Study On The Hyers-Ulam-Rassias Stability Of Functional Equations

The stability problem was raised firstly by S.Ulam in 1940.The problem for studying is:Given a group G and a metric group G' with metricp(·,·).Given ?>0,does there exist a ?>0 such that if f:G ?G' satisfies p(f(x·u),f(x)·f(y))<? for all x,y? G,then a homomorphism h:G?G' exist with ?(f(x),h(x))<? for all x? G?If the answer is affirmative,we say that the functional equation for homomorphisms is stable.Th.M.Rassias study the stability of linear mappings by considering an unbounded Cauchy difference.The stability of the control quantity as a function is called Hyers-Ulam-Rassias stability.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 new functional equations.Now,we study the stability of functional equations in different spaces:multi-?-normcd space,non-Archimedean field,fuzzy normed spaces,intuitionistic fuzzy ? normed space and(n,?)-normed spaces.In chapter 1,we adopt the fixed point method to investigate the stability of additive-cubic-quartic functional equation,quintic functional equation and sextic functional e-quation in multi-?-normed space.In chapter 2,we mainly study the stability of quintic functional equation in non-Archimedean(n,?)-normed spaces.In chapter 3,we mainly study the stability of quintic functional equation in fuzzy normed spaces.In chapter 4,we get a new equationsreciprocal-quintic functional equation and reciprocal-sextic functional equation.Next we mainly study the stability of reciprocal-quintic func-tional equation and reciprocal-sextic functional equation in non-Archimedean field.In chapter 5,we get another new space with intuitionistic fuzzy ? normed space.Next we mainly study the stability of reciprocal-quintic functional equation and reciprocal-sextic functional equation in this space.