Study On The Hyers-ulam-rassias Stability Of Functional Equations

The stability problem was raised firstly by S. Ulam in1940. The problem for studying is:Given a group G and a metric group G1with metric p(·,·). Given ε>0, does there exist a δ>0such that if f: G-> G’satisfies p(f(x·y),f(x)· f(y))<δ for all x,y∈G, then a homomorphism h:G�'G1exist with p(f(x), h(x))<ε for all x G G? Since the stability of functional equations has widely applications in Banach space geometry, harmonic analysis, relative theory, operator theory, information theory etc., so many researchers pay more attentions to the study of the stability problem of functional equations. In recent years, people have discovered new spaces for studying the stability of different functional equations.In recent years, mathematicians have investigated the stability of different functional equa-tions in fuzzy normed spaces、non-Archimedean spaces and n-normed spaces. In this thesis, we mainly study the stability of functional equations in multi-normed space, non-Archimedean spaces and non-Archimedean(n,β)-normed spaces. The main results are the following three aspects:1. The dissertation adopt the direct method and the fixed point method to investigate the stability of additive-quartic functional equation and the orthogonal stability of mixed additive-quadratic Jensen functional equation.2. On the basis of n-normed space, the dissertation generalized the definition of n-normed space, then we get a new space:(n,β)-normed spaces. Next, this paper give some relative definitions and properties of (n,β)-normed spaces and investigate the stability of Cauchy func-tional equation、Pexiderized Cauchy functional equation and the orthogonal stability of mixed additive-quadratic Jensen functional equation.3. Considering non-Archimedean field, the dissertation combine it with (n,β)-normed spaces, then we get another new space:non-Archimedean (n,β)-normed spaces. Next this paper mainly study the stability of Cauchy functional equation, Jensen functional equation and the orthogonal stability of additive-quartic functional equation in non-Archimedean (n,β)-normed spaces.