# Some remarks on a problem of C. Alsina.

Stochastica (1986)

• Volume: 10, Issue: 3, page 199-212
• ISSN: 0210-7821

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## Abstract

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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].

## How to cite

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Matkowski, J., and Sablik, M.. "Some remarks on a problem of C. Alsina.." Stochastica 10.3 (1986): 199-212. <http://eudml.org/doc/38955>.

@article{Matkowski1986,
abstract = {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].},
author = {Matkowski, J., Sablik, M.},
journal = {Stochastica},
keywords = {Ecuaciones funcionales; Ecuación de Cauchy; continuous solution; characterization of the inverse; continuous and decreasing involutions; general solution; Cauchy functional equation; uniqueness},
language = {eng},
number = {3},
pages = {199-212},
title = {Some remarks on a problem of C. Alsina.},
url = {http://eudml.org/doc/38955},
volume = {10},
year = {1986},
}

TY - JOUR
AU - Matkowski, J.
AU - Sablik, M.
TI - Some remarks on a problem of C. Alsina.
JO - Stochastica
PY - 1986
VL - 10
IS - 3
SP - 199
EP - 212
AB - 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].
LA - eng
KW - Ecuaciones funcionales; Ecuación de Cauchy; continuous solution; characterization of the inverse; continuous and decreasing involutions; general solution; Cauchy functional equation; uniqueness
UR - http://eudml.org/doc/38955
ER -

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