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Mohammed Harunor Rashid



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Mohammed Harunor Rashid


WSEAS Transactions on Mathematics


Print ISSN: 1109-2769
E-ISSN: 2224-2880

Volume 16, 2017

Notice: As of 2014 and for the forthcoming years, the publication frequency/periodicity of WSEAS Journals is adapted to the 'continuously updated' model. What this means is that instead of being separated into issues, new papers will be added on a continuous basis, allowing a more regular flow and shorter publication times. The papers will appear in reverse order, therefore the most recent one will be on top.


Volume 16, 2017



Convergence Analysis of an Extended Newton-type Method for Implicit Functions and Their Solution Mappings

AUTHORS: Mohammed Harunor Rashid

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ABSTRACT: Let P, X and Y be Banach spaces. Suppose that f : P × X → Y is continuously Frechet differentiable function depend on the point (p, x) and F : X ⇒ 2^Y is a set-valued mapping with closed graph. Consider the following parametric generalized equation of the form: 0 ∈ f(p, x) + F(x). (1) In the present paper, we study an extended Newton-type method for solving parametric generalized equation (1). Indeed, we will analyze semi-local and local convergence of the sequence generated by extended Newton-type method under the assumptions that f(p, x), the Frechet derivative Dxf(p, x) in x of f(p, x) are continuously depend on (p, x) and (f(p, ·) + F)^−1 is Lipschitz-like at (p, x).

KEYWORDS: Set-valued maps, Parametric generalized equations, Semilocal convergence, Lipschitz-like mappings, Solution mapping, Extended Newton-type method

REFERENCES:

[1] A. D. Ioffe and V. M. Tikhomirov, Theory of extremal problems, Studies in Mathematics and its Applications, North–Holland, Amsterdam, 1979

[2] A. L. Dontchev, The Graves theorem revisited, J. Convex Anal. 3, 1996, pp. 45–53.

[3] A. L. Dontchev and R. T. Rockafellar, Implicit functinos and solution mappings: A view from variational analysis, Springer Science+Business Media, LLC, New York, 2009

[4] A. L. Dontchev and R. T. Rockafellar, Robinson’s implicit functions theorem and its extensions, Math. Program Ser. B 117, 2009, pp. 129–147.

[5] A. L. Dontchev and R. T. Rockafellar, Newton’s method for generalized equations: a sequential implicit function theorem, Math. Program Ser. B 123, 2010, PP. 139–159.

[6] A. L. Dontchev and W. W. Hager, An inverse mapping theorem for set-valued maps, Proc. Amer. Math. Soc. 121, 1994, pp. 481–498.

[7] A. Pietrus, Does Newton’s method for set-valued maps converges uniformly in mild differentiability context?, Rev. Colombiana Mat. 32, 2000, pp. 49– 56.

[8] C. Jean-Alexis and A. Pietrus, On the convergence of some methods for variational inclusions, Rev. R. Acad. Cien. serie A. Mat. 102 (2), 2008, pp. 355–361.

[9] C. Li and K. F. Ng, Majorizing functions and convergence of the Gauss-Newton method for convex composite optimization, SIAM J. Optim. 18, 2007, 613– 642.

[10] I. K. Argyros and S. Hilout, Local convergence of Newton-like methods for generalized equations, Appl. Math. and Comp. 197, 2008, pp. 507–514.

[11] J. P. Aubin, Lipschitz behavior of solutions to convex minimization problems,Math. Oper. Res. 9, 1984, pp. 87–111.

[12] J. P. Aubin and H. Frankowska, Set-valued analysis, Birkhauser, Boston, 1990 ¨

[13] J. P. Dedieu and M. H. Kim, Newton’s method for analytic systems of equations with constant rank derivatives, J. Complexity 18, 2002, pp. 187–209.

[14] J. P. Dedieu and M. Shub, Newton’s method for overdetermined systems of equa- tions, Math. Comp. 69, 2000, pp. 1099–1115.

[15] M. H. Rashid, S. H. Yu, C. Li and S. Y. Wu, Convergence analysis of the Gauss-Newton method for Lipschitz-like maps,J. Optim. Theory Appl., 158(1), 2013, pp. 216–233.

[16] M. H. Rashid, On the Convergence of Extended Newton-type Method for Solving Variational Inclusions, Cogent Mathematics 1(1), 2014, pp.–(DOI 10.1080/23311835.2014.980600).

[17] M.H. Rashid, A Convergence Analysis of GaussNewton-type Method for Holder Continuous ¨ Maps,Indian Journal of Mathematics 57(2), 2014, pp. 181–198.

[18] S. Haliout, C. J. Alexis and A. Pietrus, A semilocal convergence of the secant-type method for solving a generalized equations, Possitivity 10, 2006, pp. 673– 700.

[19] X. B. Xu and C. Li, Convergence criterion of Newton’s method for singular systems with constant rank derivatives, J. Math. Anal. Appl. 345, 2008, pp. 689– 701.

WSEAS Transactions on Mathematics, ISSN / E-ISSN: 1109-2769 / 2224-2880, Volume 16, 2017, Art. #16, pp. 133-142


Copyright © 2017 Author(s) retain the copyright of this article. This article is published under the terms of the Creative Commons Attribution License 4.0

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