Abstract
In this note, we study the Ulam stability of a functional equation both in Banach and m-Banach spaces. Particular cases of this equation are, among others, equations which characterize multi-additive and multi-Jensen functions. Moreover, it is satisfied by the so-called multi-linear mappings.
Highlights
Background and MotivationAssume that X is a linear space over the field F, and Y is a linear space over the field K
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]
Given an n ∈ N such that n ≥ 2, we will say that a function f : Xn → Y is n-linear if it satisfies the linear functional equation in each of its arguments, i.e
Summary
Assume that X is a linear space over the field F, and Y is a linear space over the field K. Let us recall (see for instance [20]) that a mapping f : X → Y satisfies a linear functional equation provided f (a1x + a2y) = A1f (x) + A2f (y), x, y ∈ X (1). It is obvious that the functional equation f (x + y) = f (x) + f (y). Under the additional assumption that the characteristics of F and K are different from 2, the equation f x + y = f (x) + f (y) ,. (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]
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