Study On The Stability Of Functional Equations

In 1940,Ulam proposed the stability problem of functional equations,that is:given a group G₁ and a metric group G₂?·,p?,for all ?>0,exist a ?>0 such that f:G₁ ? G₂ satisfies the inequality p?f?x· y?,f?x?· f?y??<? for all x,y ? G₁,then there is a homomorphism h:G₁ ? G₂ with p?f?x?,h?x??<? for all x? G₁?If the answer of this question is affirmative,then we say the functional equation h?x·y?= h?x?·h?y?is stable.The stability of the functional equations mainly studies that if a mapping is almost homomorphic,then whether there is a homomorphism and which is approximation of the mapping.Since the stability of functional equations is widely used in space theory,Ba-nach algebra,relativity theory,information theory,quantum mechanics and so on,many experts and scholars are attracted to the research of this topic.In recent years,mathe-maticians have been looking for different metric spaces and different forms of functional equations to study.In this thesis,we mainly study the stability of functional equations in matrix ?-normed spaces,non-Archimedean?n,??-normed spaces,fuzzy ?-normed spaces,intuitionistic fuzzy?n,??-normed spaces and non-Archimedean field.This thesis is organised as follows:In chapter 1,we study the stability of quintic functional equation and sextic function-al equation in matrix ?-normed spaces by "directly method^ and "fixed point method",respectively.In chapter 2,we investigate the stability of additive-quadratic-cubic-quartic func-tional equation and sextic functional equation in non-Archimedean?n,??-normed spaces.In chapter 3,we investigate the stability of quadratic functional equation with invo-lution in fuzzy ?-normed spaces by "fixed point method".In chapter 4,we introduce the notion of intuitionistic fuzzy?n,??-normed spaces which is a generalization of intuitionistic fuzzy n-normed spaces and investigate the stability of quintic functional equation in this space.In chapter 5,we give two new equations:reciprocal-nonic functional equation and reciprocal-decic functional equation,then we study the stability of the two functional equations in non-Archimedean field.