Semi-local convergence of iterative methods and Banach space valued functions in abstract fractional calculus

  • Ioannis K. Argyros Cameron University
  • George A. Anastassiou University of Memphis
Keywords: iterative method, Banach space, semi-local convergence, fractional calculus, Bochner-type integral


We present a semi-local convergence analysis for a class of iterative methods under generalized conditions. Some applications are suggested including Banach space valued functions of fractional calculus, where all integrals are of Bochner-type.

Author Biography

Ioannis K. Argyros, Cameron University
Full tenured Professor of Mathematics.


C.D. Aliprantis, K.C. Border, Infinite Dimensional Analysis, Springer, New York, 2006.

S. Amat, S. Busquier, S. Plaza,Chaotic dynamics of a third-order Newton-typemethod, J. Math. Anal. Appl., 366 (2010) 1, 24–32,

G.A. Anastassiou, Strong Right Fractional Calculus for Banach space valued functions, Revista Proyecciones, 36 (2017) 1, 149–186,

G.A. Anastassiou, A strong Fractional Calculus Theory for Banach space valued functions, Nonlinear Functional Analysis and Applications (Korea), accepted for publication, 2017.

G.A. Anastassiou, Strong mixed and generalized fractional calculus for Banach space valued functions, Mat. Vesnik (2017) no. 3, pp. 176-191.

I.K. Argyros, A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Appl., 298 (2004), 374–397,

I.K. Argyros, A. Magrenan, Iterative methods and their dynamics with applications, CRC Press, New York, 2017.

Bochner integral. Encyclopedia of Mathematics. URL: oldid=38659.

M. Edelstein, On fixed and periodic points under contractive mappings, J. LondonMath. Soc., 37 (1962), 74–79,

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), 53–76.

L.V. Kantorovich, G.P. Akilov, Functional Analysis in Normed Spaces, Pergamon Press, New York, 1982.

G.E. Ladas, V. Lakshmikantham, Differential equations in abstract spaces, Academic Press, New York, London, 1972.

A. Alberto Magrenan, A new tool to study real dynamics: The convergence plane, Appl. Math. Comput., 248(2014), 215–224,

J. Mikusinski, The Bochner integral, Academic Press, New York, 1978.

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,

G.E. Shilov, Elementary Functional Analysis, Dover Publications, Inc., New York, 1996

How to Cite
Argyros, I. K., & Anastassiou, G. A. (2018). Semi-local convergence of iterative methods and Banach space valued functions in abstract fractional calculus. J. Numer. Anal. Approx. Theory, 47(1), 3-19. Retrieved from