A Functional Equation Of Davison And Its Generalization

. We consider the functional equation f(xy) + f(x + y) = f(xy + x) + f(y) and its pexiderized version. The general solutions on R and on R+ are given. 1 Introduction During the 17th ISFE (Oberwolfach, 1979) T.M.K. Davison [1] introduced the functional equation f(xy) + f(x + y) = f(xy + x) + f(y) (1) and asked for the general solution if domain and range of f are (commutative) fields. During the meeting W. Benz proposed the following result: Every continuous solution f : R ! R of (1) for all x; y 2 R is of the form f(x) = ax + b where a; b are real constants. The aim of the present paper is to present the general solution of (1) and of its pexiderized version f(xy) + g(x + y) = h(xy + x) + k(y) (2) for x; y 2 R and for x; y 2 R+ := fx : x ? 0g. Of course, equation (2) on R+ is the most general case. From its general solution the solutions of the other cases can be deduced. But to find the solution, we have to use a non-trivial extension theorem due to Rim'an [2] which is not needed ..