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

Abstract

Let Y be a real separable Banach space and let $\left({𝒦}_C\left(Y\right),d_{\infty }\right)$ be the subspace of all normal fuzzy convex and upper semicontinuous fuzzy sets of Y equipped with the supremum metric $d_{\infty }$. In this paper, we introduce several types of additive fuzzy set-valued functional equations in $\left({𝒦}_C\left(Y\right),d_{\infty }\right)$. 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.