C * - Algebra, And A Few Class Space On The Stability Of The Functional Equations

The stability problem of functional equations originated from a question of Ulamconcerning the stability of group homomorphisms in1940:G_ive a group （G₁,） and a metric group （G₂,·, d） with the metric d（·,·）. G_ive>0,does there exists a δ>0such that if f: G₁â†'G₂satisfies d（f （x y）, f （x）· f （y））<δ forall x, y∈G₁, then is there a homomorphism g: G₁â†'G₂with d（f （x）, g（x））<ε for allx∈G₁In1941, D. H. Hyers solved the stability problem of additive mapping on Banachspaces. In the following decades, many mathematicians have studied the stability ofdiferent kinds of functional equations such as exponential equation, quadratic functionalequation, cubic functional equation, generalized additive equation and so on. In1978, Th.M. Rassias solved the stability problem of linear mapping in Banach space. In1999, Y.Lee and K. Jun studied the stability of generalized Jensen equation. These stability resultshave applications in some related fields such as random analysis, financial mathematicsand actuarial mathematics.This thesis consists of two chapters.In chapter1, we consider the following Cauchy-Jensen functional equationwith r∈（0,3）\{1} and consider its stability in C-algebras.In chapter2, we consider the solution and stability of a new mixed functional equationderiving from the additive, quadratic and cubic functional equationsWe first consider the solution of above mixed type functional equation. Then we show theHyers-Ulam stability of this functional equation in Banach spaces and non-Archimedeanspaces.