Stability Of Operator Equations

In this article,the super-stability and the stability of the exponential operator equation f(x+y)=f(x)f(y) in some special spaces are studied.The stability of the equation in restrict domain is also studied.Moreover,theε-Hyers-Ulam stability of the general operator equation is introduced.Some properties of this kind of stability are discussed.The followings are the construction and main contents of this paper:In Chapter 1,we introduce the history of the theory on stability of functional equations.At the same time,we outline the development for the results in recent years and the significance of the theory in other subjects.Our main research work is summaried finally.In Chapter 2,we firstly study the super-stability of the exponential operator equation f(x+y)= f(x)f(y) in some special spaces.Motivated by Baker's result, we introduce a functional index‖E(f)‖_∞to prove that if a mapping f which is from a normed space to a normed algebra satisfying‖ab‖=‖a‖‖b‖satisfies that‖E(f)‖_∞is bounded,then either f is bounded or f is exponential.This result generlized Baker's result.Then we consider the stability of the exponential operator equation under the definition which Ger first defined.In Chapter 3,we study the super-stability of the exponential operator equation f(x+y)= f(x)f(y) in the restrict domain.Our result shows that the result that Soon Mo Jung got can be extended to the case where the target space is a commutative semisimple complex Banach algebra.We also get that if f is a unbounded operator from a normed space X to(?)~n and f is bounded in every subset of X,then the necessary and sufficient conditions for f is exponential is:whenever‖x‖+‖y‖goes to infinity,the difference between f(x+y) and f(x)f(y) goes to zero. This proposition is called the asympotic behavior of exponential operator equation. Then the Ger-stability in restrict domain of it is also researched.In Chapter 4,theε-Hyers-Ulam stability of the general operator equation Ax= 0 is introduced.We study this kind of stability of Ax=0,and get some necessary and sufficient conditions for Ax =0 isε-Hyers-Ulam stable.