The Stability Of Several Types Of Functional Equations In The Space Of Multiple Norm

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 there is 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. These results of stability could be applied to some related fields such as random analysis, financial mathematics, actuarial mathematics and so on.In this paper, we investigate an generalized quadratic functional equation Using the fixed point method, we will prove the generalized Hyers-Ulam stability of the above functional equation in multi-Banach spaces.Then we investigate the stability of the following mixed functional equation deriving from additive, Quadratic Cubic and Quartic mappings in multi-Banach spaces: using fixed point methods.