &\quad -\nabla(k^{N-1}\phi(\triangle v_k))\ \\&=\lambda Nk^{N-1}\left(\frac{f'(\varphi^{-1}(v_k))}{\sqrt{1-(\triangle v_k)^2}}-f(\varphi^{-1}(v_k))H(\varphi^{-1}(v_k), k)\right), k\in[2, n-1]_{\mathbb{Z}}, \\ &|\triangle v_k|<1, \\ &\triangle v_1=0=v_n, \end{split} \end{cases} \end{equation*} $

where $ \phi(s)=\frac{s}{\sqrt{1-s^2}} $, $ \phi: (-1, 1)\rightarrow \mathbb{R} $ is an increasing homeomorphism with $ \phi(0)=0 $, $ \lambda $ is a positive parameter, $ \triangle $ is the forward difference operator defined by $ \triangle v_k=v_{k+1}-v_k $, $ \nabla $ is the backward difference operator defined by $ \nabla v_k=v_k-v_{k-1} $, $ f\in C^{\infty}(I) $ and $ f>0 $, $ I $ is an open interval in $ \mathbb{R} $, $ \varphi(s)=\int_0^s\frac{dt}{f(t)} $, $ \varphi^{-1} $ is the inverse function of $ \varphi $, $ H:I\times[2, n-1]_{\mathbb{Z}}\rightarrow \mathbb{R} $ is a continuous function, $ [2, n-1]_{\mathbb{Z}}:={2, 3, \ldots, n-1} $, and the integer $ n\geq4 $. By using the method of lower and upper solutions, topological degree theory and Szulkin's critical point theory for convex, lower semicontinous perturbations of $ C^1 $-functionals, we determine the interval of parameter $ \lambda $ in which the above problem has zero, one, two positive solutions according to sublinear at zero.