Some remarks on a problem of C. Alsina.

• Autores: Janusz Matkowski, Maciej Sablik
• Localización: Stochastica: revista de matemática pura y aplicada, ISSN 0210-7821, Vol. 10, Nº. 3, 1986, págs. 199-212
• Idioma: inglés
• Títulos paralelos:
  • Algunas observaciones a un problema de C. Alsina.
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• Resumen

  • 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].