On Stability and Hyperstability of an Equation Characterizing Multi-Cauchy–Jensen Mappings

Abstract

Recently, functions of several variables satisfying, with respect to each variable, some functional equation (among them Cauchy's, Jensen's, quadratic and other ones) have been studied. We give a new characterization of multi-Cauchy-Jensen mappings, which states that a function fulfilling some equation on a restricted domain is multi-Cauchy-Jensen. Next, using a fixed point theorem, it is proved that a function which approximately satisfies (on restricted domain) the equation characterizing such functions is close (in some sense) to the solution of the equation. This result is a tool for obtaining a generalized Hyers-Ulam stability or hyperstability of this equation for particular control functions, which is presented in several examples.Mathematics Subject Classification. 39B52, 39B82, 39B72, 47H10.

On Ulam Stability of a Functional Equation

“…In recent years, the stability of various objects has been studied by many researchers (for more information on the notion of the Ulam stability as well as its applications we refer the reader to [4,6,7,12,18]). Furthermore, some stability results on equations (5), (6) and (7) can be found, among others, in [1,2,9,21].…”

“…(their solutions are said to be additive and Jensen mappings, respectively) are particular cases of (1). For more information about equations (2) and (3) and some applications of them ((3) is called the Jensen equation and it is connected with the notion of convexity) we refer the reader for example to [14,15].…”

“…. = A n1 = A n2 = 1 2 in the above definition we obtain the notion of a k-Cauchy and n − k-Jensen (briefly, multi-Cauchy-Jensen) function (see [1,2]).…”

Approximate multi-Jensen-cubic mappings and a fixed point theorem

“…It is worth mentioning that the fixed point theorems have been considered for various mappings and functional equations in [2], [3], [12], [16], [26] and [31]. Similar investigations have been carried out in the stability of linear recurrence; see [14], [15], and [32].…”

Characterization and Stability of Multimixed Additive-Quartic Mappings: A Fixed Point Application

“…In the last two decades, the stability problem for several variable mappings such as multiadditive, multi-Jensen, multiquadratic, multicubic, and multiquartic mappings by applying direct and fixed point methods has been studied by a number of authors which are available for example in [1,2,4,8,9,[19][20][21][22][23][24][25][26].…”