Stability and solution of two functional equations in unital algebras

Abstract

In this paper, we consider two functional equations: h(F(x, y, z) + 2x + y + z) + h(xy + z) + yh(x) + yh(z) (1) = h(F(x, y, z) + 2x + y) + h(xy) + yh(x + z) + 2h(z), h(F(x, y, z) − y + z + 2e) + 2h(x + y) + h(xy + z) + yh(x) + yh(z) (2) = h(F(x, y, z) − y + 2e) + 2h(x + y + z) + h(xy) + yh(x + z), without any regularity assumption for all x, y, z in a unital algebra A, where F : A3 → A is defined by F(x, y, z) := h(x + y + z) − h(x + y) − h(z) for all x, y, z ∈ A. Also, we find general solutions of these equations in unital algebras. Finally, we prove the Hyers-Ulam stability of (1) and (2) in unital Banach algebras.