Solution and Stability of a Generalized K–additive Functional Equation

In this paper, we obtain the general solution of a new generalized k - additive functional equation of the form kf(x+ky) - f(kx+y) =k(k(2)-1)/2 [f(x+y)+ f(x-y)] + (k-k(3)) f(x)+(k(2)-1)f(y) for all x, y is an element of X with any real k is an element of R-{0,+/- 1} and investigate its Hyers-Ulam Rassias stabilities in Banach spaces.