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The majorant method in the theory of Newton-Kantorovich approximations and the Pták error estimates. (English) Zbl 0627.65069

Let X, Y be Banach spaces, $B\left({x}_{0},r\right)$ the closed ball with the centre ${x}_{0}$ and radius r in X. Suppose that $F:B\left({x}_{0},r\right)\to Y,$ $G:B\left({x}_{0},r\right)\to Y$ are operators such that F is Fréchet differentiable on Int B(x${}_{0},r\right)$, ${F}^{\text{'}}\left({x}_{0}\right)$ is continuously invertible and $\parallel {F}^{\text{'}}\left({x}_{1}\right)-{F}^{\text{'}}\left({x}_{2}\right)\parallel \le k\left(r\right)\parallel {x}_{1}-{x}_{2}\parallel ,\parallel G\left({x}_{1}\right)-G\left({x}_{2}\right)\parallel \le ϵ\left(r\right)\parallel {x}_{1}-{x}_{2}\parallel ,$ ${x}_{1},{x}_{2}\in B\left({x}_{0},r\right)$, where k(r) and $ϵ$ (r) are nondecreasing functions on [0,R].

Put $a=\parallel {\left[{F}^{\text{'}}\left({x}_{0}\right)\right]}^{-1}\left(F\left({x}_{0}\right)+G\left({x}_{0}\right)\right)\parallel$, $b=\parallel {\left[{F}^{\text{'}}\left({x}_{0}\right)\right]}^{-1}\parallel$, $\omega \left(r\right)={\int }_{0}^{r}k\left(t\right)dt,\phi \left(r\right)=a+b{\int }_{0}^{r}\omega \left(t\right)dt-r,\psi \left(r\right)=a{\int }_{0}^{r}ϵ\left(t\right)dt·$

Theorem. Suppose that the function $\alpha \left(r\right)=\phi \left(r\right)+\psi \left(r\right)$ has a unique zero $\rho$ in [0,R] and $\alpha$ (R)$\le 0$. Then the equation $F\left(x\right)+G\left(x\right)=0$ has a solution ${x}^{*}$ in $B\left({x}_{0},\rho \right)$ and this solution is unique in the ball $B\left({x}_{0},R\right)$. Moreover, the approximations ${x}_{n+1}={x}_{n}-{\left[{F}^{\text{'}}\left({x}_{n}\right)\right]}^{-1}\left(F\left({x}_{n}\right)+G\left({x}_{n}\right)\right)$ are defined for all n, belong to $B\left({x}_{0},\rho \right)$ and satisfy the estimates $\parallel {x}_{n+1}-{x}_{n}\parallel \le {\rho }_{n+1}-{\rho }_{n},\parallel {x}^{*}-{x}_{n}\parallel \le \rho -{\rho }_{n},$ where $\left({\rho }_{n}\right)$ is monotonically increasing sequence convergent to $\rho$ and defined by the formula ${\rho }_{n+1}={\rho }_{n}-\alpha \left({\rho }_{n}\right){\left[{\phi }^{\text{'}}\left({\rho }_{n}\right)\right]}^{-1}·$ These error estimates generalize the estimates of V. Pták [Comment. Math. Univ. Carol. 16, 699-705 (1975; Zbl 0314.65023), and Numer. Math. 25, 279-285 (1976; Zbl 0304.65037)] for the usual Newton-Kantorovich method.

Reviewer: J.Kolomý

##### MSC:
 65J15 Equations with nonlinear operators (numerical methods) 47J25 Iterative procedures (nonlinear operator equations)