Several Types Of Stochastic Stability Of The Mixed 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 aδ>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 2010,M.Eshaghi Gordji and M.B.Savadkouhi studied the stability of additive quartic functional equations in random normed space,M.Mohamadi studied the stability of additive-quadratic-quartic functional equations in random normed space.This thesis consists of three chapters.In chapter 1,we provide some introductive knowledge of random normed space.In chapter 2,we consider the following mixed type additive-quartic functional equa-tion f(2x+y)+f(2x-y)=4[f(x+y)+f(x-y)]-3/7(f(2y)-2f(y))+2f(2x)-8f(x).The stability of this functional equation in random normed space is considered.In chapter 3,we consider the following mixed functional equation deriving from additive,quadratic,cubic and quartic functional equation f(x+2y)+f(x-2y)=4[f(x+y)+f(x-y)]+f(4y)-4f(3y)+6f(2y)-4f(y)-6f(x). We study the stability of this functional equation in random normed space.