Geodesic
One functional equation with displacement in the space of continuous functions
Matematičeskie zametki, Tome 22 (1977) no. 2, pp. 303-311.

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In the space of continuous functions defined on a simple continuous contour, we examine the functional equation \begin{equation} a(t)\varphi(t)+b(t)\varphi[\alpha(t)]=g(t).\tag{1} \end{equation} A criterion for Eq. (1) being Noetherian is established under the condition that there exist a finite number of fixed points on the first multiplicity in the homeomorphism $\alpha(t)$ of the contour onto itself. 
@article{MZM_1977_22_2_a14,
     author = {V. G. Kravchenko},
     title = {One functional equation with displacement in the space of continuous functions},
     journal = {Matemati\v{c}eskie zametki},
     pages = {303--311},
     publisher = {mathdoc},
     volume = {22},
     number = {2},
     year = {1977},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_22_2_a14/}
}
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V. G. Kravchenko. One functional equation with displacement in the space of continuous functions. Matematičeskie zametki, Tome 22 (1977) no. 2, pp. 303-311. http://geodesic.mathdoc.fr/item/MZM_1977_22_2_a14/