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Approximation Methods for Nonexpansive Type Mappings in Hadamard Manifolds

  • Genaro López
  • Victoria Martín-Márquez
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 49)

Abstract

Nonexpansrive type mappings defined on Hadamard manifolds and iterative methods for approximating fixed points of these mappings are surveyed. The close relationship with monotone vector fields is pointed out and some numerical examples are included.

Keywords

Hadamard manifold Fixed point Nonexpansive mapping Iterative method Monotone vector field Resolvent 

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Notes

Acknowledgements

This work was supported by DGES, Grant MTM2009-13997-C02-01 and Junta de Andaluca, Grant FQM-127. It was partially prepared while the second author was visiting the Department of Mathematics of UBC Okanagan in Kelowna, Canada. She is very grateful to Professor Bauschke for his wonderful hospitality.

References

  1. 1.
    Barbu, V.: Nonlinear differential equations of monotone types in Banach spaces. Springer Monographs in Mathematics. Springer, New York (2010)MATHCrossRefGoogle Scholar
  2. 2.
    Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam-London (1973)MATHGoogle Scholar
  3. 3.
    Brézis, H., Crandall, G., Pazy, P.: Perturbations of nonlinear maximal monotone sets in Banach spaces. Comm. Pure Appl Math. 23, 123–144 (1970)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bridson, M., Haefliger, A.: Metric spaces of non-positive curvature. Springer, Berlin (1999)MATHGoogle Scholar
  5. 5.
    Browder, F.E.: Existence and approximation of solutions of nonlinear variational inequalities. Proc. Nat. Acad. Sci. U.S.A. 56, 1080–1086 (1966)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Browder, F.E.: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch. Rational Mech. Anal. 24, 82–90 (1967)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Browder, F.E.: Nonlinear maximal monotone operators in Banach spaces. Math. Ann. 175, 89–113 (1968)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Browder, F.E., Petryshyn, W.V.: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 20, 197–228 (1967)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Bruck, R.E.: Convergence theorems for sequence of nonlinear operators in Banach spaces. Math. Z. 100, 201–225 (1967)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bruck, R.E.: Nonexpansive projections on subsets of Banach spaces. Pac. J. Math 47, 341–355 (1973)MathSciNetMATHGoogle Scholar
  11. 11.
    Bruck, R.E.: Asymptotic behavior of nonexpansive mappings. Contemp. Math. 18, 1–47 (1983)MathSciNetMATHGoogle Scholar
  12. 12.
    Bruck, R.E., Reich, S.: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J. Math. 3, 459–470 (1977)MathSciNetMATHGoogle Scholar
  13. 13.
    Chidume, C.: Geometric properties of Banach spaces and nonlinear iterations. Lecture Notes in Mathematics, 1965. Springer, London (2009)Google Scholar
  14. 14.
    Chidume, C.E.: Iterative approximation of fixed points of Lipschitzian strictly pseudo-contractive mappings. Proc. Amer. Math. Soc. 99, 283–288 (1987)MathSciNetMATHGoogle Scholar
  15. 15.
    Cioranescu, I.: Geometry of Banach spaces, duality mappings and nonlinear problems. Kluwer Academic Publishers, Dordrecht (1990)MATHGoogle Scholar
  16. 16.
    Colao, V., López., G., Marino, G., Martín-Márquez, V.: Equilibrium problems in Hadamard manifolds. J. Math. Anal. Appl. (submitted)Google Scholar
  17. 17.
    Da Cruz Neto, J.X., Ferreira, O.P., Lucambio Pérez, L.R.: Monotone point-to-set vector fields. Balkan J. Geom. Appl. 5, 69–79 (2000)Google Scholar
  18. 18.
    Da Cruz Neto, J.X., Ferreira, O.P., Lucambio Pérez, L.R.: Contributions to the study of monotone vector fields. Acta Math. Hungarica 94, 307–320 (2002)Google Scholar
  19. 19.
    Da Cruz Neto, J.X., Ferreira, O.P., Lucambio Pérez, L.R., Nmeth, S.Z.: Convex- and monotone-transformable mathematical programming problems and a proximal-like point method. J. Global Optim. 35, 53–69 (2006)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    DoCarmo, M.P.: Riemannian Geometry. Boston, Birkhauser (1992)Google Scholar
