The Stability Of Additive (?,?)-functional Equations

In 1940,Ulam raised the question of the stability of group homomorphisms.The problem description is given an approximate homomorphic map of a metric group.Does there exist a homomorphism map that approximates to it?In 1941,Hyers gave the first affirmative answer to the Ulam question,then Rassias weakened the bounded Cauchy difference of Hyers and extended Hyers's conclusion to the unbounded Cauchy difference.Therefore,the stability proved by Rassias is called Hyers-Ulam-Rassias stability.The Hyers-Ulam stability studied in this thesis is a special case of Hyers-Ulam-Rassias stability.In particular,in the past three decades,there have been many achievements in the study of the stability of various functional equations,the variety of equation types,the diversity of spaces and the wide application.Its research and development are becoming more and more perfect.In this thesis,the stability of additive(?,?)-functional equations are studied,which consists of two parts:The first part proves the Hyers-Ulam stability of set-valued Pexider functional equations in topological vector space by direct method;the second part proves the stability of the additive(?,?)-functional equation in non-Archimedean space by fixed point method and direct method.By comparing the conditions and conclusions of the lemma and theorem,it can be concluded that the positions and ranges of coefficients of the equation are different,which have different effects on the stability of the equation.