Abstract and Applied Analysis

Convergence of a Viscosity Iterative Method for Multivalued Nonself-Mappings in Banach Spaces

Jong Soo Jung

Full-text: Open access

Abstract

Let E be a reflexive Banach space having a weakly sequentially continuous duality mapping J φ with gauge function φ , C a nonempty closed convex subset of E , and T : C 𝒦 ( E ) a multivalued nonself-mapping such that P T is nonexpansive, where P T ( x ) = { u x T x : x - u x = d ( x , T x ) } . Let f : C C be a contraction with constant k . Suppose that, for each v C and t ( 0,1 ) , the contraction defined by S t x = t P T x + ( 1 - t ) v has a fixed point x t C . Let { α n } , { β n }, and { γ n } be three sequences in ( 0,1 ) satisfying approximate conditions. Then, for arbitrary x 0 C , the sequence { x n } generated by x n α n f ( x n - 1 ) + β n x n - 1 + γ n P T ( x n )    for all    n 1 converges strongly to a fixed point of T .

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 369412, 7 pages.

Dates
First available in Project Euclid: 18 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1366306750

Digital Object Identifier
doi:10.1155/2013/369412

Mathematical Reviews number (MathSciNet)
MR3034954

Zentralblatt MATH identifier
1275.47124

Citation

Jung, Jong Soo. Convergence of a Viscosity Iterative Method for Multivalued Nonself-Mappings in Banach Spaces. Abstr. Appl. Anal. 2013 (2013), Article ID 369412, 7 pages. doi:10.1155/2013/369412. https://projecteuclid.org/euclid.aaa/1366306750


Export citation

References

  • S. B. Nadler, Jr., “Multi-valued contraction mappings,” Pacific Journal of Mathematics, vol. 30, pp. 475–488, 1969.
  • A. Moudafi, “Viscosity approximation methods for fixed-points problems,” Journal of Mathematical Analysis and Applications, vol. 241, no. 1, pp. 46–55, 2000.
  • H. K. Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 279–291, 2004.
  • F. E. Browder, “Convergence of approximants to fixed points of nonexpansive non-linear mappings in Banach spaces,” Archive for Rational Mechanics and Analysis, vol. 24, pp. 82–90, 1967.
  • B. Halpern, “Fixed points of nonexpanding maps,” Bulletin of the American Mathematical Society, vol. 73, pp. 957–961, 1967.
  • J. S. Jung and S. S. Kim, “Strong convergence theorems for nonexpansive nonself-mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 33, no. 3, pp. 321–329, 1998.
  • J. S. Jung and T. H. Kim, “Strong convergence of approximating fixed points for nonexpansive nonself-mappings in Banach spaces,” Kodai Mathematical Journal, vol. 21, no. 3, pp. 259–272, 1998.
  • G. E. Kim and W. Takahashi, “Strong convergence theorems for nonexpansive nonself-mappings in Banach spaces,” Nihonkai Mathematical Journal, vol. 7, no. 1, pp. 63–72, 1996.
  • S. Reich, “Strong convergence theorems for resolvents of accretive operators in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 75, no. 1, pp. 287–292, 1980.
  • S. P. Singh and B. Watson, “On approximating fixed points,” Proceedings of Symposia in Pure Mathematics, vol. 45, no. 2, pp. 393–395, 1988.
  • W. Takahashi and G. E. Kim, “Strong convergence of approximants to fixed points of nonexpansive nonself-mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 32, no. 3, pp. 447–454, 1998.
  • H. K. Xu, “Approximating curves of nonexpansive nonself-mappings in Banach spaces,” Comptes Rendus de l'Académie des Sciences, Série I, vol. 325, no. 2, pp. 151–156, 1997.
  • H. K. Xu and X. M. Yin, “Strong convergence theorems for nonexpansive non-self-mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 24, no. 2, pp. 223–228, 1995.
  • J. S. Jung, “Strong convergence theorems for multivalued nonexpansive nonself-mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 11, pp. 2345–2354, 2007.
  • N. Shahzad and H. Zegeye, “Strong convergence results for nonself multimaps in Banach spaces,” Proceedings of the American Mathematical Society, vol. 136, no. 2, pp. 539–548, 2008.
  • T. C. Lim, “A fixed point theorem for weakly inward multivalued contractions,” Journal of Mathematical Analysis and Applications, vol. 247, no. 1, pp. 323–327, 2000.
  • T. H. Kim and J. S. Jung, “Approximating fixed points of nonlinear mappings in Banach spaces,” in Proceedings of the Workshop on Fixed Point Theory (Kazimierz Dolny, 1997), vol. 51, no. 2, pp. 149–165, 1997.
  • G. López Acedo and H. K. Xu, “Remarks on multivalued nonexpansive mappings,” Soochow Journal of Mathematics, vol. 21, no. 1, pp. 107–115, 1995.
  • D. R. Sahu, “Strong convergence theorems for nonexpansive type and non-self-multi-valued mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 37, no. 3, pp. 401–407, 1999.
  • A. Rafiq, “On Mann iteration in Hilbert spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 10, pp. 2230–2236, 2007.
  • Y. Yao, Y. C. Liou, and R. Chen, “Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 12, pp. 3311–3317, 2007.
  • L. C. Ceng and J. C. Yao, “Strong convergence of an iterative algorithm for nonself multimaps in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 10, pp. 4476–4485, 2009.
  • Y. Alber, S. Reich, and J. C. Yao, “Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces,” Abstract and Applied Analysis, vol. 2003, no. 4, pp. 193–216, 2003, Proceedings of the International Conference on Fixed Point Theory and its Applications.
  • L. C. Ceng, A. Petruşel, and J. C. Yao, “Iterative approximation of fixed points for asymptotically strict pseudocontractive type mappings in the intermediate sense,” Taiwanese Journal of Mathematics, vol. 15, no. 2, pp. 587–606, 2011.
  • A. Petruşel and J. C. Yao, “Viscosity approximation to common fixed points of families of nonexpansive mappings with generalized contractions mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 4, pp. 1100–1111, 2008.
  • W. Takahashi, J. C. Yao, and P. Kocourek, “Weak and strong convergence theorems for generalized hybrid nonself-mappings in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 11, no. 3, pp. 567–586, 2010.
  • K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, vol. 83 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1984.
  • I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, vol. 62 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1990.
  • K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990.
  • T. Husain and A. Latif, “Fixed points of multivalued nonexpansive maps,” Mathematica Japonica, vol. 33, no. 3, pp. 385–391, 1988.
  • H. K. Xu, “On weakly nonexpansive and $\ast\,\!$-nonexpansive multivalued mappings,” Mathematica Japonica, vol. 36, no. 3, pp. 441–445, 1991.
  • H. K. Xu, “Multivalued nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 43, no. 6, pp. 693–706, 2001.
  • H. K. Xu, “Metric fixed point theory for multivalued mappings,” Dissertationes Mathematicae (Rozprawy Matematyczne), vol. 389, p. 39, 2000.