Improved generalized differentiability conditions for Newton-like methods
The authors are concerned with the problem of approximating a locally unique solution \(x^{*}\) of a nonlinear equation \(F(x)=0\) where \(F\) is a Fréchet differentiable operator defined on a subset \(\mathcal{D}\) of the Banach space \(\mathcal{X}\) with values in the Banach space \(\mathcal{Y}\). The Newton-like approximating sequence is given by \[ x_{n+1}=x_{n}-A(x_{n})^{-1}F(x_{n}) \quad (n>0). \] Here \( A(x)\in\mathcal{L(X,Y)}, (x\in\mathcal{D}),\quad \mathcal{L(X,Y)}\) being the space of linear bounded operators from \(\mathcal{X}\) to \(\mathcal{Y}, \) and \( A(x)\) an approximation to the Fréchet-derivative. A theorem proves the semilocal convergence of the sequence. The method and the results are then extended to the equation \(F(x)+G(x)=0\) where \(G:\mathcal{D}\rightarrow\mathcal{Y}\) and the Newton-like method generating the corresponding sequence approximating the exact solution reads as follows \[ x_{n+1}=x_{n}-A(x_{n})^{-1}(F(x_{n})+G(x_{n})) \quad (n\geq 0). \] The conditions under which the convergence and the error estimates take place are better than the known ones. Examples are presented.
Chandrasekhar nonlinear integral equation
differential equation with Green's kernel
scientific article; zbMATH DE number 5763695
scientific article