Advances in Iterative Methods for Nonlinear Equations by Sergio Amat & Sonia Busquier

Author:Sergio Amat & Sonia Busquier
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

We are then ready to prove the following semilocal convergence result for Newton’s method under conditions (A1)–(6.19).

Theorem 6.4.5

Let be a twice continuously differentiable operator defined on a nonempty open convex domain Ω of a Banach space X with values in a Banach space Y and g(t) be the function defined in ( 6.23 ). Suppose that (A1)–( 6.19 ) are satisfied, there exists a root α > t 0 of ( 6.25 ), such that g(α) ≤ 0, and , where t ∗ is the smallest root of g(t) = 0 in . Then, Newton’s sequence {x n } converges to a solution x ∗ of F(x) = 0 starting at x 0 . Moreover, , for all , and

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