On The Stability Of Functional Equations

Abstract. In this article, we study the stability of the additive functional equation f(x + y) = f(x) + f(y) and multiplicative functional equation f(x · y) = f(x)f(y). We also consider the stability of ring homomorphism and characters. We divide this article into two chapters.In Chapter 1, we study the stability of additive functional equation f(x + y) = f(x) + f(y). We first introduce functional index A_r(-) to study approximately additive mappings. We get that if / maps a group into a Banach space and Ar(f) is bounded, then / is stable in the sense of Hyers and Ulam. This result generalized Hyers' result. So far, we have studied the stability of additive mapping defined on entire space. It is natural to ask about the stability of the additive mapping on a restricted domain. We prove that the mapping which satisfies the additive equation approximately in a restricted domain is stable in entire space. At the end of the chapter, we prove the generalized Hyers-Ulam stability of the linear mapping in Banach modules over a unital Banach algebra.In Chapter 2, we study the stability of multiplicative functional equation f(x · y) = f(x)f(y). Our first result shows that the well-known Baker's super-stability theorem for approximately multiplicative mapping can be extended to the case where the target space is a commutative semisimple complex Banach algebra. We also introduc another functional index Mr{·) to study approximately multiplicative mappings. We prove some results concerning stability of a ring homomorphism and generalizes R. Badora's theorem in two directions. We get that if / maps a ring into a Banach algebra and preserves unit and Ar(f) + Mr(f) is bounded, then f is stable. Next we give more general stability result and obtain stability theorem on Jordan and Lie homomorphisms. Most of the rest of this chapter is devoted to studying the stability of approximately multiplicative linear functionals defined on a commutative complex Banach algebra A, that is the stability of characters. We study some properties of approximate characters and give some characterizations of AMNM algebra which is with stable characters. Then we get that finite dimensional commutative complex Banach algebra is AMNM algebra and other results.