Abstract and Applied Analysis

Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces

Hiroko Manaka

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Abstract

Let E be a smooth Banach space with a norm · . Let V ( x , y ) = x 2 + y 2 - 2  x , J y for any x , y E , where · , · stands for the duality pair and J is the normalized duality mapping. We define a V -strongly nonexpansive mapping by V ( · , · ) . This nonlinear mapping is nonexpansive in a Hilbert space. However, we show that there exists a V -strongly nonexpansive mapping with fixed points which is not nonexpansive in a Banach space. In this paper, we show a weak convergence theorem and strong convergence theorems for fixed points of this elastic nonlinear mapping and give the existence theorem.

Article information

Source
Abstr. Appl. Anal., Volume 2015, Special Issue (2015), Article ID 760671, 9 pages.

Dates
First available in Project Euclid: 17 August 2015

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1439816297

Digital Object Identifier
doi:10.1155/2015/760671

Mathematical Reviews number (MathSciNet)
MR3378563

Zentralblatt MATH identifier
06929077

Citation

Manaka, Hiroko. Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces. Abstr. Appl. Anal. 2015, Special Issue (2015), Article ID 760671, 9 pages. doi:10.1155/2015/760671. https://projecteuclid.org/euclid.aaa/1439816297


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References

  • W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and its Applications, Yokohama Publishers, Yokohama, Japan, 2000.
  • Y. I. Alber and S. Guerre-Delabriere, “Principle of weakly con-tractive maps in Hilbert spaces,” in New Results in Operator Theory and Its Applications, vol. 98 of Operator Theory: Advances and Applications, pp. 7–22, Birkhäuser, Basel, Switzerland, 1997.
  • D. Butnariu and E. Resmerita, “Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces,” Abstract and Applied Analysis, vol. 2006, Article ID 84919, 39 pages, 2006.
  • T. Ibaraki and W. Takahashi, “A new projection and convergence theorems for the projections in Banach spaces,” Journal of Approximation Theory, vol. 149, no. 1, pp. 1–14, 2007.
  • H. Manaka, “Convergence theorems for a maximal monotone operator and a $V$-strongly nonexpansive mapping in a Banach space,” Abstract and Applied Analysis, vol. 2010, Article ID 189814, 20 pages, 2010.
  • S. Reich, “Iterative methods for accretive sets,” in Nonlinear Equations in Abstract Spaces, pp. 317–326, Academic Press, New York, NY, USA, 1978.
  • H. H. Bauschke, “Fenchel duality, Fitzpatrick functions and the extension of firmly nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 135, no. 1, pp. 135–139, 2007.
  • R. E. Bruck and S. Reich, “Nonexpansive projections and resolvents of accretive operators in Banach spaces,” Houston Journal of Mathematics, vol. 3, no. 4, pp. 459–470, 1977.
  • 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.
  • H. Manaka, Convergence theorems for fixed points with iterative methods in Banach spaces [Thesis], Yokohama National University, 2011.
  • W. Takahashi, “Fixed point, minimax, and Hahn-Banach theorems,” Proceedings of Symposia in Pure Mathematics, vol. 45, part 2, pp. 419–427, 1986.
  • H. K. Xu, “Inequalities in Banach spaces with applications,” Nonlinear Analysis: Theory, Methods & Applications, vol. 16, no. 12, pp. 1127–1138, 1991.
  • S. Kamimura and W. Takahashi, “Strong convergence of a proximal-type algorithm in a Banach space,” SIAM Journal on Optimization, vol. 13, no. 3, pp. 938–945, 2002.
  • S. Reich, “Approximating zeros of accretive operators,” Proceedings of the American Mathematical Society, vol. 51, no. 2, pp. 381–384, 1975.
  • F. E. Browder, “Fixed point theorems for nonlinear semicontractive mappings in Banach spaces,” Archive for Rational Mechanics and Analysis, vol. 21, no. 4, pp. 259–269, 1966.
  • H. O. Fattorini, Encyclopedia of Mathematics and its Applications, vol. 18 of Encyclopedia of Mathematics and its Applications, Addison-Wesley, New York, NY, USA, 1983.
  • I. Miyadera, Hisenkei hangun (Kinokuniya Suugaku gansyo 10), Kinokuniya shoten, 1977 (Japanese). \endinput