On the Halley method in Banach spaces
The article deals with the Halley modification of the Newton-Kantorovich method for an approximate solution of a nonlinear operator equation \(F(x) = 0\); the iteration in this modification is defined by NEWLINE\[NEWLINEx_{n+1} = x_n - [I - L_F(x_n)]^{-1}F'(x_n)^{-1}F(x_n) \qquad (n \geq 0, \;x_0 \in \Omega),NEWLINE\]NEWLINE where NEWLINE\[NEWLINEL_F(x) = \frac12 f'(x)^{-1}F''(x)F'(x)^{-1}F(x).NEWLINE\]NEWLINE It is assumed that the operator \(F\) satisfies the majorant inequality NEWLINE\[NEWLINE\|F'(x_0)^{-1}[F''(y) - F''(x)]\| \leq f''(\|y - x\| + \|x - x_0\|) - f''(\|x - x_0\|),NEWLINE\]NEWLINE where \(f\) is a convex and incresing scalar function. In terms of this function, the authors formulate some new (rather complicated and cumbersome) conditions that guarantee the existence of the Halley approximations in a ball \(U(x_0,R)\), the convergence of these approximations to a solution of the equation \(F(x) = 0\) (lying in the ball \(U(x_0,R)\)), and \textit{a priori} and \textit{a posteriori} error estimates. Some special cases are presented at the end of the article. Besides, in the article, some remarks about the comparison of the obtained results with some previous ones are also given.
scientific article; zbMATH DE number 6024873