The Mazur-Ulam Theorem On Fuzzy N-normed Linear Spaces And Functional Equations Stability

In this paper,we will study the Mazur-Ulam theorem on fuzzy n-normed linear spaces.Moreover,we discuss the stability of quartic functional equation,cubic-quartic functional equations on fuzzy normed linear spaces.In the first section,we mainly summarize the results of Mazur-Ulam theorem in normed spaces and n-normed linear spaces.We shall focus attention on Mazur-Ulam theorem in n-nomed linear spaces,which was obtained by researcher H.Y.Chu.In the second section,we study the Mazur-Ulam theorem in strong fuzzy n-normed linear spaces.We prove that every surjective mapping preserving a certain fuzzy n-distance??)1<<0(is affine.Also,we show that any mapping preserving fuzzy n-distance one and a certain fuzzy n-distance ??)1<<0(is affine.This implies that fuzzy n-isometries between two strong fuzzy n-normed linear spaces are always affine.That is Mazur-Ulam theorem is established in stong fuzzy n-normed linear spaces.In the third section,we mainly study the stability of quartic functional equations and cubic-quartic functional equations in fuzzy normed linear spaces under some conditions.The method of proof contains direct proof and the proof of fixed point theorem.