stability of the quadratic functional equation

Abstract

In the present paper a solution of the generalized<br />quadratic functional equation<br />$$<br />f(kx+ y)+f(kx+sigma(y))=2k^{2}f(x)+2f(y),phantom{+} x,yin{E}$$ is<br />given where $sigma$ is an involution of the normed space $E$ and<br />$k$ is a fixed positive integer. Furthermore we investigate the<br />Hyers-Ulam-Rassias stability of the functional equation. The<br />Hyers-Ulam stability on unbounded domains is also studied.<br />Applications of the results for the asymptotic behavior of the<br />generalized quadratic functional equation are provided.