Geodesic
On a~problem in the theory of nonlinear Fredholm operators
Matematičeskie zametki, Tome 10 (1971) no. 5, pp. 541-549

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We prove that for any Fredholm operator $A(x)$ of class $C^1$ and zero index in a Hilbert space, in a neighborhood of any star compactum $T$ lying in the domain of a we can define a completely continuous and continuously differentiable operator $C$, so that the linear operator $A'(x)+C'(x)$ has a bounded inverse for all $x\in T$. 
@article{MZM_1971_10_5_a8,
     author = {V. I. Ovchinnikov},
     title = {On a~problem in the theory of nonlinear {Fredholm} operators},
     journal = {Matemati\v{c}eskie zametki},
     pages = {541--549},
     publisher = {mathdoc},
     volume = {10},
     number = {5},
     year = {1971},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1971_10_5_a8/}
}
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V. I. Ovchinnikov. On a~problem in the theory of nonlinear Fredholm operators. Matematičeskie zametki, Tome 10 (1971) no. 5, pp. 541-549. http://geodesic.mathdoc.fr/item/MZM_1971_10_5_a8/