Abstract

Alternative iterative methods for a nonexpansive mapping in a Banach space are proposed and proved to be convergent to a common solution to a fixed point problem and a variational inequality. Rates of asymptotic regularity for such iterations are given using proof-theoretic techniques. Some applications of the convergence results are presented.

Author information

Contact details are reproduced from the original publication and may be historical.

Vittorio Colao

Dip. di Matematica, Università della Calabria, 87036 Arcavacata di Rende, Italy

colao@mat.unical.it

Laurenţiu Leuştean

Fachbereich Mathematik, Technische Universität, Schlossgartenstrasse 7, 64289 Darmstadt, Germany
and: Institute of Mathematics "Simion Stoilow", Romanian Academy, Calea Griviţei 21, 010702 Bucharest, Romania

Genaro López

Dep. de Análisis Matemático, Universidad de Sevilla, Apdo. 1160, 41080 Sevilla, Spain

glopez@us.es

Victoria Martín-Márquez

Dep. de Análisis Matemático, Universidad de Sevilla, Apdo. 1160, 41080 Sevilla, Spain

victoriam@us.es

Keywords

  • Nonexpansive mapping
  • iterative algorithm
  • fixed point
  • viscosity approximation
  • uniformly smooth Banach space
  • rates of asymptotic regularity
  • proof mining
  • variational inequality
  • accretive operator

