On Ulam Stability of Functional Equations in 2-Normed Spaces—A Survey II

Abstract

Ulam stability is motivated by the following issue: how much an approximate solution of an equation differs from the exact solutions to the equation. It is connected to some other areas of investigation, e.g., optimization, approximation theory and shadowing. In this paper, we present and discuss the published results on such stability for functional equations in the classes of function-taking values in 2-normed spaces. In particular, we point to several pitfalls they contain and provide possible simple improvements to some of them. Thus we show that the easily noticeable symmetry between them and the analogous results proven for normed spaces is, in fact, mainly apparent. Our article complements the earlier similar review published in Symmetry (13(11), 2200) because it concerns the outcomes that have not been discussed in this earlier publication.

Stability in the Sense of Hyers–Ulam–Rassias for the Impulsive Volterra Equation

“…It should be noted that a logical extension of this idea is n-normed space (see, e.g., [18]), i.e., the 2-normed space is an n-normed space, with n = 2. For more stability results (especially in 2-normed spaces), readers are advised to read [19].…”

Hyperstability for a Generalized Class of Pexiderized Functional Equations on Monoids via Páles’ Approach

“…We refer, for example, to [10][11][12][13][14][15][16]. In addition, there are dozens of published papers that are concerned with the hyperstability of functional equations; see, for instance, [17][18][19][20][21].…”

Untitled

“…Another point worth noting is that there were other versions of Theorem 1.3 in ultrametric space [3], in 2-Banach space [4], [18], and in n-Banach space [19] that helped to discuss many results on the stability of functional equations. For more details on the stability and hyperstability in 2-Banach spaces and n-Banach spaces, we refer the reader to seeing the surveys [7] and [25].…”