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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:
 [1] Pták V., Comm. Math. Univ. Carolinae 16 pp 699– (1975) [2] V. Pták, Numer. Math. 25 pp 279– (1976) · Zbl 0304.65037 [3] Kantorovich L. V., Nauka (1984) [4] Krasnosel’skij M. A., Nauka (1969) [5] Zabrejko P. P., Gosud. Univ., Yaroslavl pp 51– (1980) [6] Zabrejko P. P., Ukr. Mat. Zhurn 34 pp 365– (1982) [7] DOI: 10.1007/BF01463998 · Zbl 0434.65034
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