Study On The Ulam Stability Of Functional Equations

Mathematician Ulam put forward the problem of the stability of functional equation in 1940 and gave the definition of the stability when G1 and G2 are homomorphism.Let(G1,*)be a group and(G2,(?),d)be a metric group with d(·,·).For each ?>0,there exists ?(?)>0,if function h:G1?G2 satisfies inequality d(h(x*y),h(x)(?)h(y))<?,for each x,y ?G1,then there is a homomorphism H:G1?G2 satisfying d(h(x),H(x))<?,for all x?G1.If the answer is right,the equation H(x*y)=H(x)(?)H(y)is stable.The stablity of functional equation mainly discusses when the turbulent inequality takes the place of the function,in which situation we can assert that the solution of the related inequality approach to the solution of the equation.Functional equation is widely used in statistics,random analysis,algebra,geometry,phsychology,social science,and becomes an essential math tool for solving these problems.In recent years,math-ematicians are doing much more research in functional equations in various spaces,and studying how to solve various kinds of functional equations.In this article,we mainly discuss functional equations in ?-norm space,matrix ?-norm space,multi-Banach space,non-Archimedean(n,?)-normed space,non-Archimedean field and C*-algebra and talk about the stability,and check the solutions of high order equations.In this thesis,we arrange as follow:In chapter 1,we learn hexdecic functional equations in ?-norm and matrix ?-norm spaces with its stability.In chapter 2,we solve a new kind of 20th order functional equation,and discuss its stability in multi-Banach space and non-Archimedean(n,?)-normed space.In chapter 3,we give two new equations which are reciprocal fifteenth and reciprocal sixteenth order equations,then we research their stability in non-Archimedean field.In chapter 4,we give the fifth root and sixth root functional equation,then we learn the stability in C*-algebra.