Implicit operator

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 $F_y(x,y)$ 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 in $\Omega$, then $f(x)$ is also analytic. These assertions generalize well-known propositions about implicit functions. For degenerate cases, see Branching of solutions of non-linear equations.

References

[1]	T.H. Hildebrandt, L.M. Graves, "Implicit functions and their differences in general analysis" Trans. Amer. Math. Soc. , 29 (1927) pp. 127–153
[2]	W.I. [V.I. Sobolev] Sobolew, "Elemente der Funktionalanalysis" , H. Deutsch , Frankfurt a.M. (1979) (Translated from Russian)
[3]	M.M. Vainberg, V.A. Trenogin, "Theory of branching of solutions of non-linear equations" , Noordhoff (1974) (Translated from Russian)
[4]	L. Nirenberg, "Topics in nonlinear functional analysis" , New York Univ. Inst. Math. Mech. (1974)

References

[a1]	M.S. Berger, "Nonlinearity and functional analysis" , Acad. Press (1977)