Taiwanese Journal of Mathematics

Weak and Strong Convergence Theorems for Positively Homogenuous Nonexpansive Mappings in Banach Spaces

Wataru Takahashi and Jen-Chih Yao

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Abstract

Our purpose in this paper is first to prove a weak convergence theorem by Mann's iteration for positively homogeneous nonexpansive mappings in a Banach space. Further, using the shrinking projection method defined by Takahashi, Takeuchi and Kubota, we prove a strong convergence theorem for such mappings. From two results, we obtain weak and strong convergence theorems for linear contractive mappings in a Banach space. These results are new even if the mappings are linear and contractive.

Article information

Source
Taiwanese J. Math., Volume 15, Number 3 (2011), 961-980.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406277

Digital Object Identifier
doi:10.11650/twjm/1500406277

Mathematical Reviews number (MathSciNet)
MR2829891

Zentralblatt MATH identifier
1321.47151

Subjects
Primary: 47H05: Monotone operators and generalizations 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc. 47H20: Semigroups of nonlinear operators [See also 37L05, 47J35, 54H15, 58D07]

Keywords
Banach space nonexpansive mapping fixed point generalized nonexpansive mapping hybrid method Mann's iteration

Citation

Takahashi, Wataru; Yao, Jen-Chih. Weak and Strong Convergence Theorems for Positively Homogenuous Nonexpansive Mappings in Banach Spaces. Taiwanese J. Math. 15 (2011), no. 3, 961--980. doi:10.11650/twjm/1500406277. https://projecteuclid.org/euclid.twjm/1500406277


