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Approximation Methods for Nonexpansive Type Mappings in Hadamard Manifolds

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Fixed-Point Algorithms for Inverse Problems in Science and Engineering

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 49))

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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.

AMS 2010 Subject Classification: 47H09, 47H14, 65K05, 90C25

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References

  1. Barbu, V.: Nonlinear differential equations of monotone types in Banach spaces. Springer Monographs in Mathematics. Springer, New York (2010)

    Book  MATH  Google Scholar 

  2. Brézis, H.: Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. North-Holland, Amsterdam-London (1973)

    MATH  Google Scholar 

  3. Brézis, H., Crandall, G., Pazy, P.: Perturbations of nonlinear maximal monotone sets in Banach spaces. Comm. Pure Appl Math. 23, 123–144 (1970)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bridson, M., Haefliger, A.: Metric spaces of non-positive curvature. Springer, Berlin (1999)

    MATH  Google Scholar 

  5. Browder, F.E.: Existence and approximation of solutions of nonlinear variational inequalities. Proc. Nat. Acad. Sci. U.S.A. 56, 1080–1086 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Browder, F.E.: Nonlinear maximal monotone operators in Banach spaces. Math. Ann. 175, 89–113 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bruck, R.E.: Convergence theorems for sequence of nonlinear operators in Banach spaces. Math. Z. 100, 201–225 (1967)

    Article  MathSciNet  Google Scholar 

  10. Bruck, R.E.: Nonexpansive projections on subsets of Banach spaces. Pac. J. Math 47, 341–355 (1973)

    MathSciNet  MATH  Google Scholar 

  11. Bruck, R.E.: Asymptotic behavior of nonexpansive mappings. Contemp. Math. 18, 1–47 (1983)

    MathSciNet  MATH  Google Scholar 

  12. Bruck, R.E., Reich, S.: Nonexpansive projections and resolvents of accretive operators in Banach spaces. Houston J. Math. 3, 459–470 (1977)

    MathSciNet  MATH  Google Scholar 

  13. Chidume, C.: Geometric properties of Banach spaces and nonlinear iterations. Lecture Notes in Mathematics, 1965. Springer, London (2009)

    Google Scholar 

  14. Chidume, C.E.: Iterative approximation of fixed points of Lipschitzian strictly pseudo-contractive mappings. Proc. Amer. Math. Soc. 99, 283–288 (1987)

    MathSciNet  MATH  Google Scholar 

  15. Cioranescu, I.: Geometry of Banach spaces, duality mappings and nonlinear problems. Kluwer Academic Publishers, Dordrecht (1990)

    MATH  Google Scholar 

  16. Colao, V., López., G., Marino, G., Martín-Márquez, V.: Equilibrium problems in Hadamard manifolds. J. Math. Anal. Appl. (submitted)

    Google Scholar 

  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. 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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  20. DoCarmo, M.P.: Riemannian Geometry. Boston, Birkhauser (1992)

    Google Scholar 

  21. Ferreira, O.P., Oliveira, P.R.: Subgradient algorithm on Riemannian manifolds. J. Optim. Theory Appl. 97, 93–104 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ferreira, O.P., Oliveira, P.R.: Proximal point algorithm on Riemannian manifolds. Optimization 51, 257–270 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  24. Genel, A., Lindenstrauss, J.: An example concerning fixed points. Israel Journal of Math. 22, 81–86 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  25. Goebel, K., Kirk, W.A.: Iteration processes for nonexpansive mappings. Contemp. Math. 21, 115–123 (1983)

    MathSciNet  MATH  Google Scholar 

  26. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)

    MATH  Google Scholar 

  27. Halpern, B.: Fixed points of nonexpanding maps. Bull. Amer. Math. Soc. 73, 591–597 (1967)

    Article  MathSciNet  Google Scholar 

  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. Iwamiya, T., Okochi, H.: Monotonicity, resolvents and Yosida approximations of operators on Hilbert manifolds. Nonlinear Anal. 54, 205–214 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Jost, J.: Nonpositive curvature: geometric and analytic aspects. Lectures in Mathematics ETH Zrich. Birkhuser, Basel (1997)

    MATH  Google Scholar 

  31. Kamimura, S., Takahashi, W.: Approximating solutions of maximal monotone operators in Hilbert spaces. J. Approx. Theory. 13, 226–240 (2000)

    Article  MathSciNet  Google Scholar 

  32. Kato, T.: Nonlinear semigroups and evolution equations. J. Math. Soc. Japan 19, 508–520 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kido, K.: Strong convergence of resolvent of monotone operators in Banach spaces. Proc. Amer. Math. Soc. 103, 755–758 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kirk, W.A.: Krasnoselskii’s Iteration process in hyperbolic space. Numer. Funct. Anal. Optim. 4, 371–381 (1981/1982)

    Google Scholar 

  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. 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. Kirk, W.A., Schöneberg, R.: Some results on pseudo-contractive mappings. Pacific J. Math. 71, 89–99 (1977)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  40. Li, C., López, G., Martín-Márquez, V.: Iterative algorithms for nonexpansive mappings in Hadamard manifolds. Taiwanese J. Math. 14, 541–559 (2010)

