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)