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.