On Some Recent Developments in Ulam's Type Stability

Abstract

We present a survey of some selected recent developments (results and methods) in the theory of Ulam's type stability. In particular we provide some information on hyperstability and the fixed point methods.

On the generalized Fréchet functional equation with constant coefficients and its stability

“…Such a method has been used in, e.g., [4][5][6]10,26,30,33,34]. Moreover, the results that we provide correspond to the outcomes in [3,9,13,16,18,21,24,25,31,32] (for more details see, e.g., [8,19,22]) and complement [4,Corollary 6]. …”

Stability of a generalization of the Fréchet functional equation

“…The results in [2] as well as our main theorem have been motivated by the notion of hyperstability of functional equations (see, e.g., [3,4,5,13,20]), introduced in connection with the issue of stability of functional equations (for more details see, e.g., [14,17]). …”

On approximate homomorphisms of ternary semigroups

“…The first work on the Ulam stability of functional equations in complete non-Archimedean normed spaces (some particular cases were considered earlier; see [3] for details) was [14]. After it a lot of papers (see, for instance, [3,18] and the references given there) on the stability of different equations in such spaces have been published.…”

“…In the last few decades, several stability problems of various (functional, differential, difference, integral) equations have been investigated by many mathematicians (see [3,8] for the comprehensive accounts of the subject), but mainly in classical spaces. However, the notion of an approximate solution and the idea of nearness of two functions can be understood in various, nonstandard ways, depending on the needs and tools available in a particular situation.…”