Hyers-Ulam-Rassias Stability of Some Additive Fuzzy Set-Valued Functional Equations with the Fixed Point Alternative

Yonghong Shen,Wei Chen,Yaoyao Lan

doi:10.1155/2014/139175

Abstract

LetYbe a real separable Banach space and let𝒦CY,d∞be the subspace of all normal fuzzy convex and upper semicontinuous fuzzy sets ofYequipped with the supremum metricd∞. In this paper, we introduce several types of additive fuzzy set-valued functional equations in𝒦CY,d∞. Using the fixed point technique, we discuss the Hyers-Ulam-Rassias stability of three types additive fuzzy set-valued functional equations, that is, the generalized Cauchy type, the Jensen type, and the Cauchy-Jensen type additive fuzzy set-valued functional equations. Our results can be regarded as important extensions of stability results corresponding to single-valued functional equations and set-valued functional equations, respectively.

Highlights

IntroductionIn 1940, Ulam [1] proposed the following question concerning the stability of group homomorphisms
In 1940, Ulam [1] proposed the following question concerning the stability of group homomorphisms.Let G1 be a group and let G2 be a metric group with the metric d(⋅, ⋅)
The result of Hyers was generalized by Aoki [3] for additive mappings and by Rassias [5] for linear mappings in which the Cauchy difference is allowed to be unbounded

Summary

Introduction

In 1940, Ulam [1] proposed the following question concerning the stability of group homomorphisms. Cieplinski [9] summarized some applications of several types of fixed point theorems to the Hyers-Ulam stability of functional equations As of this method has been successfully used in the study of stability problems of many types of functional equations in abstract spaces. It should be pointed out that, in their studies, the inclusion relation is applied to characterize the set-valued functional inequality rather than an appropriate metric. Similar to the method that is used to deal with the single-valued functional equations, Kenary et al [19] proved the stability of several types of set-valued functional equations via the fixed point approach, in which the Hausdorff metric is adopted to characterize the set-valued functional inequality. Notice that the supremum metric, as a generalization of the Hausdorff metric, is applied to characterize the fuzzy set-valued functional inequality. The corresponding single-valued and set-valued functional equations acted as special cases will be included in our results

Preliminaries

Stability of the Jensen Type Additive Fuzzy Set-Valued Functional Equation

Concluding Remark