Lyusternik's theorem and the best parametrization of solutions to nonlinear equations (Q1264755)

Various aspects of parameter continuation method for solving nonlinear equations \(F(x)=0\) in the neighbourhood of the known solution \(x_{0}\) \((F(x_{0})=0)\) are developed on the base of the known Lyusternik's theorem about tangent manifold. At certain sufficient conditions existence theorems are proved, where the unique solution can be found by successive approximations. The estimates for convergence rate are given. Applications to branching theory are indicated. In the last direction the reviewer's work [\textit{B. V. Loginov} and \textit{V. E. Pospeev}, Izv. Akad. Nauk Uz. SSR, Ser. Fiz.-Mat. Nauk 15, No. 2, 64-66 (1971; Zbl 0232.47077)] must be noted.