A Class Of Functional Equations And Functional Inequalities Stability

The stability problem of functional equations originated from a question of Ulam concerning the stability of group homomorphisms in 1940:Give a group (G1,*) and a metric group (G2,·, d) with the metric d(·,·). Give (?)> 0, does there existsαδ> 0 such that if f:G1â†'G2 satisfies d(f(x*y),f(x)·f(y))<δfor all x,y∈G1, then is there a homomorphism g:G1â†'G2 with d(f(x).g(x))<εfor all x∈G1(?)In 1941. D. H. Hyers solved the stability problem of additive mapping on Banach spaces. In the following decades, many mathematicians have studied the stability of different kinds of functional equations such as exponential equation. quadratic functional equation. cubic functional equation, generalized additive equation and so on. In 1978, Th. M. Rassias solved the stability problem of linear mapping in Banach space. In 1999, Y. Lee and K. Jun studied the stability of generalized Jense equation. These stability results have applications in some related fields such as random analysis, financial mathematics and actuarial mathematics.This thesis consists of two chapters.In Chapter 1, we consider the following Cauch-Jensen functional inequation and consider its stability in Banach modules over C*-algebras.In Chapter 2, we consider the solution and stability of a new mixed functional equa-tion deriving from the additive and quadratic functional equations We first discuss the solution of above mixed functional equation. Then we consider the Hyers-Ulam stability of this functional equation in quasiβBanach spaces.