Abstract
We introduce a new iterative method in order to approximate a locally unique solution of variational inclusions in Banach spaces. The method uses only divided differences operators of order one. An existence–convergence theorem and a radius of convergence are given under some conditions on divided difference operator and Lipschitz-like continuity property of set-valued mappings. Our method extends the recent work related to the resolution of nonlinear equation in Argyros (J Math Anal Appl 332:97–108, 2007) and has the following advantages: faster convergence to the solution than all the previous known ones in Argyros and Hilout (Appl Math Comput, 2008 in press), Hilout (J Math Anal Appl 339:53–761, 2008, Positivity 10:673–700, 2006), and we do not need to evaluate any Fréchet derivative. We provide also an improvement of the ratio of our algorithm under some center-conditions and less computational cost. Numerical examples are also provided.
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References
Amat, S., Busquier, S., Candela, V.F.: A class of quasi-Newton generalized Steffensen’s methods on Banach spaces. J. Comput. Appl. Math. 149, 397–406 (2002)
Argyros, I.K.: The secant method and fixed points of nonlinear operators. Monatshefte für Math. 106, 85–94 (1988)
Argyros, I.K.: New sufficient convergence conditions for the Secant method. Czechoslov. Math. J. 55, 175–187 (2005)
Argyros, I.K.: On a two-point Newton-like method of convergent order two. Int. J. Comput Math. 88, 219–234 (2005)
Argyros, I.K.: On the Secant method for solving nonsmooth equations. J. Math. Anal. Appl. 322, 146–157 (2006)
Argyros, I.K.: Convergence and Applications of Newton-type Iterations. Springer-Verlag, New York (2008)
Argyros, I.K.: A Kantorovich-type analysis for a fast iterative method for solving nonlinear equations. J. Math. Anal. Appl. 332, 97–108 (2007)
Argyros, I.K., Hilout, S.: Local convergence of Newton-like methods for generalized equations. Appl. Math. Comput. (2008, in press)
Aubin, J.P., Frankowska, H.: Set-valued Analysis. Birkhäuser, Boston (1990)
Dontchev, A.L., Hager, W.W.: An inverse function theorem for set-valued maps. Proc. Amer. Math. Soc. 121, 481–489 (1994)
Hilout, S.: Convergence analysis of a family of Steffensen-type methods for generalized equations. J. Math. Anal. Appl. 339, 753–761 (2008)
Hilout, S., Piétrus, A.: A semilocal convergence of a secant-type method for solving generalized equations. Positivity 10, 673–700 (2006)
Mordukhovich, B.S.: Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis. Trans. Amer. Math. Soc. 343, 609-657 (1994)
Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I. Basic Theory, II. Applications (two volumes), vol. 330 and 331. Springer, Grundlehren Series (2006)
Robinson, S.M.: Generalized equations and their solutions, part I: basic theory. Math. Programming Study 10, 128–141 (1979)
Robinson, S.M.: Generalized equations and their solutions, part II: applications to nonlinear programming. Math. Programming Study 19, 200–221 (1982)
Rockafellar, R.T.: Lipschitzian properties of multifunctions. Nonlinear Analysis 9, 867–885 (1984)
Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis, a Series of Comprehensives Studies in Mathematics, vol. 317. Springer (1998)
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Argyros, I.K., Hilout, S. A Fréchet derivative-free cubically convergent method for set-valued maps. Numer Algor 48, 361–371 (2008). https://doi.org/10.1007/s11075-008-9205-8
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DOI: https://doi.org/10.1007/s11075-008-9205-8
Keywords
- Banach space
- Divided differences operators
- Generalized equation
- Aubin’s continuity
- Radius of convergence
- Fréchet derivative
- Set-valued map