Cm solutions of systems of finite difference equations

Abstract

Let ℝ be the real number axis. Suppose that G, H are Cm maps from ℝ2n+3 to ℝ. In this note, we discuss the system of finite difference equations G(x,f(x),f(x+1),…,f(x+n),g(x),g(x+1),…,g(x+n))+0 and H(x,g(x),g(x+1),…,g(x+n),f(x),f(x+1),…,f(x+n))=0 for all x∈ℝ, and give some relatively weak conditions for the above system of equations to have unique Cm solutions (m≥0).