The stability problem of the functional equation was conjectured by Ulam and was solved by Hyers in the case of additive mapping. Baker et al. investigated the superstability of the functional equation from a vector space to real numbers. In this paper, we exhibit the superstability of $m$-additive maps on complete non--Archimedean spaces via a fixed point method raised by Diaz and Margolis.

In this paper we will investigate the superstability of the generalized d’Alembert type functional equations Pm i=1 f(x + σ i(y)) = kg(x)f(y) and Pm i=1 f(x + σ i(y)) = kf(x)g(y).

We obtain the superstability of the Pexiderized multiplicative functional equation fxy gxhy and investigate the stability of this equation in the following form: 1/1 ψx, y ≤ fxy/gxhy ≤ 1 ψx, y.

In this paper, we investigate the stability and superstability of homomorphisms on C∗−ternary algebras associated with the functional equation f( x+ 2y + 2z 5 ) + f( 2x+ y − z 5 ) + f( 2x− 3y − z 5 ) = f(x).

This paper is part of a program initiated by Saharon Shelah to extend the model theory of first order logic to the nonelementary setting of abstract elementary classes (AECs). An abstract elementary class is a semantic generalization of the class of models of a complete first order theory with the elementary substructure relation. We examine the symmetry property of splitting (previously isolat...

We prove that several definitions of superstability in abstract elementary classes (AECs) are equivalent under the assumption that the class is stable, tame, has amalgamation, joint embedding, and arbitrarily large models. This partially answers questions of Shelah. Theorem 0.1. Let K be a tame AEC with amalgamation, joint embedding, and arbitrarily large models. Assume K is stable. Then the fo...