Functions That Can Be Approximated By Additive Mapping And Quadratic Mapping

In this paper,we characterize the functions with values in a Banach space and(β,p)-Banach space which can be approximated by additive mappings and a quadratic mapping,with a given error.The thesis is divided into three sections.In chapter 1,we give some fundamental knowledge of functional equations.In chapter 2,we prove the Φ-approximation by the additive mapping f(x + y)= f(x)+ f(y)in Banach space or(β,p)-Banach space,where X is linear space and Y is Banach space or Y is(β,p)-Banach space,f:X → Y is a mapping.In chapter 3,we prove the Φ-approximation by the quadratic mappings f(x + y)+ f(x-y)= 2f(x)+ 2f(y)in Banach space or(β,p)-Banach space,where X is linear space and Y is Banach space or Y is(β,p)-Banach space,f:X →F is a mapping.