Geodesic
Linear functional equations
Izvestiya. Mathematics, Tome 12 (1978) no. 2, pp. 357-370 
L. P. Kuchko. Linear functional equations. Izvestiya. Mathematics, Tome 12 (1978) no. 2, pp. 357-370. http://geodesic.mathdoc.fr/item/IM2_1978_12_2_a9/
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     author = {L. P. Kuchko},
     title = {Linear functional equations},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1978_12_2_a9/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Necessary and sufficient conditions are found for the existence and uniqueness of local $C^\infty$ solutions of multidimensional linear functional equations of the form $$ \varphi(\Lambda x)-Q\varphi(x)=\gamma(x). $$ Sufficient conditions for local $C^\infty$ solvability are also obtained for equations with variable coefficients. Bibliography: 5 titles.

[1] Belitskii G. R., “Funktsionalnye uravneniya i lokalnaya sopryazhennost otobrazhenii klassa $C^\infty$”, Matem. sb., 133:8 (1973), 565–579

[2] Kassels Dzh. V. S., Vvedenie v teoriyu diofantovykh priblizhenii, IL, M., 1961

[3] Kuchko L. P., “O lineinykh funktsionalnykh uravneniyakh”, Vestnik Kharkovskogo un-ta, ser. mekhaniko-matem., 39 (1974), 3–14 | Zbl

[4] Malgranzh B., Idealy differentsiruemykh funktsii, Mir, M., 1968

[5] Tychonoff A. N., “Ein Fixpunktsatz”, Math. Ann., 111:5 (1935), 767–776 | DOI | MR | Zbl