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., 10.1016/j.jmaa.2004.04.008, J. Math. Anal. Appl. 298 (2004), 374-397. (2004) Zbl1061.47052MR2086964DOI10.1016/j.jmaa.2004.04.008
  3. Argyros, I. K., 10.1007/s10587-005-0013-1, Czech. Math. J. 55 (2005), 175-187. (2005) Zbl1081.65043MR2121665DOI10.1007/s10587-005-0013-1
  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., 10.1090/S0002-9939-1994-1215027-7, Proc. Amer. Math. Soc. 121 (1994), 481-489. (1994) Zbl0804.49021MR1215027DOI10.1090/S0002-9939-1994-1215027-7
  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., 10.1016/j.jmaa.2003.10.008, J. Math. Anal. Appl. 290 (2004), 497-505. (2004) MR2033038DOI10.1016/j.jmaa.2003.10.008
  12. Hilout, S., Piétrus, A., 10.1007/s11117-006-0044-3, Positivity 10 (2006), 693-700. (2006) MR2280643DOI10.1007/s11117-006-0044-3
  13. Hernández, M. A., Rubio, M. J., 10.1016/S0898-1221(02)00147-5, Comput. Math. Appl. 44 (2002), 277-285. (2002) Zbl1055.65069MR1912346DOI10.1016/S0898-1221(02)00147-5
  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., 10.1090/S0002-9947-1993-1156300-4, Trans. Amer. Math. Soc. 340 (1993), 1-36. (1993) Zbl0791.49018MR1156300DOI10.1090/S0002-9947-1993-1156300-4
  17. Mordukhovich, B. S., 10.1090/S0002-9947-1994-1242786-4, Trans. Amer. Math. Soc. 343 (1994), 609-657. (1994) Zbl0826.49008MR1242786DOI10.1090/S0002-9947-1994-1242786-4
  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., 10.1016/0362-546X(85)90024-0, Nonlinear Analysis 9 (1984), 867-885. (1984) MR0799890DOI10.1016/0362-546X(85)90024-0
  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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