The Solvability of a System of Nonlinear Equations

Abstract

It is proved: if \(\phi(\tau,\xi)\) is a scalar continuous real function of arguments \(\tau\in [a_{(n-1)},\ b_{(n-1)}]\subset R^{n-1},\) \(\xi \in [a,\ b]\subset R^{1}\) and \(\phi(\tau,a)\phi(\tau,b)<0\) for all \(\tau,\) then for all \(\varepsilon >0\) there exists a continuous function \(\phi_{0}(\tau,\xi)\) such that \(|\phi(\tau,\xi)-\phi_{0}(\tau,\xi)|<\varepsilon,\) and the equation \(\phi_{0}(\tau,\xi)=0\) has a solution continuously dependent on \(\tau\). The assertion is applied to the proof of the solvability of a finite system of nonlinear equations, to the estimation of the number of solutions. We give illustrating examples.

This is a preview of subscription content, access via your institution.

REFERENCES

  1. 1

    Danilov, V.I. Lectures on Fixed Points (Rossiiskaya Ekonomicheskaya Shkola, Moscow, 2006) [in Russian].

  2. 2

    Petrovskii, I.G. Lectures on Theory of Ordinary Differential Equations (Izdatelstvo MGU, Moscow, 1984) [in Russian].

  3. 3

    Nirenberg, L. Topics in Nonlinear Functional Analysis (Lecture Notes, Courant Inst., 1974; Mir, Moscow, 1977).

  4. 4

    Bolzano, B. Rein analytischer Beweis dass Lehrsatzes, dass zwischen je zwei Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege, Math. P. 47 d. (Haase, Prag, 1817).

  5. 5

    Mokeychev, V.S. “The Brower Theorem on Fixed Points (Simple Proof, Clarifications)” (in: Modern Problems of Theory of Functions and their Applications, Proceedings of 16 Saratov Winter School, pp. 122–123 (Saratov, 2012)).

  6. 6

    Filippov, I.E., Mokeychev, V.S. “The Least Root of a Continuous Function”, Lobachevskii J. Math. 39 (2), 200–203 (2018).

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to V. S. Mokeychev.

Additional information

Russian Text © The Author(s), 2021, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2021, No. 1, pp. 3–10.

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mokeychev, V.S. The Solvability of a System of Nonlinear Equations. Russ Math. 65, 1–7 (2021). https://doi.org/10.3103/S1066369X21010011

Download citation

Keywords

  • equation
  • smallest solution
  • continuity of solution
  • non uniqueness of solution