On approximate isometries and application to stability of a functional equation

In this paper, we show a generalized stability of isometries between Banach spaces. Making use of this result, we prove the corresponding stability of the functional equation ‖f(x−y)‖=‖f(x)−f(y)‖‖f(x−y)‖=‖f(x)−f(y)‖.