Study On The Stability Of Functional Equations

In the fall of 1940,S.M.Ulam gave a wide-ranging talk before a Mathematical Col-loquium in which he discussed a number of important unsolved problems,among those was the following question concerning the stability of homomorphisms:let G1 be a group and let G2 be a metric group with a metric d(·,·),given ?>0,does there exist a ?>0 such that if a function h:G1?G2 satisfies the inequality d(h(xy),h(x)h(y))<? for all x,y?G1,then there is a homomorphism H:G1?G2 with d(h(x),H(x))<?for all x?G1?If the answer is affirmative,we say that the functional equation for homomorphisms is stable.D.H.Hyers was the first mathematician to present the result concerning the stability of functional equations.In 1941,he brilliantly answered the question of Ulam for the case where G1 and G2 are assumed to be Banach spaces.In 1978,Th.M.Rassias addressed the Hyers's stability theorem and attempted to weaken the condition for the bound of the norm of Cauchy difference and proved a considerably generalized result of Hyers by making use of a direct method,the stability phenomenon proved by Th.M.Rassias is called the Hyers-Ulam-Rassias stability.Now,we study the stability of functional equations in non-Archimedean(n,?)-normed spaces,(n,?)-normed spaces,multi-?-normed spaces,random C*-algebra.In chapter 1,we investigate the stability of m-variable additive functional equation in non-Archimedean(n,?)-normed spaces by direct method.In chapter 2,we mainly study the stability of generalized mixed type additive-quadratic-cubic-quartic functional equation in(n,?)-normed spaces by direct method.In chapter 3,we adopt the fixed point method to investigate the stability of septic functional equation in multi-?-normed spaces.In chapter 4,we adopt the fixed point method to investigate the stability of gener-alized mixed type additive-quadratic functional equation in random C*-algebra.