Weak $ (p, q) $-Jordan centralizer and derivation on rings and algebras

In the present paper, the authors discuss two new concepts that will be known as a weak $ (p, q) $-Jordan centralizer and a weak $ (p, q) $-Jordan derivation on an arbitrary ring $ R $ and they prove that every weak $ (p, q) $-Jordan derivation is a derivation on any prime ring $ R $. Furthermore, every weak $ (p, q) $-Jordan centralizer is a centralizer on a semiprime ring $ R $. Later, they discuss the continuity of weak $ (p, q) $-Jordan centralizer on a semisimple Banach algebra and prove that every weak $ (p, q) $ -Jordan centralizer on a semisimple Banach algebra is a linear continuous operator. Moreover, these results are validated with actual examples.