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

Let Y be a real separable Banach space and let CY,d∞ be the subspace of all normal fuzzy convex and upper semicontinuous fuzzy sets of Y equipped with the supremum metric d∞. In this paper, we introduce several types of additive fuzzy set-valued functional equations in CY,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

Hyers-Ulam stability of additive set-valued functional equations

In this paper, we define the following additive set-valued functional equations f(alpha chi + beta y) = rf (chi) sf (y), (1) f(x + y + z) = 2f (x + y/2) + f(z) (2) for some real numbers alpha > 0, beta > 0, r, s is an element of R with alpha + beta = r + s not equal 1, and prove the Hyers-Ulam stability of the above additive set-valued functional equations. (C) 2011 Elsevier Ltd. All rights reserved.The second author was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788)