Fixed Points, Inner Product Spaces, and Functional Equations

Rassias introduced the following equality [inline-graphic not available: see fulltext], [inline-graphic not available: see fulltext], for a fixed integer [inline-graphic not available: see fulltext]. Let [inline-graphic not available: see fulltext] be real vector spaces. It is shown that, if a mapping [inline-graphic not available: see fulltext] satisfies the following functional equation [inline-graphic not available: see fulltext] for all [inline-graphic not available: see fulltext] with [inline-graphic not available: see fulltext], which is defined by the above equality, then the mapping [inline-graphic not available: see fulltext] is realized as the sum of an additive mapping and a quadratic mapping. Using the fixed point method, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.