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This article has two aims. First, we provide the solution to a problem posed by the author in a previous paper. Second, we consider a problem posed by Kannappan and Kurepa (Aequat Math 4: 163-175, 1970). Our results show that additive functions linked by certain types of functional equations are combinations of linear functions and derivations of various orders. We show that this is not generally the case for the problem of Kannappan and Kurepa, and we modify their problem accordingly. [ABSTRACT FROM AUTHOR]

In this note we provide the solution to a problem posed by the first author in a previous paper. In particular, we prove a result relating the number of nonzero coefficients of a certain functional equation to the order of any derivation satisfying that equation. [ABSTRACT FROM AUTHOR]

We continue the study of additive functions fk:R→F(1≤k≤n) linked by an equation of the form ∑k=1npk(x)fk(qk(x))=0, where the pk and qk are polynomials, R is an integral domain of characteristic 0, and F is the fraction field of R. A method is presented for solving all such equations. We also consider the special case ∑k=1nxmkfk(xjk)=0 in which the pk and qk are monomials. In this case we show that if there is no duplication, i.e. if (mk,jk)≠(mp,jp) for k≠p, then each fk is the sum of a linear function and a derivation of order at most n-1. Furthermore, if this functional equation is not homogeneous then the maximal orders of the derivations are reduced in a specified way. [ABSTRACT FROM AUTHOR]