Several Types Of Mixed Fuzzy Stability Of The Functional Equation,

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 exists a 6> 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))< e for 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 solution and fuzzy stability of a new mixed functional equation deriving from the quadratic and cubic functional equations 2f(2x+y)+2f(2x-y)= 4f(x+y)+4f(x-y)+Af(2x)+f(2y)-8f(x)-8f(y). We first discuss the solution of above mixed functional equation. Then we consider the Hyers-Ulam fuzzy stability of this functional equation in fuzzy Banach spaces.In chapter 2, we consider the following mixed functional equation deriving from additive, quadratic, cubic and quartic functional equations for a fixed integer k> 1, wheref= f(y)+f(-y) and discussed its fuzzy stability.