Geodesic
Necessary and sufficient conditions for the invertibility of the non-linear difference operator $(\mathscr Dx)(t)=x(t+1)-f(x(t))$ in the space of bounded continuous functions on the real axis
Sbornik. Mathematics, Tome 192 (2001) no. 4, pp. 565-576

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Necessary and sufficient conditions for the invertibility in the space of bounded continuous functions on $\mathbb R$ of the non-linear difference operator $$ (\mathscr Dx)(t)=x(t+1)-f(x(t)), \qquad t\in\mathbb R, $$ with $f\colon\mathbb R\to\mathbb R$ a continuous map, are obtained. 
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     author = {V. E. Slyusarchuk},
     title = {Necessary and sufficient conditions for the invertibility of the non-linear difference operator $(\mathscr Dx)(t)=x(t+1)-f(x(t))$ in the space of bounded continuous functions on the real axis},
     journal = {Sbornik. Mathematics},
     pages = {565--576},
     publisher = {mathdoc},
     volume = {192},
     number = {4},
     year = {2001},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2001_192_4_a4/}
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V. E. Slyusarchuk. Necessary and sufficient conditions for the invertibility of the non-linear difference operator $(\mathscr Dx)(t)=x(t+1)-f(x(t))$ in the space of bounded continuous functions on the real axis. Sbornik. Mathematics, Tome 192 (2001) no. 4, pp. 565-576. http://geodesic.mathdoc.fr/item/SM_2001_192_4_a4/