Hyers-Ulam stability of quadratic forms in 2-normed spaces

Abstract

In this paper, we obtain Hyers-Ulam stability of the functional equationsf (x + y, z + w) + f (x − y, z − w) = 2f (x, z) + 2f (y, w),f (x + y, z − w) + f (x − y, z + w) = 2f (x, z) + 2f (y, w)andf (x + y, z − w) + f (x − y, z + w) = 2f (x, z) − 2f (y, w)in 2-Banach spaces. The quadratic forms ax2 + bxy + cy2, ax2 + by2 and axy are solutions of the above functional equations, respectively.

On Ulam Stability of Functional Equations in 2-Normed Spaces—A Survey

“…Several other stability results in 2-normed spaces (also non-Archimedean and random) have been presented in [65] (for the Pexiderized Cauchy functional equation), in [17,66] (for the Cauchy equation), in [67] (for a generalized radical cubic functional equation related to quadratic functional equation), in [68] (for the radical quartic functional equation), in [69] (for the quadratic functional equation), in [70] (for the generalized Cauchy functional equation), in [71] (for a functional equation called the Cauchy-Jensen functional equation), in [72] (for a general p-radical functional equation) [73] (for radical sextic functional equation), in [74] (for several functional equations of quadratic-type), in [75,76] (for the functional equation of p-Wright affine functions), in [77] (for a system of additive-cubicquartic functional equations with constant coefficients in non-Archimedean 2-normed spaces), in [78] (for a functional inequality in non-Archimedean 2-normed spaces), in [79] (for a cubic functional equation in random 2-normed spaces), in [80] (for the Pexiderized quadratic functional equation in the random 2-normed spaces) and in [81] (for radical functional equations in 2-normed spaces and p-2-normed spaces). However, as these results are more involved and of a different character than those presented so far, we will discuss them in more details in another publication.…”

On Ulam Stability of Functional Equations in 2-Normed Spaces—A Survey II

“…In this paper, we have presented and discussed the results on Ulam stability in 2normed spaces provided in articles [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58]. In this way, we complement the paper [23], where the results from [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] have been surveyed.…”

“…Actually, it has been assumed in Theorem 2.1 of [48] that > 0, but it is easily seen that the theorem is also true for = 0; this can be deduced from Theorem 2.1 of [48] (i.e., from our Theorem 18) with → 0, or from the proof of it.…”

“…Namely, it is necessary for the set g(R) to include at least two linearly independent vectors because otherwise, the statement is not true. In fact, if g(R) ⊂ Rv with some v ∈ Y, then for every mapping f : R → Y with f (R) ⊂ Rv condition (53) holds with φ(x, y, u) ≡ 0, while f does not necessarily satisfy (48) for all x, y ∈ R (in this case φ(x, z) ≡ 0, whence (54) means that f = h).…”

“…In this paper, we complement the content of [23], where the (less complicated) results from [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] have been surveyed. Here, we present and discuss the (more involved) outcomes on Ulam stability in 2-normed spaces provided in [40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58].…”