Some remarks on a problem of C. Alsina.
Matkowski, J. ; Sablik, M.
Stochastica, Tome 10 (1986), p. 199-212 / Harvested from Biblioteca Digital de Matemáticas
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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,+∞). 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].

Publié le : 1986-01-01
DMLE-ID : 1723 
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     title = {Some remarks on a problem of C. Alsina.},
     journal = {Stochastica},
     volume = {10},
     year = {1986},
     pages = {199-212},
     zbl = {0649.39006},
     mrnumber = {MR0957486},
     language = {en},
     url = {http://dml.mathdoc.fr/item/urn:eudml:doc:38955}
}

Matkowski, J.; Sablik, M. Some remarks on a problem of C. Alsina.. Stochastica, Tome 10 (1986) pp. 199-212. http://gdmltest.u-ga.fr/item/urn:eudml:doc:38955/