Geodesic
The solvability of a system of nonlinear equations
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2021), pp. 3-10

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It is proved: if $\phi(\tau,\xi)$ is 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\ \forall \tau, $ then for each $\varepsilon >0$ exists a continuous $\phi_{0}(\tau,\xi),$ that $|\phi(\tau,\xi)- \phi_{0}(\tau,\xi)|\varepsilon $ and the equation $\phi_{0}(\tau,\xi)=0$ has continuously depends on $\tau$ solution. The statement is suitable to a proof of a solvability finite system nonlinearity equations, to an estimation of a number of solutions. We give illustrating examples.
Mots-clés : equation
Keywords: smallest solution, non uniqueness of solution. 
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     author = {V. S. Mokeychev},
     title = {The solvability of a system of nonlinear equations},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {3--10},
     publisher = {mathdoc},
     number = {1},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2021_1_a0/}
}
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V. S. Mokeychev. The solvability of a system of nonlinear equations. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 1 (2021), pp. 3-10. http://geodesic.mathdoc.fr/item/IVM_2021_1_a0/