Mathematics Subject Classification

47H06, 47H09, 47H10, 47J20, 03F60

References

  1. V. Barbu: Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff, Leyden (1976).https://doi.org/10.1007/978-94-010-1537-0Google Scholar
  2. F. E. Browder: Existence and approximation of solutions of nonlinear variational inequalities, Proc. Natl. Acad. Sci. USA 56 (1966) 1080–1086.https://doi.org/10.1073/pnas.56.4.1080Google Scholar
  3. F. E. Browder: Convergence of approximants to fixed points of nonlinear maps in Banach spaces, Arch. Ration. Mech. Anal. 24 (1967) 82–90.https://doi.org/10.1007/BF00251595Google Scholar
  4. F. E. Browder, W. V. Petryshyn: The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Amer. Math. Soc. 72 (1966) 571–575.https://doi.org/10.1090/S0002-9904-1966-11544-6Google Scholar
  5. R. E. Bruck: Nonexpansive projections on subsets of Banach spaces, Pac. J. Math. 47 (1973) 341–355.https://doi.org/10.2140/pjm.1973.47.341Google Scholar
  6. C. E. Chidume, C. O. Chidume: Iterative approximation of fixed points of nonexpansive mappings, J. Math. Anal. Appl. 318 (2006) 288–295.https://doi.org/10.1016/j.jmaa.2005.05.023Google Scholar
  7. I. Cioranescu: Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer, Dordrecht (1990).https://doi.org/10.1007/978-94-009-2121-4Google Scholar
  8. P. L. Combettes, S. A. Hirstoaga: Approximating curves for nonexpansive and monotone operators, J. Convex Analysis 13(3&4) (2006) 633–646.https://doi.org/10.68381/jca13049Google Scholar
  9. M. Edelstein: A remark on a theorem of M. A. Krasnoselskii, Amer. Math. Monthly 14 (1970) 65–73.Google Scholar
  10. J. Garcia-Falset, W. Kaczor, T. Kuczumov, S. Reich: Weak convergence theorems for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 43 (2001) 377–401.https://doi.org/10.1016/S0362-546X(99)00200-XGoogle Scholar
  11. A. Genel, J. Lindenstrauss: An example concerning fixed points, Israel J. Math. 22(1) (1975) 81–86.https://doi.org/10.1007/BF02757276Google Scholar
  12. K. Goebel, W. A. Kirk: Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics 28, Cambridge University Press, Cambridge (1990).https://doi.org/10.1017/CBO9780511526152Google Scholar
  13. K. Goebel, S. Reich: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York (1984).Google Scholar
  14. B. Halpern: Fixed points of nonexpanding maps, Bull. Amer. Math. Soc. 73 (1967) 591–597.https://doi.org/10.1090/S0002-9904-1967-11761-0Google Scholar
  15. S. Ishikawa: Fixed points and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc. 59 (1976) 65–71.https://doi.org/10.1090/S0002-9939-1976-0412909-XGoogle Scholar
  16. T. H. Kim, H. K. Xu: Strong convergence of modified Mann iterations, Nonlinear Anal. 61 (2005) 51–60.https://doi.org/10.1016/j.na.2004.11.011Google Scholar
  17. U. Kohlenbach: A quantitative version of a theorem due to Borwein-Reich-Shafrir, Numer. Funct. Anal. Optimization 22 (2001) 641–656.https://doi.org/10.1081/NFA-100105311Google Scholar
  18. U. Kohlenbach: Uniform asymptotic regularity for Mann iterates, J. Math. Anal. Appl. 279 (2003) 531–544.https://doi.org/10.1016/S0022-247X(03)00028-3Google Scholar
  19. U. Kohlenbach, L. Leuştean: Mann iterates of directionally nonexpansive mappings in hyperbolic spaces, Abstr. Appl. Anal. 2003 (2003) 449–477.https://doi.org/10.1155/S1085337503212021Google Scholar
  20. U. Kohlenbach: Applied Proof Theory: Proof Interpretations and their Use in Mathematics, Springer Monographs in Mathematics, Springer, Berlin (2008).Google Scholar
  21. M. A. Krasnosel’skij: Two remarks on the method of successive approximations, Usp. Mat. Nauk. 10(1(63)) (1955) 123–127 (in Russian).Google Scholar
  22. L. Leuştean: A quadratic rate of asymptotic regularity for CAT(0)-spaces, J. Math. Anal. Appl. 325(1) (2007) 386–399.https://doi.org/10.1016/j.jmaa.2006.01.081Google Scholar
  23. L. Leuştean: Rates of asymptotic regularity for Halpern iterations of nonexpansive mappings, J. UCS 13 (2007) 1680–1691.Google Scholar
  24. T. C. Lim, H. K. Xu: Fixed point theorems for asymptotically nonexpansive mappings, Nonlinear Anal. 22 (1994) 1345–1355.https://doi.org/10.1016/0362-546X(94)90116-3Google Scholar
  25. P. Lions: Approximation de points fixes de contractions, C. R. Acad. Sci., Paris, Sér. A 284 (1977) 1357–1359.Google Scholar
  26. L. S. Liu: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995) 114–125.https://doi.org/10.1006/jmaa.1995.1289Google Scholar
  27. G. López, V. Martín-Márquez, H. K. Xu: Halpern’s iteration for nonexpansive mappings, in: Nonlinear Analysis and Optimization I: Nonlinear Analysis, A. Leizarowitz et al. (ed.), Contemporary Mathematics 513, AMS Providence (2010) 187–207.https://doi.org/10.1090/conm/513/10085Google Scholar
  28. P. E. Maingé: A hybrid extragradient-viscosity method for monotone operators and fixed point problems, SIAM J. Control Optim. 47(3) (2008) 1499–1515.https://doi.org/10.1137/060675319Google Scholar
