Fuzzy approximation of Euler-Lagrange quadratic mappings

Hark-Mahn Kim,Juri Lee,John M Rassias

doi:10.1186/1029-242x-2013-358

Abstract

Abstract In this article, we consider the Hyers-Ulam stability of the Euler-Lagrange quadratic functional equation f ( k x + l y ) + f ( k x − l y ) = k l [ f ( x + y ) + f ( x − y ) ] + 2 ( k − l ) [ k f ( x ) − l f ( y ) ] in fuzzy Banach spaces, where k, l are nonzero rational numbers with k ≠ l .

Highlights

The theory of fuzzy spaces has much progressed as the theory of randomness has developed
A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [ ] for mappings f : X → Y, where X is a normed space and Y is a Banach space
We prove the generalized Hyers-Ulam stability of the Euler-Lagrange quadratic functional equation f + f = kl f (x + y) + f (x – y) + (k – l) kf (x) – lf (y) in fuzzy Banach spaces, where k, l are nonzero rational numbers with k = l

Summary

Introduction

The theory of fuzzy spaces has much progressed as the theory of randomness has developed. We say that a mapping f : X → Y between fuzzy normed spaces X and Y is continuous at x ∈ X if for each sequence {xn} converging to each x ∈ X, the sequence {f (xn)} converges to f (x ). A Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [ ] for mappings f : X → Y , where X is a normed space and Y is a Banach space. A mapping f : X → Y with f ( ) = between linear spaces satisfies the functional equation ( ) if and only if f is quadratic. We can find a unique quadratic mapping Q : X → Y satisfying the equation DklQ(x, y) = and the inequality.

We remark that if k

Now suppose that

Then if x

Proof We consider the set of functions

Thus we can get