Existence, uniqueness and stability of \(C^m\) solutions of iterative functional equations. (English) Zbl 0999.39020
Summary: In this paper we discuss a relatively general kind of iterative functional equation \(G(x,f(x),\dots,f^n(x))= 0\) (for all \(x\in J\)), where \(J\) is a connected closed subset of the real number axis \(\mathbb{R}\), \(G\in C^m(J^{n+1},\mathbb{R})\), and \(n\geq 2\). 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 of \(C^m\) solutions of the above equation for any integer \(m\geq 0\) under relatively weak conditions, and generalize related results in reference in different aspects.
MSC:
| 39B12 | Iteration theory, iterative and composite equations |
| 39B82 | Stability, separation, extension, and related topics for functional equations |
Keywords:
iterative functional equation; \(C^m\) map; function space; compact convex set; fixed point; diagonal method; stabilityReferences:
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