3-Variable Jensen ?-Functional Inequalities And Equations

In the process of studying mathematics,people almost invariably ask a question.When do they approximate to a mathematical object of nature,which must be near the mathematical object that does have this property?When we define the object as a functional equation,the above problem can be more accurately changed into:when we use a functional inequality to replace a functional equation,when is the solution of this inequality satisfied in the neighborhood of the solution of the functional equation?The stability of functional equations derives from the mathematical symposium held at Wisconsin University in 1940,and the stability problem of group homomorphisms proposed by Ulam S M.This is the source of the stability problem of functional equations,and the main study is that if a function is approximated to an equation.This function is very close to the solution of the original equation.Hyers is the first mathematician to study the stability of functional equations by direct method.Then Rassias T M weakens the bounded Cauchy difference of Hyers and extends the result.Because Ulam,Hyers and Rassias have made outstanding contributions to the stability research of functional equations,the stability of such functional equations is called Hyers-Ulam-Rassias stability.In this paper,the stability of the Jensen functional inequality and equations is studied and the stability of the solution of the equation is also given,It is divided into four chapters.In the first chapter,we introduce the research background and the status and progress of the research at home and abroad.In the second chapter,we prove the Hyers-Ulam stability of Jensen additive functional inequalities and corresponding equations in complex Banach spaces.In the next chapter,we use the fixed point method to prove the Hyers-Ulam stability of Jensen additive functional inequality.In the fourth chapter,we prove the stability of the Jensen functional inequality and the corresponding equation in ?-homogeneous F-space,and we use the direct method.