On the Ulam stability of an \(n\)-dimensional quadratic functional equation. (English) Zbl 1329.39034

Shen, Yonghong; Chen, Wei

J. Nonlinear Sci. Appl. 9, No. 1, 332-341 (2016).

Summary: We construct a new \(n\)-dimensional quadratic functional equation with constant coefficients \[ \sum_{i,j=1}^n f(x_i+x_j) = 2 \sum_{1\leq i < j \leq n} f(x_i-x_j) + 4f \left( \sum_{i=1}^n x_i \right). \] And then, we study the Ulam stability of the preceding equation in a real normed space and a non-Archimedean space, respectively.

MSC:

39B82	Stability, separation, extension, and related topics for functional equations
39B52	Functional equations for functions with more general domains and/or ranges
46S10	Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis

Keywords:

Ulam stability; \(n\)-dimensional quadratic functional equation; normed space; non-Archimedean space