Geodesic
On global existence of an~implicit function
Sbornik. Mathematics, Tome 79 (1994) no. 2, pp. 287-313

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The problem of global existence of an implicit function is studied, i.e., the properties of Banach spaces $X$, $Y$, $Z$ and functions $$ F\colon X\times Y\to Z, $$ for which a smooth solution $y=\varphi(x)$ of the equation $F(x,y) = 0$ is possible with given initial condition $y_0=\varphi(x_0)$, where $ F(x_0,y_0)=0$. It is shown that excessive smoothness of $F$ with respect to $y$ is necessary for the existence of a smooth global solution (in comparison with a local solution). 
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     author = {I. G. Tsar'kov},
     title = {On global existence of an~implicit function},
     journal = {Sbornik. Mathematics},
     pages = {287--313},
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     volume = {79},
     number = {2},
     year = {1994},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1994_79_2_a3/}
}
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I. G. Tsar'kov. On global existence of an~implicit function. Sbornik. Mathematics, Tome 79 (1994) no. 2, pp. 287-313. http://geodesic.mathdoc.fr/item/SM_1994_79_2_a3/