Geodesic
Picard's theorem for ordinary differential equations in locally convex spaces
Izvestiya. Mathematics , Tome 41 (1993) no. 3, pp. 465-487

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A class of infinite-dimensional Frechet spaces is constructed, including certain subspaces of $C^\infty[-1,1]$, in which Picard's theorem on solvability of an ODE with smooth right-hand side is valid in the usual formulation. Every continuous linear operator on these spaces has an exponential. 
@article{IM2_1993_41_3_a3,
     author = {S. G. Lobanov},
     title = {Picard's theorem for ordinary differential equations in locally convex spaces},
     journal = {Izvestiya. Mathematics },
     pages = {465--487},
     publisher = {mathdoc},
     volume = {41},
     number = {3},
     year = {1993},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1993_41_3_a3/}
}
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S. G. Lobanov. Picard's theorem for ordinary differential equations in locally convex spaces. Izvestiya. Mathematics , Tome 41 (1993) no. 3, pp. 465-487. http://geodesic.mathdoc.fr/item/IM2_1993_41_3_a3/