n-DERIVATIONS AND FUNCTIONAL INEQUALITIES WITH APPLICATIONS

We prove that every bounded n-derivation of a commutative factorizable Banach algebra maps into its radical. Also, the nilpotency of eigenvectors of any bounded n-derivation corresponding to its eigenvalues is derived. We introduce the notion of approximate n-derivations on a Banach algebra A and show that the separating space of an approximate n-derivation (n > 2 ) is not necessarily an ideal, unless the Banach algebra A is factorizable. From this and some results on bounded n-derivations. we prove that every approximate n-derivation of a semisimple factorizable Banach algebra is automatically continuous and every approximate n-derivation of a commutative semisimple factorizable Banach algebra is identically zero. Some applications of our results are also provided.