The authors study the existence and multiplicity of solutions for boundary value problems of the type \[ \nabla [\phi (\Delta x_k)]+f_k(x_k, \Delta x_k)=0\quad (2\leq k\leq n-1),\qquad l({\mathbf x}, \Delta {\mathbf x})=0, \tag{1} \] where \(\phi: (-a, a)\to {\mathbb R}\) denotes an increasing homeomorphism such that \(\phi(0)=0\) and \(0<a<\infty\), \(l({\mathbf x}, \Delta {\mathbf x})=0\) denotes the Dirichlet, periodic, or Neumann boundary conditions and \(f_k (2\leq k \leq n-1)\) are continuous functions.
Firstly, the authors study forced equations with discrete \(\phi\)-Laplacian and singular \(\phi\) for the three boundary conditions and give fixed point reformations for (1). Secondly, they prove that the Dirichlet problem is always solvable. This is done using a fixed point reduction and Brouwer degree theory. For the other boundary conditions, the authors prove existence when the right hand member \(f=(f_2, f_3, \dots, f_{n-1})\) only satisfies some sign conditions. They also extend the method of upper and lower solutions to this type of problem for periodic and Neumann conditions. Finally, they prove an Ambrosetti-Prodi type multiplicity result and give Lazer-Solimini type results.