Generalized Ulam-Hyers Stability of Jensen Functional Equation in Serstnev PN Spaces

Abstract

We establish a generalized Ulam-Hyers stability theorem in a Serstnev probabilistic normed space (briefly, Serstnev PN-space) endowed with Pi(M). In particular, we introduce the notion of approximate Jensen mapping in PN-spaces and prove that if an approximate Jensen mapping in a Serstnev PN-space is continuous at a point then we can approximate it by an everywhere continuous Jensen mapping. As a version of a theorem of Schwaiger, we also show that if every approximate Jensen type mapping from the natural numbers into a Serstnev PN-space can be approximated by an additive mapping, then the norm of Serstnev PN-space is complete.