View source for Implicit operator

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A solution $y=f(x)$ of a non-linear operator equation $F(x,y)=0$, in which $x$ plays the role of parameter and $y$ that of the unknown. Let $X$, $Y$ and $Z$ be Banach spaces and let $F(x,y)$ be a non-linear operator that is continuous in a neighbourhood $\Omega$ of $(x_0,y_0)\in X\dotplus Y$ and that maps $\Omega$ into a neighbourhood of zero in $Z$. If the [[Fréchet derivative|Fréchet derivative]] $F_y(x,y)$ is continuous on $\Omega$, if the operator $[F_y(x_0,y_0)]^{-1}$ exists and is continuous and if $F(x_0,y_0)=0$, then there are numbers $\epsilon>0$ and $\delta>0$ such that for $||x-x_0||<\delta$ the equation $F(x,y)=0$ has a unique solution $y=f(x)$ in the ball $||y-y_0||<\epsilon$. Here if, additionally, $F(x,y)$ is $n$ times differentiable in $\Omega$, then $f(x)$ is $n$ times differentiable. If $F(x,y)$ is an [[Analytic operator|analytic operator]] in $\Omega$, then $f(x)$ is also analytic. These assertions generalize well-known propositions about implicit functions. For degenerate cases, see [[Branching of solutions|Branching of solutions]] of non-linear equations.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  T.H. Hildebrandt,   L.M. Graves,   "Implicit functions and their differences in general analysis"  ''Trans. Amer. Math. Soc.'' , '''29'''  (1927)  pp. 127–153</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  W.I. [V.I. Sobolev] Sobolew,   "Elemente der Funktionalanalysis" , H. Deutsch , Frankfurt a.M.  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.M. Vainberg,   V.A. Trenogin,   "Theory of branching of solutions of non-linear equations" , Noordhoff  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  L. Nirenberg,   "Topics in nonlinear functional analysis" , New York Univ. Inst. Math. Mech.  (1974)</TD></TR></table>



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.S. Berger,   "Nonlinearity and functional analysis" , Acad. Press  (1977)</TD></TR></table>