  21. 21.
    Ferreira, O.P., Oliveira, P.R.: Subgradient algorithm on Riemannian manifolds. J. Optim. Theory Appl. 97, 93–104 (1998)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimization 51, 257–270 (2002)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Ferreira, O.P., Lucambio Pérez, L.R., Németh, S.Z.: Singularities of monotene vector fields and an extragradient-type algorithm. J. Global Optim. 31, 133–151 (2005)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Genel, A., Lindenstrauss, J.: An example concerning fixed points. Israel Journal of Math. 22, 81–86 (1975)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Goebel, K., Kirk, W.A.: Iteration processes for nonexpansive mappings. Contemp. Math. 21, 115–123 (1983)MathSciNetMATHGoogle Scholar
  26. 26.
    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)MATHGoogle Scholar
  27. 27.
    Halpern, B.: Fixed points of nonexpanding maps. Bull. Amer. Math. Soc. 73, 591–597 (1967)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Hoyos Guerrero, J.J.: Differential Equations of Evolution and Accretive Operators on Finsler Manifolds. Ph. D. Thesis, University of Chicago (1978)Google Scholar
  29. 29.
    Iwamiya, T., Okochi, H.: Monotonicity, resolvents and Yosida approximations of operators on Hilbert manifolds. Nonlinear Anal. 54, 205–214 (2003)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Jost, J.: Nonpositive curvature: geometric and analytic aspects. Lectures in Mathematics ETH Zrich. Birkhuser, Basel (1997)MATHGoogle Scholar
  31. 31.
    Kamimura, S., Takahashi, W.: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory. 13, 226–240 (2000)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Kato, T.: Nonlinear semigroups and evolution equations. J. Math. Soc. Japan 19, 508–520 (1967)MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Kido, K.: Strong convergence of resolvent of monotone operators in Banach spaces. Proc. Amer. Math. Soc. 103, 755–758 (1988)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    Kirk, W.A.: Krasnoselskii’s Iteration process in hyperbolic space. Numer. Funct. Anal. Optim. 4, 371–381 (1981/1982)Google Scholar
  35. 35.
    Kirk, W.A.: Geodesic Geometry and Fixed Point Theory. Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), 195–225, Univ. Sevilla Secr. Publ., Seville (2003)Google Scholar
  36. 36.
    Kirk, W.A.: Geodesic geometry and fixed point theory. In: II International Conference on Fixed Point Theory and Applications, 113–142, Yokohama Publ., Yokohama (2004)Google Scholar
  37. 37.
    Kirk, W.A., Schöneberg, R.: Some results on pseudo-contractive mappings. Pacific J. Math. 71, 89–99 (1977)MathSciNetMATHGoogle Scholar
  38. 38.
    Li, S.L., Li, C., Liu, Y.C., Yao, J.C.: Existence of solutions for variational inequalities on Riemannian manifolds. Nonlinear Anal. 71, 5695–5705 (2009)MathSciNetMATHCrossRefGoogle Scholar
  39. 39.
    Li, C., López, G., Martín-Márquez, V.: Monotone vector fields and the proximal point algorithm on Hadamard manifolds. J. Lond. Math. Soc. 79, 663–683 (2009)MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Li, C., López, G., Martín-Márquez, V.: Iterative algorithms for nonexpansive mappings in Hadamard manifolds. Taiwanese J. Math. 14, 541–559 (2010)MathSciNetMATHGoogle Scholar
  41. 41.
    Li, C., López, G., Martín-Márquez, V., Wang, J.H.: Resolvents of set-valued monotone vector fields on Hadamard manifolds. Set-Valued Var. Anal., DOI: 10.1007/s11228-010-0169-1Google Scholar
  42. 42.
    López, G., Martín-Márquez, V., Xu, H.K.: Halpern’s iteration for nonexpansive mappings. In: Nonlinear Analysis and Optimization I: Nonlinear Analysis. Contemp. Math., AMS, 513, 187–207 (2010)Google Scholar
  43. 43.
    Maingé, P.E.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47, 1499–1515 (2008)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Mann, W.R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4, 506–510 (1953)MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Martín-Márquez, V.: Nonexpansive mappings and monotone vector fields in Hadamard manifolds. Commun. Appl. Anal. 13, 633–646 (2009)MathSciNetMATHGoogle Scholar
  46. 46.
