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Monotone Convergence of Extended Iterative Methods and Fractional Calculus with Applications

Anastassiou G. A., Argyros I. K., Kumar S.

Fundamenta Informaticae

2017

Vol. 151, nr 1/4

241--253

We present monotone convergence results for general iterative methods in order to approximate a solution of a nonlinear equation defined on a partially ordered linear topological space. The main novelty of the paper is that the operators appearing in the iterative method are not necessarily linear. This way we expand of the applicability of iterative methods. Some applications are also provided from fractional calculus using Caputo and Canavati type fractional derivatives and other areas.

New and generalized convergence conditions for the Newton-Kantorovich method

Argyros I. K.

Journal of Applied Analysis

2003

Vol. 9, nr 2

287-299

We present new semilocal convergence theorems for Newton methods in a Banach space. Using earlier general conditions we find more precise error estimates on the distances involved using the majorant principle. Moreover we provide a better information on the location of the solution. In the special case of Newton's method under Lipschitz conditions we show that the famous Newton-Kantorovich hypothesis having gone unchallenged for a long time can be weakened under the same hypotheses/computational cost.