The Stability Of Cauchy - Quadratic And Jensen - Quadratic Functional Equations

This article gives the definition of Jensen-Quartic functional equation and CauchyQuartic functional equation, obtaining the general solution of those two equations, finding the relation between those two equations. Meanwhile, we proved the Ulam stability in Banach space and fuzzy normed space. To Cauchy-Quartic functional equation, we use “direct method” to prove its Ulam stability and hyperstability, and using fixed point method to prove its fuzzy stability in fuzzy normed space; To Jensen-Quartic functional equation, we using “direct method” to prove its Ulam stability and proved its fuzzy stability in fuzzy normed space. We get some useful conclusion about Quartic function and Cauchy-Quartic.In the first chapter, we gives the definition of Jensen-Quartic functional equation and Cauchy-Quartic functional equation, obtaining the general solution of those two equations, finding the relation between those two equations, in fact, those two functional equation can exchange under some condition; In the second chapter, using general method to prove the Ulam stability of these two equations on Banach spaces, meanwhile, we get a collusion about the hyperstability about Cauchy-Quartic functional equation; In the third chapter, we use direct method and fix point method to prove the Ulam stability in fuzzy Banach space.