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Algorithms 2015, 8(4), 832-849; doi:10.3390/a8040832

Newton-Type Methods on Generalized Banach Spaces and Applications in Fractional Calculus

1
Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA
2
Department of Mathematics Sciences, Cameron University, Lawton, OK 73505, USA
*
Author to whom correspondence should be addressed.
Academic Editor: Alicia Cordero
Received: 23 June 2015 / Revised: 13 September 2015 / Accepted: 29 September 2015 / Published: 9 October 2015
(This article belongs to the Special Issue Numerical Algorithms for Solving Nonlinear Equations and Systems)
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Abstract

We present a semilocal convergence study of Newton-type methods on a generalized Banach space setting to approximate a locally unique zero of an operator. Earlier studies require that the operator involved is Fréchet differentiable. In the present study we assume that the operator is only continuous. This way we extend the applicability of Newton-type methods to include fractional calculus and problems from other areas. Moreover, under the same or weaker conditions, we obtain weaker sufficient convergence criteria, tighter error bounds on the distances involved and an at least as precise information on the location of the solution. Special cases are provided where the old convergence criteria cannot apply but the new criteria can apply to locate zeros of operators. Some applications include fractional calculus involving the Riemann-Liouville fractional integral and the Caputo fractional derivative. Fractional calculus is very important for its applications in many applied sciences. View Full-Text
Keywords: Generalized Banach space; Newton-type method; semilocal convergence; Riemann-Liouville fractional integral; Caputo fractional derivative Generalized Banach space; Newton-type method; semilocal convergence; Riemann-Liouville fractional integral; Caputo fractional derivative
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Anastassiou, G.A.; Argyros, I.K. Newton-Type Methods on Generalized Banach Spaces and Applications in Fractional Calculus. Algorithms 2015, 8, 832-849.

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