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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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