ON A MEASURE ZERO STABILITY PROBLEM OF A CYCLIC EQUATION

Let G be a commutative group, Y a real Banach space and f : G -> Y. We prove the Ulam-Hyers stability theorem for the cyclic functional equation 1/vertical bar H vertical bar Sigma(h is an element of H)f(x + h . y) = f(x) + f(y) for all x, y is an element of Omega, where H is a finite cyclic subgroup of Aut( G) and Omega subset of G x G satisfies a certain condition. As a consequence, we consider a measure zero stability problem of the functional equation 1/N Sigma(N)(k=1) f(z + omega(k) zeta) = f(z) + f(zeta) for all (z, zeta) is an element of Omega, where f : C -> Y; omega = e(2 pi i/N) and Omega subset of C-2 has four-dimensional Lebesgue measure 0.