Study On The Stability Of Functional Equations In β-normed、Random Normed Spaces,Etc

In 1940,the mathematician S.M.Ulam first raised the question of homomorphism stability in a symposium:let G1 be a group and let G2 be a metric group with a metric d(·,·).Given ε>0,does there exist a δ>0 such that if a function h:G1→G2 satisfies the inequality d(h(xy),h(x)h(y))<δ for all x,y∈G1,then there is a homomorphism H:G1→G2 with d(h(x),H(x))<ε for all x∈G1?If the answer is affirmative,we say that the functional equation for homomorphisms is stable.In 1941,mathematician D.H.Hyers assumed that G1 and G2 were Banach spaces,and answered the question of Ulam firmly.In 1978,Th.M.Rassias extended the result of Hyers by turning the control constant in Hyers method into an unbounded function,which was called the Hyers-Ulam-Rassias stability.Now,we will study the stability of functional equations in non-Archimedean β-normed spaces,(n,β)-normed spaces,matrixβ-normed spaces and random normed spaces.The main results are as follows:In chapter 1,we use the direct method and the fixed point method to investigate the stability of undecic functional equation in non-Archimedean β-normed spaces and(n,β)-normed spaces,respectively.In chapter 2,we adopt the direct method to consider the stability of dodecic func-tional equation in intuitionistic fuzzy normed spaces.In chapter 3,we study the stability of general quadratic and quartic type functional equation in matrix β-normed spaces by direct method.In chapter 4,we apply the fixed point method and direct method to investigate the stability of quadratic-additive mixed functional equation with parameters in random normed spaces.