Geodesic
On maximum infinitesimal order solutions of nonlinear equations in sectorial neighbourhood of zero
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, no. 7 (2011), pp. 66-70

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A nonlinear operator equation $B(\lambda)x=R(x,\lambda)+b(\lambda)$ with conditions $R(0,0)\equiv0$, $b(0)=0$ is considered. The linear operator $B(\lambda)$ hasn't a continuous inverse operator at $\lambda=0,$ but it has a bounded inverse operator when $\lambda\in S$, where $S$ is a set named a sectorial neighbourhood of zero. The question of existence of infinitesimal continuous solutions $x(\lambda)\rightarrow0$ at $\lambda\in S$ when $\lambda\rightarrow0.$ The proved theorems propose a constructive way the solution of the maximum infinitesimal order.
Keywords: an implicit function theorem, a nonlinear operator equation, a sectorial neighbourhood, a maximum infinitesimal order solution. 
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     author = {R. Y. Leontyev},
     title = {On maximum infinitesimal order solutions of nonlinear equations in sectorial neighbourhood of zero},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matemati\v{c}eskoe modelirovanie i programmirovanie},
     pages = {66--70},
     publisher = {mathdoc},
     number = {7},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VYURU_2011_7_a8/}
}
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R. Y. Leontyev. On maximum infinitesimal order solutions of nonlinear equations in sectorial neighbourhood of zero. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, no. 7 (2011), pp. 66-70. http://geodesic.mathdoc.fr/item/VYURU_2011_7_a8/