Two Types Of Functional Equations On The Stability Of A Few Class Space

The stability problem is a classical question in the theory of functional equations.The stability problem concerning the stability of group homomorphisms was firstly raisedby Ulam in1940:Give a group (G1,) and a metric group (G2,·, d) with the metric d(·.·). Give>0,does there exists a δ>0such that if h: G1â†'G2satisfies d(h(x y), h(x)· h(y))<δ forall x, y∈G1, then is there a homomorphism H: G1â†'G2with d(h(x), H(x))<ε for allx∈G1?In1941, 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 Jense 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 Hyers-Ulam stability of a mixed additive-quadraticfunctional equation deriving from the Jensen additive functional equationin β-Banach space and quasi-Banach space. We firstly discuss the Hyers-Ulam stability ofabove mixed additive-quadratic functional equation in β-Banach space. Then we considerthe Hyers-Ulam stability of this functional equation in quasi-Banach spaces.In chapter2, we consider the following n-dimensional functional equation (here V={I M:1∈I}, M={1,2,..., n}, M/I=Ic)deriving from the quarticfunctional equationsf(2x+y)+f(2x y)=4f(x+y)+4f(x y)+24f(x)6f(y),and discussed its Hyers-Ulam stability in non-Archimedean Banach Modules and randomBanach Modules.