Semilocal convergence of Newton-like methods under general conditions with applications in fractional calculus

George A. Anastassiou; Ioannis K. Argyros

doi:10.33993/jnaat442-1055

Authors

George A. Anastassiou University of Memphis, USA
Ioannis K. Argyros Cameron University, USA

DOI:

https://doi.org/10.33993/jnaat442-1055

Keywords:

generalized Banach space, Newton-like method, semilocal convergence, Riemann-Liouville fractional integral, Caputo fractional derivative

Abstract views: 253

Abstract

We present a semilocal convergence study of Newton-like methods on a generalized Banach space setting to approximate a locally unique zero of an operator. Earlier studies such as [5], [6], [7], [14] 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-like methods to include fractional calculus and problems from other areas. 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.

Downloads

Download data is not yet available.

Author Biography

Ioannis K. Argyros, Cameron University, USA

Full tenured Professor of Mathematics.

References

S. Amat, S. Busquier, S. Plaza, Chaotic dynamics of a third-order Newton-type method, J. Math. Anal. Appl., 366 (2010) 1, pp. 24-32, http://doi.org/10.1016/j.jmaa.2010.01.047 DOI: https://doi.org/10.1016/j.jmaa.2010.01.047

G. Anastassiou, Fractional Differentiation Inequalities, Springer, New York, 2009. DOI: https://doi.org/10.1007/978-0-387-98128-4

G.A. Anastassiou, Fractional Representation Formulae and Right Fractional Inequalities, Mathematical and Computer Modelling, 54 (2011) 11–12, pp. 3098-3115, http://doi.org/10.1016/j.mcm.2011.07.040 DOI: https://doi.org/10.1016/j.mcm.2011.07.040

G. Anastassiou, Intelligent Mathematics: Computational Analysis, Springer, Heidelberg, 2011. DOI: https://doi.org/10.1007/978-3-642-17098-0

I.K. Argyros, Newton-like methods in partial ly ordered linear spaces, Approx. Theory Appl., 9 (1993) 1, pp. 1-9, http://doi.org/10.1007/BF02836146

I.K. Argyros, Results on control ling the residuals of perturbed Newton-like methods on Banach spaces with a convergence structure, Southwest J. Pure Appl. Math.,1 (1995), pp. 32-38.

I.K. Argyros, Convergence and Applications of Newton-type iterations, Springer-Verlag Publ., New York, 2008.

K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Mathematics, Vol. 2004, 1st edition, Springer, New York, Heidelberg, 2010. DOI: https://doi.org/10.1007/978-3-642-14574-2_8

J.A. Ezquerro, J.M. Gutierrez, M.A. Hernandez, N. Romero, M.J. Rubio, The Newton method: From Newton to Kantorovich (Spanish), Gac. R. Soc. Mat. Esp.,13 (2010), pp. 53-76.

J.A. Ezquerro, M. A. Hernandez, Newton-type methods of high order and domains of semilocal and global convergence , Appl. Math. Comput., 214 (2009) 1, pp. 142-154, http://doi.org/10.1016/j.amc.2009.03.072 DOI: https://doi.org/10.1016/j.amc.2009.03.072

L.V. Kantorovich, G.P. Akilov, Functional Analysis in Normed Spaces, Pergamon Press, New York, 1964.

A. A. Magrenan, Different anomalies in a Jarratt family of iterative root finding methods, Appl. Math. Comput., 233 (2014), pp. 29-38, http://doi.org/10.1016/j.amc.2014.01.037 DOI: https://doi.org/10.1016/j.amc.2014.01.037

A. A. Magrenan, A new tool to study real dynamics: The convergence plane, Appl. Math. Comput., 248 (2014), pp. 215-224, http://doi.org/10.1016/j.amc.2014.09.061 DOI: https://doi.org/10.1016/j.amc.2014.09.061

P.W. Meyer, Newton’s method in generalized Banach spaces, Numer. Func. Anal. Optimiz., 9 (1987) 3-4, pp. 249-259, http://doi.org/10.1080/01630568708816234 DOI: https://doi.org/10.1080/01630568708816234

F.A. Potra, V. Ptak, Nondiscrete induction and iterative processes, Pitman Publ., London, 1984.

P.D. Proinov, New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems, J. Complexity, 26 (2010), 3-42, http://doi.org/10.1016/j.jco.2009.05.001 DOI: https://doi.org/10.1016/j.jco.2009.05.001