In this paper, we study Sturm-Liouville boundary value problems for second order difference equations on a half line. By using the discrete upper and lower solutions, the Schäuder fixed point theorem, and the degree theory, the existence of one and three solutions are investigated. An interesting feature of our existence theory is that the solutions may be unbounded.

The main aim of this paper is to study the P 1 nonconforming finite element approximations of the variational inequality arisen from the Signorini problem. We describe the finite dimensional closed convex cone approximation in a meanvalue-oriented sense. In this way, the optimal convergence rate O(h) can be obtained by a refined analysis when the exact solution belongs to H 2 (Ω) without any assumption. Furthermore, we also study the optimal convergence for the case u ∈ H 1+ν (Ω) with 1 2 < ν < 1.