An implicit function theorem: Comment

  • S. Kumagai
Technical Comment

Abstract

In Ref. 1, Jittorntrum proposed an implicit function theorem for a continuous mappingF:Rn ×RmRn, 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):ARnRn for allyB is consciously or unconsciously treated as implying thatF(·,y) mapsA one-to-one ontoF(A, y) for allyB, 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 the condition is truly if and only if.

Key Words

Implicit function theorem nonlinear network analysis computer-aided design 

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References

  1. 1.
    Jittorntrum, K.,An Implicit Function Theorem, Journal of Optimization Theory and Applications, Vol. 25, No. 4, 1978.Google Scholar
  2. 2.
    Ohtsuki, T., Fujisawa, T., andKumagai, S.,Existence Theorem and a Solution Algorithm for Piecewise-Linear Resistor Networks, SIAM Journal on Mathematical Analysis, Vol. 8, No. 1, 1977.Google Scholar
  3. 3.
    Kumagai, S.,On the Existence of Solution of Large-Scale Nonlinear Networks, Transaction of the Institute of Electronics and Communication Engineers of Japan, Vol. 59-A, No. 11, 1976.Google Scholar

Copyright information

© Plenum Publishing Corporation 1980

Authors and Affiliations

  • S. Kumagai
    • 1
  1. 1.Department of Electronic Engineering, Faculty of EngineeringOsaka UniversityOsakaJapan

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