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Results in Mathematics

On Ulam Stability of a Functional Equation Krzysztof Ciepli´ nski 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 multiJensen functions. Moreover, it is satisfied by the so-called multi-linear mappings. Mathematics Subject Classification. 39B82, 39B52, 39B72, 41A99. Keywords. Ulam stability, functional equation, multi-linear mapping, system of functional equations.

1. Background and Motivation 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 (a1 x + a2 y) = A1 f (x) + A2 f (y),

x, y ∈ X

(1)

for some a1 , a2 ∈ F and A1 , A2 ∈ K. It is obvious that the functional equation f (x + y) = f (x) + f (y)

(2)

and, under the additional assumption that the characteristics of F and K are different from 2, the equation  x + y  f (x) + f (y) = , (3) f 2 2 (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]. 0123456789().: V,-vol

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K. Ciepli´ nski

Results Math

Given an n ∈ N (throughout this note N stands for the set of all positive integers and N0 := N ∪ {0}) such that n ≥ 2, we will say that a function f : X n → Y is n-linear (roughly, multi-linear) if it satisfies the linear functional equation in each of its arguments, i.e. f (x1 , . . . , xi−1 , ai1 xi1 + ai2 xi2 , xi+1 , . . . , xn ) = Ai1 f (x1 , . . . , xi−1 , xi1 , xi+1 , . . . , xn ) +Ai2 f (x1 , . . . , xi−1 , xi2 , xi+1 , . . . , xn ), i ∈ {1, . . . , n}, x1 , . . . , xi−1 , xi1 , xi2 , xi+1 , . . . , xn ∈ X with some ai1 , ai2 ∈ F and Ai1 , Ai2 ∈ K. It is clear that multi-additive functions, introduced by S. Mazur and W. Orlicz (see for example [15], where one can also find their application to the representation of polynomial mappings), and multi-Jensen functions, defined in 2005 by W. Prager and J. Schwaiger (see for instance [21]) with the connection with generalized polynomials, are multi-linear. Moreover, with k ∈ N such that 1 ≤ k < n, a11 = a12 = . . . = ak1 = ak2 = 1, ak+11 = ak+12 = . . . = an1 = an2 = 12 and A11 = A12 = . . . = Ak1 = Ak2 = 1, Ak+11 = Ak+12 = . . . = An1 = An2 = 12 in the above definition we obtain the notion of a k-Cauchy and n − k-Jensen (briefly, multi-Cauchy-Jensen) function (see [1,2]). Let a11 , a12 , . . . , an1 , an2 ∈ F and Ai1 ,...,in ∈ K for i1 , . . . , in ∈ {1, 2} be given scalars. In this paper, we deal with the following quite general functional equation f (a11 x11 + a12 x12 , . . . , an1 xn1 + an2 xn2 ) = 

i1 ,...,in ∈{1,2} Ai1 ,...,in f (x1i1 , . . . , xnin ).

(4)

Now, we present three functional equations w