On the Ulam type stability of several types of quadratic fuzzy set-valued functional equations

Yonghong Shen,Yaoyao Lan,Wei Chen

doi:10.1186/s13660-014-0530-2

Abstract

Let Y be a real separable Banach space and \((\mathcal {F}_{KC}(Y),d_{\infty})\) be the space of all normal fuzzy convex and upper semicontinuous fuzzy sets with compact support in Y, where \(d_{\infty}\) stands for the supremum metric in \(\mathcal{F}_{KC}(Y)\). In the present paper, several types of quadratic fuzzy set-valued functional equations are introduced based on the space mentioned above. We prove the Hyers-Ulam stability of the standard quadratic fuzzy set-valued functional equation by using the fixed point technique. Simultaneously, we also establish some Ulam type stability results of the Deeba and Appolonius type fuzzy set-valued functional equations by employing the direct method, respectively. The stability results of the corresponding single-valued and set-valued functional equations acting as special cases will be included in our results.

Highlights

1 Introduction Nowadays, the Ulam type stability is gradually becoming one of the most active research topics in the theory of functional equations. The study of such stability problems of functional equations originated from a question of Ulam [ ] concerning the stability of group homomorphisms, i.e.: Let G be a group and let G be a metric group with the metric d(·, ·)
Afterwards, Hyers [ ] gave a first affirmative partial answer to the question of Ulam for Banach spaces. This result was generalized by Aoki [ ] for additive mappings and independently by Rassias [ ] for linear mappings by considering an unbounded Cauchy deference
The main objective of this paper is to further extend and establish some new Ulam type stability results of the quadratic functional equations mentioned above, in which the quadratic mapping is assumed to be a fuzzy set-valued mapping

Summary

Introduction

The Ulam type stability is gradually becoming one of the most active research topics in the theory of functional equations. We shall establish the Hyers-Ulam stability of the standard quadratic fuzzy set-valued functional equation by using the fixed point technique.

Objectives

Results

Conclusion