Implicit operator

A solution of a non-linear operator equation , in which plays the role of parameter and that of the unknown. Let , and be Banach spaces and let be a non-linear operator that is continuous in a neighbourhood of and that maps into a neighbourhood of zero in . If the Fréchet derivative is continuous on , if the operator exists and is continuous and if , then there are numbers and such that for the equation has a unique solution in the ball . Here if, additionally, is times differentiable in , then is times differentiable. If is an analytic operator in , then 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)