Systems with discrete singular φ-Laplacian and maximal monotone boundary conditions

We are concerned with solvability of nonlinear systems involving a discrete singular φ -Laplacian operator of type aligned u ↦ Δ [ φ (Δ u(n-1))] (n∈ \1, , T\), aligned associated with a general two point boundary condition having the form aligned ( φ (Δ u(0)),-φ (Δ u(T))) ∈ γ (u(0),u(T+1)), aligned where γ :RN× RN→ 2RN× RN is a maximal monotone operator with 0RN × RN∈ γ (0RN × RN) . The mapping φ is a potential homeomorphism from an open ball of radius a centered at the origin Bₐ ⊂ RN onto RN and Δ stands for the usual forward difference operator. When the perturbing nonlinearity in the system has not a potential structure we obtain existence of solutions by a priori estimates. Also, when the nonlinearity is of gradient type and γ is a subdifferential, we provide a variational approach of the system in the frame of critical point theory for convex, lower semicontinuous perturbations of C¹ -functionals. Then we derive the existence of solutions either as minimizers or saddle points of the corresponding energy functional.