# Article

Full entry | PDF   (0.2 MB)
Keywords:
secant-like method; generalized equations; Aubin continuity; radius of convergence; divided difference
Summary:
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.
References:
[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. MR 1643344 | Zbl 0895.65024
[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. DOI 10.1016/j.jmaa.2004.04.008 | MR 2086964 | Zbl 1061.47052
[3] Argyros, I. K.: New sufficient convergence conditions for the secant method. Czech. Math. J. 55 (2005), 175-187. DOI 10.1007/s10587-005-0013-1 | MR 2121665 | Zbl 1081.65043
[4] Argyros, I. K.: Approximate Solution of Operator Equations with Applications. World Scientific Publ. Comp., New Jersey, USA (2005). MR 2174829 | Zbl 1086.47002
[5] Argyros, I. K.: An improved convergence analysis of a superquadratic method for solving generalized equations. Rev. Colombiana Math. 40 (2006), 65-73. MR 2286853 | Zbl 1189.65130
[6] Aubin, J. P., Frankowska, H.: Set-Valued Analysis. Birkhäuser, Boston (1990). MR 1048347 | Zbl 0713.49021
[7] Cătinas, E.: On some iterative methods for solving nonlinear equations. Rev. Anal. Numér. Théor. Approx. 23 (1994), 17-53. MR 1325892 | Zbl 0818.65050
[8] Dontchev, A. L.: Uniform convergence of the Newton method for Aubin continuous maps. Serdica Math. J. 22 (1996), 385-398. MR 1455391 | Zbl 0865.90115
[9] Dontchev, A. L., Hager, W. W.: An inverse mapping theorem for set-valued maps. Proc. Amer. Math. Soc. 121 (1994), 481-489. DOI 10.1090/S0002-9939-1994-1215027-7 | MR 1215027 | Zbl 0804.49021
[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. MR 1981104
[11] Geoffroy, M. H., Piétrus, A.: Local convergence of some iterative methods for solving generalized equations. J. Math. Anal. Appl. 290 (2004), 497-505. DOI 10.1016/j.jmaa.2003.10.008 | MR 2033038
[12] Hilout, S., Piétrus, A.: A semilocal convergence analysis of a secant-type method for solving generalized equations. Positivity 10 (2006), 693-700. DOI 10.1007/s11117-006-0044-3 | MR 2280643
[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. DOI 10.1016/S0898-1221(02)00147-5 | MR 1912346 | Zbl 1055.65069
[14] Hernández, M. A., Rubio, M. J.: $\omega$-conditioned divided differences to solve nonlinear equations. Monografías del Semin. Matem. García de Galdeano 27 (2003), 323-330. MR 2026031 | Zbl 1056.47055
[15] Ioffe, A. D., Tihomirov, V. M.: Theory of Extremal Problems. North Holland, Amsterdam (1979). MR 0528295 | Zbl 0407.90051
[16] Mordukhovich, B. S.: Complete characterization of openness metric regularity and Lipschitzian properties of multifunctions. Trans. Amer. Math. Soc. 340 (1993), 1-36. DOI 10.1090/S0002-9947-1993-1156300-4 | MR 1156300 | Zbl 0791.49018
[17] Mordukhovich, B. S.: Stability theory for parametric generalized equations and variational inequalities via nonsmooth analysis. Trans. Amer. Math. Soc. 343 (1994), 609-657. DOI 10.1090/S0002-9947-1994-1242786-4 | MR 1242786 | Zbl 0826.49008
[18] Piétrus, A.: Generalized equations under mild differentiability conditions. Rev. Real. Acad. Ciencias de Madrid 94 (2000), 15-18. MR 1829498
[19] Piétrus, A.: Does Newton's method for set-valued maps converges uniformly in mild differentiability context? Rev. Colombiana Mat. 32 (2000), 49-56. MR 1905206
[20] Rockafellar, R. T.: Lipschitzian properties of multifunctions. Nonlinear Analysis 9 (1984), 867-885. DOI 10.1016/0362-546X(85)90024-0 | MR 0799890
[21] Rockafellar, R. T., Wets, R. J.-B.: Variational Analysis. A Series of Comprehensives Studies in Mathematics, Springer, 317 (1998). MR 1491362 | Zbl 0888.49001

Partner of