On perturbations of two general equations in several variables

Abstract

In this paper, we deal with perturbations of two general functional equations in several variables. Namely, we prove the generalized, in the spirit of Bourgin, Ulam stability of these equations in Banach spaces. In order to do this, we use the fixed point method. Moreover, as corollaries from our main results, we get several outcomes on approximate solutions of a few important classic equations. They include, among others, the functional equations which characterize multi-additive and multi-quadratic mappings. In consequence, the perturbation of homomorphisms of Banach spaces and quadratic mappings is also treated.

Stability of a General Functional Equation in m-Banach Spaces

“…Its stability was studied in [9,11,17], and our result complements and generalizes the corresponding outcomes obtained in these two papers.…”

“…( 3) is Ulam stable in m-Banach spaces. This complements some results from [11] as well as generalizes those from [9]. Moreover, as corollaries from our main result, we get several outcomes on approximate solutions of a few known functional equations being special cases of (3).…”

“…( 3) is Ulam stable in m-Banach spaces. This complements some outcomes from [11] (where the case of Banach spaces was studied) as well as generalizes those from [9] (where 2-Banach spaces were considered). In the proof of our main result (Theorem 2), we have used the direct method, while in [9,11] a variant of the fixed point method was applied.…”

“…This equation was recently introduced and investigated in [9,11] as a natural generalization of (2) (it is obvious that (3) with a 1,i 1 ,...,i n = . .…”

“…Moreover, as corollaries from our main result, we get several outcomes on approximate solutions of a few known functional equations being special cases of (3). Let us finally mention that in papers [9,11] a variant of the fixed point method is applied, while we use another technique: the direct/Hyers method.…”