Existence of Solutions to Nonlinear 2nth-order Discrete Boundary Value Problem with Parameter Dependence

This paper presents several sufficient conditions for the existence of at least one solution and at least one nonzero solution to the boundary value problem $$\begin{aligned} \Delta ^{n} (p(k) \Delta ^{n}x(k-n)) = \lambda f(k, x(k)), & k \in {\mathbb N}[n,T] \\ x(0)=x(1)=\ldots =x(n-1) =0, & \\ x(T+1)=x(T+2)= \ldots = x(T+n)=0, & \end{aligned}$$ with a parameter $$\lambda $$ . The technical approach is based on variational methods and the Mountain pass lemma.