This paper studies the smoothness of solutions of the higher dimensional polynomial-like iterative equation. The methods given by Zhang Weinian([7]) and by Kulczvcki M, Tabor j.([3]) are improved by constructing a new operator for the structure of the equation in order to apply fixed point theorems. Existence, uniqueness and stability of continuously differentiable solutions are given.
A functional equation of nonlinear iterates is discussed on the circle S^1 for its continuous solutions and differentiable solutions. By lifting to R, the existence, uniqueness and stability of those solutions are obtained. Techniques of continuation are used to guarantee the preservation of continuity and differentiability in lifting.
It is known that small perturbations of a Fredholm operatorL have nulls of dimension not larger than dimN(L). In this paper for any given positive integer κ ? dimN(L) we prove that there is a perturbation ofL which has an exactly κ-dimensional null. Actually, our proof gives a construction of the perturbation. We further apply our result to concrete examples of differential equations with degenerate homoclinic orbits, showing how many independent homoclinic orbits can be bifurcated from a perturbation.
ZHANG WeinianDepartment of Mathematics, Sichuan University, Chengdu 610064, China
