Mirroring in lattice equations and a related functional equation

Abstract

We use a functional form of the mirroring technique to fully characterize equivalence classes of unbounded stationary solutions of lattice reaction-diffusion equations with eventually negative and decreasing nonlinearities. We show that solutions which connect a stable fixed point of the nonlinearity with infinity can be characterized by a single parameter from a bounded interval. Within a two-dimensional parametric space, these solutions form a boundary to an existence region of solutions which diverge in both directions. Additionally, we reveal a natural relationship of lattice equations with an interesting functional equation which involves an unknown function and its inverse.