Cauchy-Rassias stability of a generalized Trif's mapping in Banach modules and its applications

Let X and Y be vector spaces. It is shown that a mapping f:X→Y satisfies the functional equation (0.1)mnmn-2Ck-2fx1+⋯+xmnmn+mn-2Ck-1∑i=1mnfxi+⋯+xi+m-1m=k∑1⩽i1<⋯