Resumen de Some remarks on a problem of C. Alsina.

Janusz Matkowski, Maciej Sablik

• Equation [1] f(x+y) + f (f(x)+f(y)) = f (f(x+f(y)) + f(f(x)+y)) has been proposed by C. Alsina in the class of continuous and decreasing involutions of (0,+8). General solution of [1] is not known yet. Nevertheless we give solutions of the following equations which may be derived from [1]:

  [2] f(x+1) + f (f(x)+1) = 1, [3] f(2x) + f(2f(x)) = f(2f(x + f(x))).

  Equation [3] leads to a Cauchy functional equation:

  [4] phi(f(x)+x) = phi(f(x)) + phi(x), restricted to the graph of the function f, of the type not yet considered. We describe a general solution as well as we give some conditions sufficient for the uniqueness of solutions of [2] and [4].