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38 Newton-Raphson Iteration in Banach Spaces. Statement of the Theorems

Elsevier B.V.
DOI: 10.1016/s0079-8169(08)62724-5
  • Mathematics


Publisher Summary Throughout this chapter a specific setup has been used without repeating it in the assumptions of the individual theorems. A Banach space X is considered and a point to ξo є X. If an operator f map a neighborhood of ξo into a normed linear space Y and it can be assumed fo= f (ξo) ≠ 0. It is assumed that the F-gradient of f exists at ξo. Then the procedure leading from ξ to ξ1 can be iterated indefinitely; the resulting sequence ξv (v ≥ 1) lies in Ko and tends to a zero. It is assumed that f (ξ) exists in a neighborhood of every point of Tτ and P (ξ) = grad f (ξ) exists on Tτ and there satisfies a certain well defined relation.

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