A new convergence theorem for the secant method in Banach space and applications (Q2782915)

The author again considers the approximate solution of nonlinear operator equations in a Banach space setting, here by means of the secant method. Using Lipschitz conditions on the second Fréchet-derivative, semilocal convergence results are obtained.NEWLINENEWLINENEWLINEThe example given (\(F(x)= x^4/2400+ x-3\) in the Banach space \(\mathbb{R}\), starting with the secant in \(x_{-1}= 3\), \(x_0= 0\)) is the ame as in the author's paper [ibid. 8, No. 1, 79-87 (2001; reviewed above)] and seems to be somewhat simple when considering nonlinear equations in a Banach space setting.NEWLINENEWLINENEWLINEOther results relying also on the second Fréchet-derivative (and the same example again) can be found in another paper of the author [Panam. Math. J. 10, No. 3, 51-59 (2000; Zbl 0963.65057)].