• Received : 2008.10.09
  • Accepted : 2009.04.08
  • Published : 2009.12.31


We provide a local convergence analysis for Newton-like methods for the solution of generalized equations in a Banach space setting. Using some ideas of ours introduced in [2] for nonlinear equations we show that under weaker hypotheses and computational cost than in [7] a larger convergence radius and finer error bounds on the distances involved can be obtained.


  1. Aubin, J. -P., Lipschitz behavior of solutions to convex minimization problems, Math. Oper. Res. 9 (1984), 87-111. https://doi.org/10.1287/moor.9.1.87
  2. Argyros, I. K., A unifying local-semilocal convergence analysis and applications for two-point Newton-like methods in Banach space, J. Math. Anal. Applic. 298 (2004), 374-397. https://doi.org/10.1016/j.jmaa.2004.04.008
  3. Argyros, I. K., Newton Methods, Nova Science Publ. Inc., New York, 2005.
  4. Argyros, I. K., On the secant method for solving nonsmooth equations, J. Math. Anal. Appl. (to appear, 2006).
  5. Dontchev, A. L. and Hager, W. W., An inverse function theorem for set-valued maps, Proc. Amer. Math. Soc. 121 (1994), 481-484. https://doi.org/10.1090/S0002-9939-1994-1215027-7
  6. Dontchev, A. L., Local convergence of the Newton method for generalized equations, C. R. Acad. Sci. Paris, t. 332, Serie I (1996), 327-331, Mathematical Analysis.
  7. Geoffory, M. H. and Pietrus, A., A local convergence of some iterative methods for generalized equations, J. Math. Anal. Applic. 290 (2004), 497-505. https://doi.org/10.1016/j.jmaa.2003.10.008
  8. Ioffe, A. D. and Tikhomirov, V. M., Theory of Extremal Problems, North Holland, Amsterdam, 1979.
  9. Kantorovich, L. V. and Akilov, G. P., Functional Analysis in Normed Spaces, Pergamon Press, Oxford, 1982.
  10. Rockafellar, R. T., Lipschitz properties of multifunctions, Nonlinear Analysis, 9 (1985), 867-885. https://doi.org/10.1016/0362-546X(85)90024-0
  11. Xiao, B. and Harker, P. T., A nonsmooth Newton method for variational inequalities I: Theory, Math. Programming, 65 (1994), 151-194. https://doi.org/10.1007/BF01581695