Geodesic
Small solutions of nonlinear equations in sectorial neighbourhoods
The Bulletin of Irkutsk State University. Series Mathematics, Tome 3 (2010) no. 1, pp. 36-41

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We consider nonlinear operator equation $B(\lambda )x+R(x,\lambda )=0$. Linear operator $B(\lambda )$ does not have bounded inverse operator at $\lambda=0$. Nonlinear operator $R(x,\lambda)$ is continuous in neighborhood of zero, $R(0,0)=0$. We have deduced sufficient conditions of existence of the continuous solution $x(\lambda)\rightarrow0$ as $\lambda\rightarrow0$ in some open set $S$ of linear normalized space $\Lambda$. Zero belongs to frontier of set $\Lambda$. We have proposed way of construction the solution of maximum infinitesimal order in neighborhood of zero. The initial estimate is null element.
Keywords: nonlinear operator equation, minimal branch.
Mots-clés : ramification of solutions 
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     title = {Small solutions of nonlinear equations in sectorial neighbourhoods},
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R. Yu. Leontyev. Small solutions of nonlinear equations in sectorial neighbourhoods. The Bulletin of Irkutsk State University. Series Mathematics, Tome 3 (2010) no. 1, pp. 36-41. http://geodesic.mathdoc.fr/item/IIGUM_2010_3_1_a3/