Xu, Bing; Zhang, Weinian Hyers-Ulam stability for a nonlinear iterative equation. (English) Zbl 1018.39017 Colloq. Math. 93, No. 1, 1-9 (2002). Hyers-Ulam stability of the nonlinear iterative functional equation \(G(f^{n_1}(x), \dots, f^{n_k}(x)) =F(x)\) is considered. \(F\) is assumed to be given and \(f\) an unknown function. Both \(F\) and \(f\) are self-maps of \(I\), a subset of a Banach space; \(G:I^k\to I\), where, as usual, \(I^k=I\times \cdots\times I\), \(f^0(x)=x\), \(f^{i+1}(x) =f(f^i(x))\) for \(i>0\). Also, \(n_1,\dots, n_k\) are positive integers for \(i=1,\dots, k\).By constructing a suitable uniformly convergent sequence of functions, it is established that the functional equation has a unique solution near its approximate solution. Reviewer: F.J.Papp (Ann Arbor) Cited in 7 Documents MSC: 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges 39B12 Iteration theory, iterative and composite equations Keywords:nonlinear iterative equation; Hyers-Ulam stability; approximate solution; uniform convergence; orientation-preserving homeomorphism; nonlinear iterative functional equation; Banach space PDFBibTeX XMLCite \textit{B. Xu} and \textit{W. Zhang}, Colloq. Math. 93, No. 1, 1--9 (2002; Zbl 1018.39017) Full Text: DOI