Hyers-Ulam Stability Study Of Additive Mapping In Banach Space

In 1940,Ulam proposed the stability problem of group homomorphism,which is the source of the stability of functional equations.It is mainly concerned with a function approximation that satisfies a given equation,and whether this function is close to the solution of the original equation.Because it has a wide range of applications in Banach geometry,information theory,relativity theory,operator theory and harmonic analysis,many researchers begin to pay attention to the study of stability of functional equations.The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem.In this paper,we mainly study the stability of functional equations and inequalities,especially,we discuss the Hyers-Ulam stability of functional equations and inequalities in vector Banach spaces and fuzzy Lie Banach spaces,in the above two space proved functional equations and inequalities with functional equations and inequalities of Hyers-Ulam stability of a reference sample,has a certain theoretical value for the study of stability of functional inequalities.This paper contains four chapters.In the first chapter,we mainly introduced the research background,the current research status and the main work of functional inequality stability problem.At the same time,the main research contents and research methods are introduced.In the second chapter,we first review the basic results of vector Banach space,prove Hyers-Ulam stability of the additive functional inequality||f(ax+by+cz)+f(bx+ay+bz)+f(cx+cy+az||?||(a+b+c)f(x+y+z)|| of vector Banach space.In the third chapter,we first introduce the definition of fuzzy Lie Banach space and the basic results of previous work and fixed point theory,the stability of the following equationsf(2x-y-z)+f(x-z)+f(x+y+2z)=f(4x)is studied by the method of fixed point.In the fourth chapter,we present the conclusion and prospect.