An implicit function theorem: Comment

@article{Kumagai1980AnIF,
  title={An implicit function theorem: Comment},
  author={Sadatoshi Kumagai},
  journal={Journal of Optimization Theory and Applications},
  year={1980},
  volume={31},
  pages={285-288}
}
  • S. Kumagai
  • Published 1 June 1980
  • Mathematics
  • Journal of Optimization Theory and Applications
In Ref. 1, Jittorntrum proposed an implicit function theorem for a continuous mappingF:Rn ×Rm →Rn, withF(x0,y0)=0, that requires neither differentiability ofF nor nonsingularity of ∂xF(x0,y0). In the proof, the local one-to-one condition forF(·,y):A ⊂Rn →Rn for ally ∈B is consciously or unconsciously treated as implying thatF(·,y) mapsA one-to-one ontoF(A, y) for ally ∈B, and the proof is not perfect. A proof can be given directly, and the theorem is shown to be the strongest, in the sense that… Expand
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An implicit function theorem
Suppose thatF:D⊂Rn×Rm→Rn, withF(x0,y0)=0. The classical implicit function theorem requires thatF is differentiable with respect tox and moreover that ∂1F(x0,y0) is nonsingular. We strengthen thisExpand
Existence Theorems and a Solution Algorithm for Piecewise-Linear Resistor Networks
This paper deals with nonlinear networks which can be characterized by the equation ${\bf f}({\bf x}) = {\bf y}$, where ${\bf f}$ is a continuous piecewise-linear mapping from $R^n $ into itself. TheExpand