  29. W. R. Mann: Mean value methods in iteration, Proc. Amer. Math. Soc. 4(3) (1953) 506–510.https://doi.org/10.1090/S0002-9939-1953-0054846-3Google Scholar
  30. G. Marino, H. K. Xu: A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 318(1) (2006) 43–52.https://doi.org/10.1016/j.jmaa.2005.05.028Google Scholar
  31. V. Martín-Márquez: Fixed Point Approximation Methods for Nonexpansive Mappings: Optimization Problems, PhD Thesis, University of Seville (2010).Google Scholar
  32. A. Moudafi: Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl. 241 (2000) 46–55.https://doi.org/10.1006/jmaa.1999.6615Google Scholar
  33. K. Nakajo, W. Takahashi: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003) 372–379.https://doi.org/10.1016/S0022-247X(02)00458-4Google Scholar
  34. J. G. O’Hara, P. Pillay, H. K. Xu: Iterative approaches to convex feasibility problems in Banach Spaces, Nonlinear Anal. 64 (2006) 2002–2042.https://doi.org/10.1016/j.na.2005.07.036Google Scholar
  35. S. Reich: Asymptotic behavior of contractions in Banach spaces, J. Math. Anal. Appl. 44 (1973) 57–70.https://doi.org/10.1016/0022-247X(73)90024-3Google Scholar
  36. S. Reich: Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979) 274–276.https://doi.org/10.1016/0022-247X(79)90024-6Google Scholar
  37. S. Reich: Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980) 287–292.https://doi.org/10.1016/0022-247X(80)90323-6Google Scholar
  38. S. Reich: Product formulas, nonlinear semigroups, and accretive operators, J. Funct. Anal. 36(2) (1980) 147–168.https://doi.org/10.1016/0022-1236(80)90097-XGoogle Scholar
  39. S. Reich: Some problems and results in fixed point theory, Contemp. Math. 21 (1983) 179–187.https://doi.org/10.1090/conm/021/729515Google Scholar
  40. H. Schaefer: Über die Methode sukzessiver Approximationen, Jahresber. Dtsch. Math.-Ver. 59 (1957) 131–140.Google Scholar
  41. N. Shioji, W. Takahashi: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125 (1997) 3641–3645.https://doi.org/10.1090/S0002-9939-97-04033-1Google Scholar
  42. Y. Song, R. Chen: Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings, Appl. Math. Comput. 180 (2006) 275–287.https://doi.org/10.1016/j.amc.2005.12.013Google Scholar
  43. T. Suzuki: A sufficient and necessary condition for Halpern-type strong convergence to fixed point of nonexpansive mappings, Proc. Amer. Math. Soc. 135(1) (2007) 99–106.https://doi.org/10.1090/S0002-9939-06-08435-8Google Scholar
  44. T. Suzuki: Moudafi’s viscosity approximations with Meir-Keeler contractions, J. Math. Anal. Appl. 325 (2007) 342–352.https://doi.org/10.1016/j.jmaa.2006.01.080Google Scholar
  45. R. Wittmann: Approximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992) 486–491.https://doi.org/10.1007/BF01190119Google Scholar
  46. I. Yamada: The hybrid steepest descent method for the variational inequality over the intersection of fixed point sets of nonexpansive mappings, in: Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications (Haifa, 2000), D. Butnariu, Y. Censor, S. Reich (eds.), Elsevier, Amsterdam (2001) 473–504.https://doi.org/10.1016/S1570-579X(01)80028-8Google Scholar
  47. I. Yamada, N. Ogura: Hybrid steepest descent method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings, Numer. Funct. Anal. Optimization 25 (2004) 619–655.https://doi.org/10.1081/NFA-200045815Google Scholar
  48. H. K. Xu: Iterative algorithms for nonlinear operators, J. Lond. Math. Soc., II. Ser. 66 (2002) 240–256.https://doi.org/10.1112/S0024610702003332Google Scholar
  49. H. K. Xu: An iterative approach to quadratic optimization, J. Optimization Theory Appl. 116(3) (2003) 659–678.https://doi.org/10.1023/A:1023073621589Google Scholar
  50. H. K. Xu: Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004) 279–291.https://doi.org/10.1016/j.jmaa.2004.04.059Google Scholar
  51. H. K. Xu: An alternative regularization method for nonexpansive mappings with applications, in: Nonlinear Analysis and Optimization I: Nonlinear Analysis, A. Leizarowitz et al. (ed.), Contemporary Mathematics 513, AMS, Providence (2010) 239–263.https://doi.org/10.1090/conm/513/10087Google Scholar
  52. H. K. Xu, T. H. Kim: Convergence of hybrid steepest descent methods for variational inequalities, J. Optimization Theory Appl. 119 (2003) 185–201.https://doi.org/10.1023/B:JOTA.0000005048.79379.b6Google Scholar

Suggested citation

V. Colao, L. Leuştean, G. López, V. Martín-Márquez. “Alternative Iterative Methods for Nonexpansive Mappings, Rates of Convergence and Applications.” Journal of Convex Analysis 18 (2011), No. 2, 465–487. https://doi.org/10.68381/jca18030

Download BibTeXPublished by Heldermann Verlag.

Published by Heldermann Verlag, 2011. Rights now held by Banach Press.