Using the fixed point method, we investigate the Hyers-Ulam stability of the system of additive-cubic-quartic functional equations with constant coefficients in non-Archimedean 2-normed spaces. Also, we give an example to show that some results in the stability of functional equations in (Archimedean) normed spaces are not valid in non-Archimedean normed spaces. MSC: 39B82; 46S10; 39B52; 47S10; 47H10

Abstract. We prove the stability for the systems of quadratic-cubic and additive-quadratic-cubic functional equations with constant coefficients on the probabilistic normed spaces (briefly PN spaces).

We establish a generalized Ulam-Hyers stability theorem in aŠerstnev probabilistic normed space briefly,Šerstnev PN-space endowed with Π M . In particular, we introduce the notion of approximate Jensen mapping in PN-spaces and prove that if an approximate Jensen mapping in aŠerstnev 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 aŠerstnev PN-space can be approximated by an additive mapping, then the norm ofŠerstnev PN-space is complete.

In this paper, we prove the generalized Hyers-Ulam-Rassias stability for a system of functional equations, called general system of nonlinear functional equations, in non-Archimedean normed spaces and Menger probabilistic non-Archimedean normed spaces.

We introduce two reasonable versions of approximately additive functions in a Šerstnev probabilistic normed space endowed with triangle function. More precisely, we show under some suitable conditions that an approximately additive function can be approximated by an additive mapping in above mentioned spaces.