The Solvability of a System of Nonlinear Equations


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.

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Correspondence to V. S. Mokeychev.

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

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Mokeychev, V.S. The Solvability of a System of Nonlinear Equations. Russ Math. 65, 1–7 (2021).

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