Iterative Convergence with Banach Space Valued Functions in Abstract Fractional Calculus

Anastassiou, George A.; Argyros, Ioannis K.

doi:10.1515/awutm-2017-0012

Abstract

The goal of this paper is to present a semi-local convergence analysis for some iterative methods under generalized conditions. The operator is only assumed to be continuous and its domain is open. Applications are suggested including Banach space valued functions of fractional calculus, where all integrals are of Bochner-type.

References

[1] C.D. Aliprantis and K.C. Border, Infinite Dimensional Analysis, Springer, New York, 2006.
Search in Google Scholar Back to article
[2] S. Busquier S. Amat S. Plaza, Chaotic dynamics of a third-order Newton-type method, J. Math. Anal. Applic., 366(1), (2010), 164-174.10.1016/j.jmaa.2010.01.047
Search in Google Scholar Back to article
[3] G.A. Anastassiou, A strong Fractional Calculus Theory for Banach space valued functions, Nonlinear Functional Analysis and Applications (Korea), accepted for publication, (2017)10.4067/S0716-09172017000100009
Search in Google Scholar Back to article
[4] G.A. Anastassiou, Principles of general fractional analysis for Banach space valued functions, submitted for publication, (2017)10.1007/978-3-319-66936-6_4
Search in Google Scholar Back to article
[5] 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), 374–397.10.1016/j.jmaa.2004.04.008
Search in Google Scholar Back to article
[6] A. Magrenan I.K. Argyros, Iterative methods and their dynamics with applications, CRC Press, New York, 2017.
Search in Google Scholar Back to article
[7] Encyclopedia of Mathematics, Bochner integral, URL: http://www.encyclopediaofmath.org/index.php?title=Bochner_integral&oldid=38659
Search in Google Scholar Back to article
[8] M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc., 37, (1962), 74–79.10.1112/jlms/s1-37.1.74
Search in Google Scholar Back to article
[9] J.M. Gutierrez J.A. Ezquerro M.A. Hernandez, The Newton method: From Newton to Kantorovich (Spanish), Gac. R. Soc. Mat. Esp., 13, (2010), 53–76.
Search in Google Scholar Back to article
[10] G.P. Akilov L.V. Kantorovich, Functional Analysis in Normed Spaces, Pergamon Press, New York, 1982.10.1016/B978-0-08-023036-8.50010-2
Search in Google Scholar Back to article
[11] G.E. Ladas and V. Lakshmikantham, Differential equations in abstract spaces, Academic Press, New York, London, 1972
Search in Google Scholar Back to article
[12] S. Maruster, Ann. West. Univ. Timisoara, Math. and Comput. Sci., 54, (1), (2016), 1–9.10.1515/awutm-2016-0009
Search in Google Scholar Back to article
[13] J. Mikusinski, The Bochner integral, Academic Press, New York, 1978.10.1007/978-3-0348-5567-9
Search in Google Scholar Back to article
[14] V. Ptak F.A. Potra, Nondiscrete induction and iterative processes, Pitman Publ., London, 1984.
Search in Google Scholar Back to article
[15] P.D. Proinov, New general convergence theory for iterative processes and its applications to Newton-Kantorovich type theorems, J. Complexity, 26, (2010), 3–42.10.1016/j.jco.2009.05.001
Open DOI Search in Google Scholar Back to article
[16] G.E. Shilov, Elementary Functional Analysis, Dover Publications, Inc., New York, 1996.
Search in Google Scholar Back to article

Iterative Convergence with Banach Space Valued Functions in Abstract Fractional Calculus

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