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References

  • Y. I. Alber, Metric and generalized projections in Banach spaces: Properties and applications, in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type (A. G. Kartsatos Ed.), Marcel Dekker, New York, 1996, pp. 15-20.
  • Y. I. Alber and S. Reich, An iterative method for soling a class of nonlinear operator equations in Banach spaces, PanAmer. Math. J., 4 (1994), 39-54.
  • K. Aoyama, F. Kohsaka and W. Takahashi, Three generalizations of firmly nonexpansive mappings: Their relations and continuity properties, J. Nonlinear Convex Anal., 10 (2009), 131-147.
  • K. Aoyama and W. Takahashi, Strong convergence theorems for a family of relatively nonexpansive mappings in Banach spaces, Fixed Point Theory, 8 (2007), 143-160.
  • F. E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat. Acad. Sci. USA, 54 (1965), 1041-1044.
  • F. E. Browder, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bull. Amer. Math. Soc., 74 (1968), 660-665.
  • R. E. Bruck, On the convex approximation property and the asymptotic behaviour of nonlinear contractions in Bnach spaces, Israel J. Math., 38 (1981), 304-314.
  • J. Diestel, Geometry of Banach spaces, Selected Topics, Lecture Notes in Mathematics, 485, Springer, Berlin, 1975.
  • T. Honda, T. Ibaraki and W. Takahashi, Duality theorems and convergence theorems for nonlineaqr mappings in Banach spaces, Int. J. Math. Statis., 6 (2010), 46-64.
  • T. Honda and W. Takahashi, Nonlinear projections and generalized conditional expectations in Banach spaces, Taiwanese J. Math., to appear.
  • T. Honda and W. Takahashi, Norm one linear projections and generalized conditional expectations in Banach spaces, Sci. Math. Jpn, 69 (2009), 303-313.
  • T. Ibaraki, Y. Kimura and W. Takahashi, Convergence theorems for generalized projections and maximal monotone operators in Banach spaces, Abst. Appl. Anal., 2003 (2003), 621-629.
  • T. Ibaraki and W. Takahashi, Weak and strong convergence theorems for new resolvents of maximal monotone operators in Banach spaces, Advances in Mathematical Economics, 10 (2007), 51-64.
  • T. Ibaraki and W. Takahashi, A new projection and convergence theorems for the projections in Banach spaces, J. Approx. Theory, 149 (2007), 1-14.
  • T. Ibaraki and W. Takahashi, Generalized nonexpansive mappings and a proximal-type algorithm in Banach spaces, Contemp. Math., to appear.
  • H. Iiduka and W. Takahashi, Weak convergence theorem by Cesàro means for nonexpansive mappings and inverse-strongly monotone mappings, J. Nonlinear Convex Anal., 7 (2006), 105-113.
  • S. Itoh and W. Takahashi, The common fixed point theory of singlevalued mappings and multivalued mappings, Pacific J. Math., 79 (1978), 493-508.
  • S. Kamimura, F. Kohsaka and W. Takahashi, Weak and strong convergence theorems for maximal monotone operators in a Banach space, Set-Valued Anal., 12 (2004), 417-429.
  • S. Kamimura and W. Takahashi, Approximating solutions of maximal monotone operators in Hilbert spaces, J. Approx. Theory, 106 (2000), 226-240.
  • S. Kamimura and W. Takahashi, Weak and strong convergence of solutions to accretive operator inclusions and applications, Set-Valued Anal., 8 (2000), 361-374.
  • S. Kamimura and W. Takahashi, Strong convergence of a proximal-type algorithm in a Banach apace, SIAM J. Optim., 13 (2002), 938-945.
  • F. Kohsaka and W. Takahashi, Strong convergence of an iterative sequence for maximal monotone operators in a Banach space, Abstr. Appl. Anal., 2004 (2004), 239- 249.
  • F. Kohsaka and W. Takahashi, Weak and strong convergence theorems for minimax problems in Banach spaces, in Nonlinear Analysis and Convex Analysis (W. Takahashi and T. Tanaka, Eds.), Yokohama Publishers, 2004, pp. 203-216.
  • F. Kohsaka and W. Takahashi, Generalized nonexpansive retractions and a proximal-type algorithm in Banach spaces, J. Nonlinear Convex Anal., 8 (2007), 197-209.
  • F. Kohsaka and W. Takahashi, Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM J. Optim., 19 (2008), 824-835.
  • F. Kohsaka and W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math., 91 (2008), 166-177.
  • S. Matsushita and W. Takahashi, Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl., 2004 (2004), 37-47.
  • S. Matsushita and W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory, 134 (2005), 257-266.
  • S. Matsushita and W. Takahashi, Approximating fixed points of nonexpansive mappings in a Banach space by metric projections, Applied Math. Comput., 196 (2008), 422-425.
  • U. Mosco, convergence of convex sets and of solutions of variational inequalities, Adv. Math., 3 (1969), 510-585.
  • A. Moudafi, Weak convergence theorems for nonexpansive mappings and equilibrium problems, J. Nonlinear Convex Anal., 9 (2008), 37-43.
  • A. Moudafi and M. Théra, Proximal and dynamical approaches to equilibrium problems, Lecture Notes in Economics and Mathematical Systems, 477, Springer, 1999, pp. 187-201.
  • K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279 (2003), 372-379.
  • S. Ohsawa and W. Takahashi, Strong convergence theorems for resolvento of maximal monotone operator, Arch. Math., 81 (2003), 439-445.
  • Z. Opial, Weak covergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., 73 (1967), 591-597.
  • S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), 274-276.
  • S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl., 75 (1980), 287-292.
  • S. Reich, A weak convergence theorem for the alternative method with Bregman distance, in Theory and Applications of Nonlinear Operators of Accretive and Monotone Type (A. G. Kartsatos Ed.), Marcel Dekker, New York, 1996, pp. 313-318.
  • M. V. Solodov and B. F. Svaiter, Forcing strong convergence of proximal point iterations in a Hilbert space, Math. Program., 87 (2000), 189-202.
  • A. Tada and W. Takahashi, Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, J. Optim. Theory Appl., 133 (2007), 359-370.
  • S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 331 (2007), 506-515.
  • S. Takahashi and W. Takahashi, Strong convergence theorems for a generalized equilibrium problem anda nonexpansive mapping in a Hilbert space, Nonlinear Anal., 69 (2008), 1025-1033.
  • W. Takahashi, Iterative methods for approximation of fixed points and their applications, J. Oper. Res. Soc. Japan, 43 (2000), 87-108.
  • W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000.
  • W. Takahashi, Convex Analysis and Approximation of Fixed Points (Japanese), Yokohama Publishers, Yokohama, 2000.
  • W. Takahashi, Introduction to Nonlinear and Convex Analysis (Japanese), Yokohama Publishers, Yokohama, 2005.
  • W. Takahashi, Viscosity approximation methods for resolvents of accretive operators in Banach spaces, J. Fixed Point Theory Appl., 1 (2007), 135-147.
  • W. Takahashi, Proximal point algorithms and four resolvents of nonlinear operators of monotone type in Banach spaces, Taiwanese J. Math., 12 (2008), 1883-1910.
  • W. Takahashi, Viscosity approximation methods for countable families of nonexpansive mappings in Banach spaces, Nonlinear Anal., 70 (2009), 719-734.
  • W. Takahashi, Fixed point theorems for new nonexpansive mappings in a Hilbert space, J. Nonlinear Convex Anal., 11 (2010), 78-88.
  • W. Takahashi and G. E. Kim, Approximating fixed points of nonexpansive mappings in Banach spaces, Math. Japon., 48 (1998), 1-9.
  • W. Takahashi, Y. Takeuchi and R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 341 (2008), 276-286.
  • W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417-428.
  • W. Takahashi and Y. Ueda, On Reich's strong convergence theorems for resolvents of accretive operators, J. Math. Anal. Appl., 104 (1984), 546-553.
  • W. Takahashi and J. C. Yao, Fixed point theorems and ergodic theorems for nonlinear mappings in a Hilbert space, Taiwanese J. Math., to appear.
  • H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal., 16 (1991), 1127-1138.