    Martín-Márquez, V.: Fixed point approximation methods for nonexpansive mappings: optimization problems. Ph. D. Thesis, University of Seville (2010)Google Scholar
  47. 47.
    Minty, G.J.: On the monotonicity of the gradient of a convex function. Pacific J. Math. 14, 243–247 (1964)MathSciNetMATHGoogle Scholar
  48. 48.
    Moudafi, A.: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46–55 (2000)MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    Németh, S.Z.: Five kinds of monotone vector fields. Pure Math. Appl. 9, 417–428(1999)Google Scholar
  50. 50.
    Németh, S.Z.: Geodesic monotone vector fields. Lobachevskii J. Math. 5, 13–28 (1999)MathSciNetMATHGoogle Scholar
  51. 51.
    Németh, S.Z.: Monotone vector fields. Publ. Math. Debrecen 54, 437–449 (1999)MATHGoogle Scholar
  52. 52.
    Németh, S.Z.: Monotonicity of the complementary vector field of a nonexpansive map. Acta Math. Hungarica 84, 189–197 (1999)MATHCrossRefGoogle Scholar
  53. 53.
    Németh, S.Z.: Variational inqualities on Hadamard manifolds. Nonlinear Anal. 52, 1491–1498 (2003)MathSciNetMATHCrossRefGoogle Scholar
  54. 54.
    Papa Quiroz, E.A., Oliveira, P.R.: Proximal point methods for quasiconvex and convex functions with Bregman distances on Hadamard manifolds. J. Convex Anal. 16, 49–69 (2009)MathSciNetMATHGoogle Scholar
  55. 55.
    Papa Quiroz, E.A., Quispe, E. M., Oliveira, P.R.: Steepest descent method with a generalized Armijo search for quasiconvex functions on Riemannian manifolds. J. Math. Anal. Appl. 341, 467–477 (2008)MathSciNetMATHCrossRefGoogle Scholar
  56. 56.
    Pascali, D., Sburlan, S.: Nonlinear Mappings of Monotone Type. Sythoff & Noordhoff, Alphen aan den Rijn, The Netherlands (1978)Google Scholar
  57. 57.
    Phelps, R.R.: Convex sets and nearest points. Proc. Amer. Math. Soc. 8, 790–797 (1957)MathSciNetMATHCrossRefGoogle Scholar
  58. 58.
    Pia̧tek, B.: Halpern iteration in CAT(κ) spaces. Acta Math. Sinica (English Series) 27, 635–646 (2011)Google Scholar
  59. 59.
    Rapcsk, T.: Sectional curvature in nonlinear optimization. J. Global Optim. 40, 375–388 (2008)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Reich, S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math Anal. Appl. 67, 274–276 (1979)MathSciNetMATHCrossRefGoogle Scholar
  61. 61.
    Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J Math. Anal. Appl. 75, 287–292 (1989)CrossRefGoogle Scholar
  62. 62.
    Reich, S., Shafrir, I.: The asymptotic behavior of firmly nonexpansive mappings. Proc. Amer. Math. Soc. 101, 246–250 (1987)MathSciNetMATHCrossRefGoogle Scholar
  63. 63.
    Reich, S., Shafrir, I.: Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal. 15, 537–558 (1990)MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)MathSciNetMATHCrossRefGoogle Scholar
  65. 65.
    Saejung, S.: Halpern’s iteration in CAT(0) spaces. Fixed Point Theory Appl., Art. ID 471781, 13 pp. (2010)Google Scholar
  66. 66.
    Sakai, T.: Riemannian Geometry. Translations of Mathematical Monographs 149. American Mathematical Society, Providence, RI (1996)Google Scholar
  67. 67.
    Singer, I.: The Theory of Best Approximation and Functional Analysis. CBMS-NSF Regional Conf. Ser. in Appl. Math., 13, SIAM, Philadelphia, PA (1974)Google Scholar
  68. 68.
    Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Mathematics and Its Applications, 297. Kluwer Academic Publisher, Dordrecht (1994)Google Scholar
  69. 69.
    Walter, R.: On the metric projection onto convex sets in Riemannian spaces. Arch. Math. 25, 91–98 (1974)MATHCrossRefGoogle Scholar
  70. 70.
    Wang, J.H., López, G., Martín-Márquez, V., Li, C.: Monotone and accretive vector fields on Riemannian manifolds. J. Optim. Theory Appl. 146, 691–708 (2010) DOI: 10.1007/s10957-010-9688-zMathSciNetMATHCrossRefGoogle Scholar
  71. 71.
    Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004)MathSciNetMATHCrossRefGoogle Scholar
  72. 72.
    Zeidler, E.: Nonlinear Functional Analysis and Applications, II/B. Nonlinear Monotone Operators. Springer, New York (1990)Google Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematical AnalysisUniversity of SevilleSevilleSpain

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