On some local topological semigroups

Summary

We determine all continuous functionsf, defined on a real intervalI with 0∈ I, taking values in ℝ and such that the operationA _f :I × I → ℝ given by

$$A_f (x,y) = xf(y) + yf(x)$$

is locally associative, i.e., for allx, y, z ∈ I, ifA _f (x, y) ∈ I andA _f (y, z) ∈ I, thenA _f (A _f (x, y), z) = A _f (x, A _f (y, z)). The problem leads to the following functional equation

$$f(A_f (x,y)) = f(x)f(y) + cxy$$

wherec is a real constant andx, y ∈ I are such thatA _f (x, y) ∈ I. We solve this equation generalizing thus some earlier results obtained by N. Brillouet and J. Dhombres [3] who solved it in the caseI = ℝ andc = 0, as well as those obtained by P. Volkmann and H. Weigel [7] who were dealing with an equivalent form of this equation in the caseI = ℝ andc >; 0. Some partial results concerning the problem can be found in our paper [6].