Generalization of functional equation for the square root spiral

摘要

In the paper we give the general solution of the linear functional equation ϕ(p−1(p(x) + c)) = ϕ(x) + h(x). Furthermore, we discuss its Hyers–Ulam–Rassias stability and the stability in the sense of Ger for the equation ϕ(p−1(p(x) + c)) = ϕ(x). Our results are applied to the square root spiral so that the related results by Heuvers et al. [K.J. Heuvers, D.S. Moak, B. Boursaw, The functional equation of the square root spiral, in: Th.M. Rassias (Ed.), Functional Equations and Inequalities, Kluwer, 2000, pp. 111–117] and results by Jung and Sahoo [Stability of functional equation for square root spirals, Appl. Math. Lett. 15 (2002) 435–438] are generalized.