On the generalized Hyers–Ulam stability of multi-quadratic mappings

On an equation characterizing multi-Jensen-quadratic mappings and its Hyers–Ulam stability via a fixed point method

“…Our results are significant supplements and/or generalizations of some results from [1,12,13,14,16,18,23,27,29,30,31].…”

On an equation characterizing multi-additive-quadratic mappings and its Hyers–Ulam stability

“…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 [28], where their application to the representation of polynomial functions is also presented); for k = 0 we obtain the notion of multi-quadratic function (see [20]), and an 1-additive and 1-quadratic mapping is just an additivequadratic mapping defined by Park et al [33].…”

Solution and Stability of the Multiquadratic Functional Equation

“…A general version of the biquadratic functional equation is the multiquadratic functional equation. Recall from [22] that a mapping : → , where is a commutative group, is a linear space, and ≥ 2 is an integer, is called multiquadratic if it is quadratic in each variable. On the other hand, for more details about the multiadditive (resp., the multi-Jensen mappings) (i.e., mappings satisfying Cauchy's (resp., Jensen's) functional equation in each variable) and the stability for them, one can see [23][24][25][26][27][28] and the references given there.…”

“…For example, Park [29] proved the stability of the multiquadratic functional equation in Banach spaces. Ciepliński [22] proved the stability of this functional equation in complete non-Archimedean spaces as well as in Banach spaces but using the fixed point method. However, to our knowledge, not many results are known about the solution of this functional equation.…”