On a Cauchy–Jensen functional equation and its stability

Abstract

In this paper, we obtain the general solution and stability of the Cauchy-Jensen functional equation

On a functional equation connected with bi-linear mappings and its Hyers-Ulam stability

“…Let us point out that these equations characterize bi-additive, bi-Jensen and Cauchy-Jensen mappings, respectively (see [2,6,16]). …”

“…In particular, the stability of functional equations (1.6), (1.7) and (1.8) has been studied, among others in [2,[6][7][8][16][17][18].…”

On the Ulam stability of the Cauchy-Jensen equation and the additive-quadratic equation

“…Letting j → ∞, F satisfies (1.1). By Theorem 4 in [7], F is a Cauchy-Jensen mapping. Setting l = 0 and taking m → ∞ in (2.15), one can obtain the inequality (2.9).…”

“…Let F : Y × Y → X be another mapping satisfying (1.1) and (2.2). By [7], there exist bi-additive mappings B, B : Y × Y → X and additive mappings A, A : Y → X such that F (x, y) = B(x, y) + A(x) and F (x, y) = B (x, y) + A (x) for all x, y ∈ Y . Since r > log 2 6, we obtain that…”

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

“…Let us note that for k = n the above definition leads to the so-called multi-additive mappings (some basic facts on such mappings can be found for instance in [30], where their application to the representation of polynomial functions is also presented); for k = 0 we obtain the notion of multi-Jensen function (which was introduced in 2005 by Prager and Schwaiger (see [33]) in the connection with generalized polynomials), and an 1-Cauchy and 1-Jensen mapping is just a Cauchy-Jensen mapping defined by Park and Bae [32].…”