The Stability Of A Kind Of Functional Equation In Two Kinds Of Space

The stability problem of functional equations originated from a question of Ulam concerning the stability of group homomorphisms in1940:Give a group (G1,*) and a metric group (G2,·, d) with the metric d(·,·). Give ε>0, does there exists a8>0such that if f:G1â†'G2satisfies d(f(x*y), f(x)·f(y))<δ for all x, y∈G1, then is there a homomorphism g:G1â†'G2with d(f(x),g(x))<ε for all x∈G1?In1941, 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. In1978, Th. M. Rassias solved the stability problem of linear mapping in Banach space. In1992, Czerwik solved the stability problem of quadratic functional equation in normed spaces. In1999, Y. Lee and K. Jun studied the stability of generalized Jense equation. In2002, K.-W.Jun and H.-M.Kim solved the stability problem of cubic functional 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 chapter1, we consider the solution and fuzzy stability of a mixed functional equation deriving from the quadratic and cubic functional equations f(2x+y)+f(2x-y)=f(x+y)-f(-x-y)+f(x-y)-f(y-x)+10f(x)-2f(-x)+f(y)+f(-y). We consider the Hyers-Ulam fuzzy stability of this functional equation in non-Archimedean Banach spaces.In chapter2, we consider fuzzy Banach spaces of the following functional equations f(2x+y)+f(2x-y)-f(x+y)+f(-x-y)-f(x-y)+f(y-x)-10f{x)+2/(-x)-f(y)-f(-y)=0. We first definition a fuzzy almost function. Then we consider the Hyers-Ulam fuzzy stability of this functional equation in fuzzy Banach spaces.