Hyers–Ulam stability of a generalized Apollonius type quadratic mapping

Let X, Y be linear spaces. It is shown that if a mapping Q:X →Y satisfies the following functional equation: Q n i=1 zi − n i=1 xi +Q n i=1 zi − n i=1 yi = 1 2 Q n i=1 xi − n i=1 yi + 2Q n i=1 zi − ( n i=1 xi ) +( n i=1 yi ) 2 (0.1) then the mapping Q:X→Y is quadratic. We moreover prove the Hyers–Ulam stability of the functional equation (0.1) in Banach spaces. © 2005 Elsevier Inc. All rights reserved