Stability Analysis Of Drygas Functional Equation

In 1940,Ulam proposed the basic problem of the stability of group homomorphism.The problem can be described as whether a group homomorphism can be found to approximate any given group homomorphism.In 1941,Hyers gave an affirmative answer to the Ulam stability problem of additive groups in Banach space for the first time.Subsequently,Hyers theorem is generalized by Aoki in the direction of additive mapping,and Rassias extended it from bounded Cauchy difference to unbounded Cauchy difference.Therefore,the stability of this type of equation is called Hyers-Ulam-Rassias stability.In the past thirty years,the concept of stability of functional equations has evolved from a pure point of view into a systematic field.Many scholars have done a lot of research on functional equations,which has made it more and more complete.The problem of stability is still a classic subject in the study of functional equations nowadays.This paper studies the stability of functional equations,which is mainly composed of two parts: the first part uses the direct method to prove the Hyers-Ulam stability of the ternary Drygas functional equation from the Abel group to the Banach space;the second part is to prove the fuzzy version of generalized Ulam stability of Pexider additive functional inequalities by direct method.This paper is divided into three chapters.In the first chapter,the research background and development status of functional equations as well as inequality problems are mainly introduced.In the second chapter,we introduce the relevant definition of Banach space,and prove the stability of the two Drygas inequalities using the direct method in Banach space.In the third chapter,We introduce the relevant definitions of fuzzy Banach spaces,and use the direct method to prove the generalized Hyers-Ulam stability of Pexider's additive functional inequalities,and then give relevant inferences.