    MathSciNet  MATH  Google Scholar 

  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/s11228-010-0169-1

    Google Scholar 

  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. Maingé, P.E.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47, 1499–1515 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  44. Mann, W.R.: Mean value methods in iteration. Proc. Amer. Math. Soc. 4, 506–510 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  45. Martín-Márquez, V.: Nonexpansive mappings and monotone vector fields in Hadamard manifolds. Commun. Appl. Anal. 13, 633–646 (2009)

    MathSciNet  MATH  Google Scholar 

  46. Martín-Márquez, V.: Fixed point approximation methods for nonexpansive mappings: optimization problems. Ph. D. Thesis, University of Seville (2010)

    Google Scholar 

  47. Minty, G.J.: On the monotonicity of the gradient of a convex function. Pacific J. Math. 14, 243–247 (1964)

    MathSciNet  MATH  Google Scholar 

  48. Moudafi, A.: Viscosity approximation methods for fixed-points problems. J. Math. Anal. Appl. 241, 46–55 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  49. Németh, S.Z.: Five kinds of monotone vector fields. Pure Math. Appl. 9, 417–428(1999)

    Google Scholar 

  50. Németh, S.Z.: Geodesic monotone vector fields. Lobachevskii J. Math. 5, 13–28 (1999)

    MathSciNet  MATH  Google Scholar 

  51. Németh, S.Z.: Monotone vector fields. Publ. Math. Debrecen 54, 437–449 (1999)

    MATH  Google Scholar 

  52. Németh, S.Z.: Monotonicity of the complementary vector field of a nonexpansive map. Acta Math. Hungarica 84, 189–197 (1999)

    Article  MATH  Google Scholar 

  53. Németh, S.Z.: Variational inqualities on Hadamard manifolds. Nonlinear Anal. 52, 1491–1498 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  56. Pascali, D., Sburlan, S.: Nonlinear Mappings of Monotone Type. Sythoff & Noordhoff, Alphen aan den Rijn, The Netherlands (1978)

    Google Scholar 

  57. Phelps, R.R.: Convex sets and nearest points. Proc. Amer. Math. Soc. 8, 790–797 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  58. Pia̧tek, B.: Halpern iteration in CAT(κ) spaces. Acta Math. Sinica (English Series) 27, 635–646 (2011)

    Google Scholar 

  59. Rapcsk, T.: Sectional curvature in nonlinear optimization. J. Global Optim. 40, 375–388 (2008)

    Article  MathSciNet  Google Scholar 

  60. Reich, S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math Anal. Appl. 67, 274–276 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  61. Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J Math. Anal. Appl. 75, 287–292 (1989)

    Article  Google Scholar 

  62. Reich, S., Shafrir, I.: The asymptotic behavior of firmly nonexpansive mappings. Proc. Amer. Math. Soc. 101, 246–250 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  63. Reich, S., Shafrir, I.: Nonexpansive iterations in hyperbolic spaces. Nonlinear Anal. 15, 537–558 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  64. Rockafellar, R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  65. Saejung, S.: Halpern’s iteration in CAT(0) spaces. Fixed Point Theory Appl., Art. ID 471781, 13 pp. (2010)

    Google Scholar 

  66. Sakai, T.: Riemannian Geometry. Translations of Mathematical Monographs 149. American Mathematical Society, Providence, RI (1996)

    Google Scholar 

  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. Udriste, C.: Convex Functions and Optimization Methods on Riemannian Manifolds. Mathematics and Its Applications, 297. Kluwer Academic Publisher, Dordrecht (1994)

    Google Scholar 

  69. Walter, R.: On the metric projection onto convex sets in Riemannian spaces. Arch. Math. 25, 91–98 (1974)

    Article  MATH  Google Scholar 

  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-z

    Article  MathSciNet  MATH  Google Scholar 

  71. Xu, H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  72. Zeidler, E.: Nonlinear Functional Analysis and Applications, II/B. Nonlinear Monotone Operators. Springer, New York (1990)

    Google Scholar 

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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.

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Authors and Affiliations

  1. Department of Mathematical Analysis, University of Seville, 41012, Seville, Spain

    Genaro López

Authors
  1. Genaro López
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  2. Victoria Martín-Márquez
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Correspondence to Genaro López .

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Editors and Affiliations

  1. Okanagan Campus, Department of Mathematics and Statistic, University of British Columbia, Kelowna, V1V 1V7, British Columbia, Canada

    Heinz H. Bauschke

  2. School of Mathematics & Statistics, Division of Information Technology, University of South Australia, Mawson Lakes Blvd., Mawson Lakes, 5095, South Australia, Australia

    Regina S. Burachik

  3. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Place Jussieu 4, Paris, 75005, France

    Patrick L. Combettes

  4. Dept. Physics, Lab. Atomic & Solid State, Cornell University, Clark Hall, Ithaca, 14853-2501, New York, USA

    Veit Elser

  5. Institut für Numerische und Angewandte M, Universität Göttingen, Lotzestr. 16-18, Göttingen, 37073, Germany

    D. Russell Luke

  6. Faculty of Mathematics, Dept. Combinatorics & Optimization, University of Waterloo, Waterloo, N2L 3G1, Ontario, Canada

    Henry Wolkowicz

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López, G., Martín-Márquez, V. (2011). Approximation Methods for Nonexpansive Type Mappings in Hadamard Manifolds. In: Bauschke, H., Burachik, R., Combettes, P., Elser, V., Luke, D., Wolkowicz, H. (eds) Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Optimization and Its Applications(), vol 49. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9569-8_14

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