The Stability Of Several Types Of Functional Equations On The Group

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 [2]. In the following decades, many mathematicians have studied the stability of different kinds of functional equations (see [3]-[8]). These stability results have applications in some related fields such as random analysis, financial mathematics and actuarial mathematics.In this thesis, we consider the stability of several functional equations on group. This thesis consists of three chapters.In chapter 1, we consider the stability of Swiatak functional equationsWe first discuss the stability of the above functional equation; Then we give another prove of the stability of this functional equation in abelian group.In chapter 2, we solve the conditional functional equations of the forms orwhereγis a given function,f, g, and h are unknown functions. Moreover, we study the stability problems for the equations, in the sense of Hyers and Ulam in the sense of Bourgin.In chapter 3, we study the stability of the multi-quadratic functional equationWe first consider the stability of the above equation on abelian group, then we discuss it on non-Archimedean space.