Abstract
We present a semilocal convergence for some iterative methods on a Banach space with a convergence structure to locate zeros of operators which are not necessarily Fréchet-differentiable as in earlier studies such as (Argyros in J Approx Theory Appl 9(1):1–10, 1993; Argyros in Southwest J Pure Appl Math 1:32–38, 1995; Argyros in Convergence and applications of Newton-type iterations, Springer, New York, 2008; Meyer in Numer Funct Anal Optim 13(5 and 6):463–473, 1992). This way we expand the applicability of these methods. If the operator involved is Fréchet-differentiable one approach leads to more precise error estimates on the distances involved than before (Argyros in Convergence and applications of Newton-type iterations, Springer, New York, 2008; Meyer in Numer Funct Anal Optim 13(5 and 6):463–473, 1992) and under the same hypotheses. Special cases are presented and some examples from fractional calculus.
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Anastassiou, G.A., Argyros, I.K. Convergence for iterative methods on Banach spaces of a convergence structure with applications to fractional calculus. SeMA 71, 23–37 (2015). https://doi.org/10.1007/s40324-015-0044-y
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DOI: https://doi.org/10.1007/s40324-015-0044-y
Keywords
- Banach space with a convergence structure
- Semilocal convergence
- Newton-like method
- Fractional Caputo derivative
- Fractional Taylor formula