Zeros of semilinear systems with applications to nonlinear partial difference equations on graphs†

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Let Q be a m × m real matrix and f j : ℝ → ℝ, j = 1, …, m, be some given functions. If x and f(x) are column vectors whose j-coordinates are x j and f j (x j ), 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:C 0(𝒢) → C 0(𝒢), where C 0(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:V × ℝ → 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 ▵2. In (Neuberger, Elliptic partial difference equations on graphs, Experiment. Math. 15 (2006), pp. 91–107) was considered the nonlinear elliptic partial difference equations ▵2 u = F(u), for the metric d = 1.