Geodesic
On the Functional Equation f phi f = f
Publications de l'Institut Mathématique, _N_S_29 (1981) no. 43, p. 61 
Valentina Harizanov. On the Functional Equation f phi f = f. Publications de l'Institut Mathématique, _N_S_29 (1981) no. 43, p. 61 . http://geodesic.mathdoc.fr/item/PIM_1981_N_S_29_43_a8/
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     title = {On the {Functional} {Equation} f phi f = f},
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     number = {43},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PIM_1981_N_S_29_43_a8/}
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Voir la notice de l'article provenant de la source eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts

In this note we determine the general solution of the equation $f\varphi f=f$, where $f:X\to Y$ is a given function and $\varphi:Y\to X$ is an unkown function ($X$ and $Y$ are arbitrary nonempty sets). The general solution of that equation is given by the formula (4), where $\varphi_0:Y\to X$ is a particular solution, $k:Y\to X$ and $h:X\to X$ are arbitrary functions, $F:X^3\times Y^3\to X$ is defined by (3).