Pólya and Szegő [53, Teil I, Aufgabe 99] proved that every approximate sequence of reals is near an additive sequence. Bourgin [11] showed that every approximate ring homomorphism from a Banach algebra onto a unital Banach algebra is necessarily a ring homomorphism. We deal with Pólya-Szegő’s result for a general functional equation and a system of general functional equations in several variables. To do this, we shall use a different direct method from the previous studies. In consequence, Bourgin’s result for approximate homomorphisms and Lie homomorphisms on Banach algebras are discussed. Several examples for comparison with previous studies are included.
