C～m Solutions Of A Kind Of Systems Of Finite Difference Equations

Let R be the real number set, and m > 0 be an integer number. Suppose that G, H are C" maps from R2n+3 to R. In this paper, we discuss the Cm solutions of the system of finite difference equationsfor all x R, (1) where f, g Cm(R, R) are unknown functions to be solved.Let m > 0 and G, H Cm(R2n+3,R). For real number , define a map , using the method of approximating fixed points by small shift of maps, then the problem of solutions of (1) can be translated into that of fixed points of gh- Choosing a suitable metric on the functional space, we construct a compact convex subset Xm x Xm of Cm(R, R) x Cm(R, R) such that is a continuous self-map. Therefore, by Schauder-Tyclionoff Fixed Point Theorem, we obtain some theorems on the existence and uniqueness of Cm solutions of (1) under some relatively weak conditions.