Local convergence of a relaxed two-step Newton like method with applications
The article deals with two-step Newton-like approximations \[ \begin{aligned} x_{n+1} & = x_n - \mu_n F'(x_n)^{-1}[y_n,x_n;F]F'(x_n)^{-1}F(x_n), \\ y_n & = x_n + \lambda_nF'(x_n)^{-1}F(x_n), \,n = 0,1,2,\dots,\end{aligned} \] (\(\mu_n, \lambda_n \in {\mathbb R}\) are given parameters, \([x_1,x_2;F] \in L(X,Y)\) is a divided difference, \([x_1,x_2;F](x - y) = F(x) - F(y)\)) for approximative solving the nonlinear operator equation \(F(x) = 0\) with a nonlinear operator \(F\) between Banach spaces \(X\) and \(Y\). The authors assume that the following conditions hold \[ \begin{aligned} &F'(x_0)^{-1} \in L(Y,X);\\ &[F'(x_0)^{-1}](F'(x_1) - F'(x_2))\| \leq \beta\|x_1 - x_2\|;\\ &\|[F'(x_0)^{-1}](F''(x_1) - F''(x_2))\| \leq w(\|x_1 - x_2\|)\\ &\|F'(x_0)^{-1}([x_1,x_2;F] - [v_1,v_2;F])\| \leq w_1(\|x_1 - v_1\|,\|x_2 - v_2\|) \end{aligned} \] (\(w(\cdot)\), \(w_1(\cdot,\cdot)\) are continuous functions, which are nondecreasing in their arguments). The authors describe conditions under which the approximations under consideration are well defined close the solution \(x_*\), remain in a ball \(B(x_*,r)\) and converge to \(x_*\); moreover, for these approximations they obtain recurrent error estimates. Three numerical examples are presented.
scientific article; zbMATH DE number 6836687
scientific article