Abstract

Let Q be a m x m real matrix and fj:ℝ→ℝ, j=1, ...,m, be some given functions. If x and f(x) are column vectors whose j-coordinates are xj and fj(xj), respectively, then we apply the finite dimensional version of the mountain pass theorem to provide conditions for the existence of solutions of the semilinear system Qx = f(x) for Q symmetric and positive semi-definite. The arguments we use are a simple adaptation of the ones used by Neuberger. An application of the above concerns partial difference equations on a finite, connected simple graph. A derivation of a graph G is just any linear operator D:C0(G)→C0(G), where C0(G) is the real vector space of real maps defined on the vertex set V of the graph. Given a derivation D and a function F:Vxℝ→ℝ, one has associated a partial difference equation Dμ=F(v,μ), and one searches for solutions μ∈C 0(G). Sufficient conditions in order to have non-trivial solutions of partial difference equations on any finite, connected simple graph for D symmetric and positive semi-definite derivation are provided. A metric (or weighted) graph is a pair (G, d), where G is a connected finite degree simple graph and d is a positive function on the set of edges of the graph. The metric d permits to consider some classical derivations, such as the Laplacian operator δ2. In (Neuberger, Elliptic partial difference equations on graphs, Experiment. Math. 15 (2006), pp. 91-107) was considered the nonlinear elliptic partial difference equations δ2u=F(u), for the metric d = 1. © 2008 Taylor & Francis.