Homomorphism And Derivations Of The Stability Problem

In 1940, S. M. Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed the stability of homomorphisms. In 1941, D. H. Hyers [2] considered the case of approximately additive mappings f:E1â†'E2 between Banach spaces when f satisfies the following inequality ||f(x+y)-f(x)-f(y)||≤(?) for all x,y∈E1. It was shown that there is a unique additive mapping L satisfying ||f(x)-L(x)||≤(?) limit and L is given byNo continuity conditions are required for this result, but if f(tx) is continuous in the real variable t for each fixed x∈E1, then L is linear; if f is continuous at a single point of E1, then L:E1â†'E2 is also continuous.1978, Th. M. Rassias [11] succeeded in extend-ing the result of Hyers'Theorem by weakening the condition for the Cauchy difference controlled by (||x||p+||y||p),P∈[0,1) to be unbounded. Later, many mathematicians have considered the stability of different kinds of functional equations. These results have applications in random analysis, financial mathematics and actuarial mathematics.This dissertation consists of four chapters:The first chapter investigates fuzzy homomorphisms stability of the following Cauchy-Jensen functional equation in a C*-ternary algebra. for all A∈T1={μ∈C:|μ|= 1} and for all x,y,z∈X.The second chapter studies the stability of approximate quadratic homomorphisms about the quadratic functional equation f(x+y)+f(x-y)= 2f(x)+2f(y). And we give some theorems and corollaries.In the third chapter, we study the n-Jordan homomorphisms of the Cauchy-Jensen functional equation on JC*-algebra. Then we succeed in extending the result of Jordan homomorphisms. In the fourth chapter, we study the n-Lie homomorphisms and n-Lie derivations of the Cauchy-Jensen functional equation on Lie C*-algebra and consider their stability problem respectively.