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Generalization of functional equation for the square root spiral. (English) Zbl 1108.39029

Applying the method in K. J. Heuvers, D. S. Moak and B. Boursaw [Math. Appl., Dordr. 518, 111–117 (2000; Zbl 0976.39018)], the authors solve the functional equation \(\varphi(p^{-1}(p(x)+c))=\varphi(x)+h(x)\), where \(p, h\) are given functions, \(p^{-1}\) denotes the inverse of \(p\), \(\varphi\) is the unknown function and \(c \neq 0\) is a constant.
For some technical reasons, they prove the further generalized stability and the stability in the sense of Ger of the special case of the above functional equation, where \(h=0\). Applying their results to the \(n\)-th root spiral \(\varphi(\root n\of{x^n+1})=\varphi(x)+\arctan\frac{1}{x}\), they generalize the results of the cited paper above and S. M. Jung and P. K. Sahoo [Appl. Math. Lett. 15, No. 4, 435–438 (2002; Zbl 1016.39020)].

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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