Some Further Generalizations of the Hyers Ulam Rassias Stability of Functional Equations1

In this paper we study the Hyers Ulam Rassias stability theory by considering the cases where the approximate remainder is defined by fŽx y. fŽx. fŽy. Žx, y. Ž x, y G., Ž1. fŽx y. gŽx. hŽy. Žx, y. Ž x, y G., Ž2. 2fŽŽx y.1 2. fŽx. fŽy. Žx, y. Ž x, y G., Ž3. where ŽG, . is a certain kind of algebraic system, E is a real or complex Hausdorff topological vector space, and f, g, h are mappings from G into E. We prove theorems for the Hyers Ulam Rassias stability of the above three kinds of functional equations and obtain the corresponding error formulas.