Approximate quadratic forms on restricted domains (Q2794929)

The concept of stability for a functional equation arises when one replaces a functional equation by an inequality, which acts as a perturbation of the equation.NEWLINENEWLINEIn the 1940's S. M. Ulam posed the first stability problem. In the following year, \textit{D. H. Hyers} [Proc. Natl. Acad. Sci. USA 27, 222--224 (1941; Zbl 0061.26403; JFM 67.0424.01)] gave a partial affirmative answer to the question of Ulam. Hyers' theorem was generalized by \textit{T. Aoki} [J. Math. Soc. Japan 2, 64--66 (1950; Zbl 0040.35501)] for additive mappings and by \textit{T. M. Rassias} [Proc. Am. Math. Soc. 72, 297--300 (1978; Zbl 0398.47040)] for linear mappings by considering an unbounded Cauchy difference. \textit{A. Najati} and \textit{S.-M. Jung} [J. Inequal. Appl. 2010, Article ID 503458, 10 p. (2010; Zbl 1215.39036)] introduced a generalized quadratic functional equation and the authors solved this quadratic functional equation for the \(2\)-variable case.NEWLINENEWLINEIn the present work, the authors consider the quadratic functional equation \(f(rx+sy,z+w)+rsf(x-y,z-w)=rf(x,z)+sf(y,w)\), where \(r, s\) are nonzero real numbers with \(r+s=1\) and find out the general solution and the Hyers-Ulam stability of this type of functional equations.