On a secant-like method for solving generalized equations

Ioannis K. Argyros; Said Hilout

Mathematica Bohemica (2008)

  • Volume: 133, Issue: 3, page 313-320
  • ISSN: 0862-7959

Abstract

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In the paper by Hilout and Piétrus (2006) a semilocal convergence analysis was given for the secant-like method to solve generalized equations using Hölder-type conditions introduced by the first author (for nonlinear equations). Here, we show that this convergence analysis can be refined under weaker hypothesis, and less computational cost. Moreover finer error estimates on the distances involved and a larger radius of convergence are obtained.

How to cite

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Argyros, Ioannis K., and Hilout, Said. "On a secant-like method for solving generalized equations." Mathematica Bohemica 133.3 (2008): 313-320. <http://eudml.org/doc/250543>.

@article{Argyros2008,
abstract = {In the paper by Hilout and Piétrus (2006) a semilocal convergence analysis was given for the secant-like method to solve generalized equations using Hölder-type conditions introduced by the first author (for nonlinear equations). Here, we show that this convergence analysis can be refined under weaker hypothesis, and less computational cost. Moreover finer error estimates on the distances involved and a larger radius of convergence are obtained.},
author = {Argyros, Ioannis K., Hilout, Said},
journal = {Mathematica Bohemica},
keywords = {secant-like method; generalized equations; Aubin continuity; radius of convergence; divided difference; secant-like method; generalized equations; Aubin continuity; radius of convergence; divided difference; error estimates},
language = {eng},
number = {3},
pages = {313-320},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a secant-like method for solving generalized equations},
url = {http://eudml.org/doc/250543},
volume = {133},
year = {2008},
}

TY - JOUR
AU - Argyros, Ioannis K.
AU - Hilout, Said
TI - On a secant-like method for solving generalized equations
JO - Mathematica Bohemica
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 133
IS - 3
SP - 313
EP - 320
AB - In the paper by Hilout and Piétrus (2006) a semilocal convergence analysis was given for the secant-like method to solve generalized equations using Hölder-type conditions introduced by the first author (for nonlinear equations). Here, we show that this convergence analysis can be refined under weaker hypothesis, and less computational cost. Moreover finer error estimates on the distances involved and a larger radius of convergence are obtained.
LA - eng
KW - secant-like method; generalized equations; Aubin continuity; radius of convergence; divided difference; secant-like method; generalized equations; Aubin continuity; radius of convergence; divided difference; error estimates
UR - http://eudml.org/doc/250543
ER -

References

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  1. Argyros, I. K., A new convergence theorem for Steffensen's method on Banach spaces and applications, Southwest J. Pure Appl. Math. 1 (1997), 23-29. (1997) Zbl0895.65024MR1643344
  2. Argyros, I. K., 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. (2004) Zbl1061.47052MR2086964
  3. Argyros, I. K., New sufficient convergence conditions for the secant method, Czech. Math. J. 55 (2005), 175-187. (2005) Zbl1081.65043MR2121665
  4. Argyros, I. K., Approximate Solution of Operator Equations with Applications, World Scientific Publ. Comp., New Jersey, USA (2005). (2005) Zbl1086.47002MR2174829
  5. Argyros, I. K., An improved convergence analysis of a superquadratic method for solving generalized equations, Rev. Colombiana Math. 40 (2006), 65-73. (2006) Zbl1189.65130MR2286853
  6. Aubin, J. P., Frankowska, H., Set-Valued Analysis, Birkhäuser, Boston (1990). (1990) Zbl0713.49021MR1048347
  7. Cătinas, E., On some iterative methods for solving nonlinear equations, Rev. Anal. Numér. Théor. Approx. 23 (1994), 17-53. (1994) Zbl0818.65050MR1325892
  8. Dontchev, A. L., Uniform convergence of the Newton method for Aubin continuous maps, Serdica Math. J. 22 (1996), 385-398. (1996) Zbl0865.90115MR1455391
  9. Dontchev, A. L., Hager, W. W., An inverse mapping theorem for set-valued maps, Proc. Amer. Math. Soc. 121 (1994), 481-489. (1994) Zbl0804.49021MR1215027
  10. Geoffroy, M. H., Hilout, S., Piétrus, A., Acceleration of convergence in Dontchev's iterative method for solving variational inclusions, Serdica Math. J. 29 (2003), 45-54. (2003) MR1981104
  11. Geoffroy, M. H., Piétrus, A., Local convergence of some iterative methods for solving generalized equations, J. Math. Anal. Appl. 290 (2004), 497-505. (2004) MR2033038
  12. Hilout, S., Piétrus, A., A semilocal convergence analysis of a secant-type method for solving generalized equations, Positivity 10 (2006), 693-700. (2006) MR2280643
  13. Hernández, M. A., Rubio, M. J., Semilocal convergence of the secant method under mild convergence conditions of differentiability, Comput. Math. Appl. 44 (2002), 277-285. (2002) Zbl1055.65069MR1912346
  14. Hernández, M. A., Rubio, M. J., ω -conditioned divided differences to solve nonlinear equations, Monografías del Semin. Matem. García de Galdeano 27 (2003), 323-330. (2003) Zbl1056.47055MR2026031
  15. Ioffe, A. D., Tihomirov, V. M., Theory of Extremal Problems, North Holland, Amsterdam (1979). (1979) Zbl0407.90051MR0528295
  16. Mordukhovich, B. S., Complete characterization of openness metric regularity and Lipschitzian properties of multifunctions, Trans. Amer. Math. Soc. 340 (1993), 1-36. (1993) Zbl0791.49018MR1156300
  17. Mordukhovich, B. S., Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis, Trans. Amer. Math. Soc. 343 (1994), 609-657. (1994) Zbl0826.49008MR1242786
  18. Piétrus, A., Generalized equations under mild differentiability conditions, Rev. Real. Acad. Ciencias de Madrid 94 (2000), 15-18. (2000) MR1829498
  19. Piétrus, A., Does Newton's method for set-valued maps converges uniformly in mild differentiability context? Rev, Colombiana Mat. 32 (2000), 49-56. (2000) MR1905206
  20. Rockafellar, R. T., Lipschitzian properties of multifunctions, Nonlinear Analysis 9 (1984), 867-885. (1984) MR0799890
  21. Rockafellar, R. T., Wets, R. J.-B., Variational Analysis, A Series of Comprehensives Studies in Mathematics, Springer, 317 (1998). (1998) Zbl0888.49001MR1491362

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