Abstract
In this paper we discuss a relatively general kind of iterative functional equation G(x,f(x), ...,f n(x)) = 0 (for allx ∈J), whereJ is a connected closed subset of the real number axis ℝ,G∈C m(J n+1, ℝ) andn ≥ 2. Using the method of approximating fixed points by small shift of maps, choosing suitable metrics on functional spaces and finding a relation between uniqueness and stability of fixed points of maps of general spaces, we prove the existence, uniqueness and stability ofCm solutions of the above equation for any integer m ≥ 0 under relatively weak conditions, and generalize related results in reference in different aspects.
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Mai, J., Liu, X. Existence, uniqueness and stability ofC m solutions of iterative functional equations. Sci. China Ser. A-Math. 43, 897–913 (2000). https://doi.org/10.1007/BF02879796
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DOI: https://doi.org/10.1007/BF02879796