id: 05763695 dt: j an: 05763695 au: Argyros, Ioannis K.; Hilout, Saïd ti: Improved generalized differentiability conditions for Newton-like methods. so: J. Complexity 26, No. 3, 316-333 (2010). py: 2010 pu: Elsevier Science (Academic Press), San Diego, CA la: EN cc: ut: Newton-like methods; semilocal convergence; Newton-Kantorovich method; numerical examples; nonlinear operator equation; majorizing sequence; Chandrasekhar nonlinear integral equation; radiative transfer; differential equation with Green’s kernel; Fréchet differentiable operator; Banach space; error estimates ci: li: doi:10.1016/j.jco.2009.12.001 ab: 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. rv: Erwin Schechter (Moers)