Approximately Quadratic Mappings on Restricted Domains
  In this paper, we introduce a generalized quadratic functional equation $f(rx + sy) = rf(x) + sf(y) - rsf(x - y)$ where $r, s$ are nonzero real numbers with $r + s = 1.$ We show that this functional equation is quadratic if $r, s$ are rational numbers. We also investigate its stability problem on restricted domains. These results are applied to study of an asymptotic behavior of these generalized quadratic mappings. 
Abbas Najati And Soon-Mo Jung
math.FA