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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, $$B(x_ 0,r)$$ the closed ball with the centre $$x_ 0$$ and radius r in X. Suppose that $$F:B(x_ 0,r)\to Y,$$ $$G:B(x_ 0,r)\to Y$$ are operators such that F is Fréchet differentiable on Int B(x$${}_ 0,r)$$, $$F'(x_ 0)$$ is continuously invertible and $$\| F'(x_ 1)-F'(x_ 2)\| \leq k(r)\| x_ 1-x_ 2\|,\| G(x_ 1)-G(x_ 2)\| \leq \epsilon (r)\| x_ 1-x_ 2\|,$$ $$x_ 1,x_ 2\in B(x_ 0,r)$$, where k(r) and $$\epsilon$$ (r) are nondecreasing functions on [0,R].
Put $$a=\| [F'(x_ 0)]^{-1}(F(x_ 0)+G(x_ 0))\|$$, $$b=\| [F'(x_ 0)]^{-1}\|$$, $$\omega (r)=\int^{r}_{0}k(t)dt,\phi (r)=a+b\int^{r}_{0}\omega (t)dt-r,\psi (r)=a\int^{r}_{0}\epsilon (t)dt.$$
Theorem. Suppose that the function $$\alpha (r)=\phi (r)+\psi (r)$$ has a unique zero $$\rho$$ in [0,R] and $$\alpha$$ (R)$$\leq 0$$. Then the equation $$F(x)+G(x)=0$$ has a solution $$x^*$$ in $$B(x_ 0,\rho)$$ and this solution is unique in the ball $$B(x_ 0,R)$$. Moreover, the approximations $$x_{n+1}=x_ n-[F'(x_ n)]^{-1}(F(x_ n)+G(x_ n))$$ are defined for all n, belong to $$B(x_ 0,\rho)$$ and satisfy the estimates $$\| x_{n+1}-x_ n\| \leq \rho_{n+1}-\rho_ n,\| x^*-x_ n\| \leq \rho -\rho_ n,$$ where $$(\rho_ n)$$ is monotonically increasing sequence convergent to $$\rho$$ and defined by the formula $$\rho_{n+1}=\rho_ n-\alpha (\rho_ n)[\phi '(\rho_ n)]^{-1}.$$ 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 Numerical solutions to equations with nonlinear operators 47J25 Iterative procedures involving nonlinear operators
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##### References:
  Pták V., Comm. Math. Univ. Carolinae 16 pp 699– (1975)  V. Pták, Numer. Math. 25 pp 279– (1976) · Zbl 0304.65037  Kantorovich L. V., Nauka (1984)  Krasnosel’skij M. A., Nauka (1969)  Zabrejko P. P., Gosud. Univ., Yaroslavl pp 51– (1980)  Zabrejko P. P., Ukr. Mat. Zhurn 34 pp 365– (1982)  DOI: 10.1007/BF01463998 · Zbl 0434.65034
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