Evaluation of solutions of one class of finite-dimensional nonlinear equations

In this article, we obtain a theorem on a priori estimates for solutions of nonlinear equations in a finite-dimensional space. This theorem is proved under certain conditions which are borrowed from the conditions that are satisfied by finite-dimensional approximations of one class of nonlinear initial-boundary-value problems. The main result establishes sufficient conditions for the existence of a solution to A(u)=f, where A is a nonlinear operator. An example is given to illustrate the applicability of the main result to nonlinear analysis and mathematical physics