Abstract
This paper investigates a class of nonlinear operator equations in a finite-dimensional Hilbert space. Specifically, equations of the form u+L(u) = g are considered, where L is a nonlinear continuous operator acting on the space H, and g is a given element of this space. The main focus is on deriving a priori estimates for solutions and studying the conditions for their existence. For a sufficiently broad class of nonlinear operators, uniform estimates of solutions are established, expressed in terms of the parameters of the original problem and the properties of the operator under consideration. It is shown that these estimates do not depend on the index of the finite-dimensional approximation, which allows them to be used in the study of the corresponding infinite-dimensional problems. The results justify the method of finite-dimensional approximations and to carry out the limiting transition to the original problem. This makes it possible to obtain a priori estimates for solutions of infinite-dimensional nonlinear operator equations arising in problems of mathematical physics and functional analysis. The proposed approach can be applied to the study of a wide class of nonlinear problems and to the construction of approximate methods for their solution.