Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2013, Special Issue (2013), Article ID 541079, 8 pages.

Strong Convergence Theorems for Solutions of Equations of Hammerstein Type

Chih-Sheng Chuang

Full-text: Open access

Abstract

We consider an auxiliary operator, defined in a real Hilbert space in terms of K and F , that is, monotone and Lipschitz mappings (resp., monotone and bounded mappings). We use an explicit iterative process that converges strongly to a solution of equation of Hammerstein type. Furthermore, our results improve related results in the literature.

Article information

Source
J. Appl. Math., Volume 2013, Special Issue (2013), Article ID 541079, 8 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394807748

Digital Object Identifier
doi:10.1155/2013/541079

Mathematical Reviews number (MathSciNet)
MR3056209

Zentralblatt MATH identifier
1282.47075

Citation

Chuang, Chih-Sheng. Strong Convergence Theorems for Solutions of Equations of Hammerstein Type. J. Appl. Math. 2013, Special Issue (2013), Article ID 541079, 8 pages. doi:10.1155/2013/541079. https://projecteuclid.org/euclid.jam/1394807748


Export citation

References

  • A. Hammerstein, “Nichtlineare integralgleichungen nebst an-wendungen,” Acta Mathematica, vol. 54, no. 1, pp. 117–176, 1930.
  • H. Brézis and F. E. Browder, “Some new results about Hammerstein equations,” Bulletin of the American Mathematical Society, vol. 80, pp. 567–572, 1974.
  • H. Brezis and F. E. Browder, “Existence theorems for nonlinear integral equations of Hammerstein type,” Bulletin of the American Mathematical Society, vol. 81, pp. 73–78, 1975.
  • H. Brézis and F. E. Browder, “Nonlinear integral equations and systems of Hammerstein type,” Advances in Mathematics, vol. 18, no. 2, pp. 115–147, 1975.
  • F. E. Browder, D. G. de Figueiredo, and C. P. Gupta, “Maximal monotone operators and nonlinear integral equations of Hammerstein type,” Bulletin of the American Mathematical Society, vol. 76, pp. 700–705, 1970.
  • F. E. Browder and C. P. Gupta, “Monotone operators and nonlinear integral equations of Hammerstein type,” Bulletin of the American Mathematical Society, vol. 75, pp. 1347–1353, 1969.
  • V. Dolezal, Monotone Operators and Applications in Control and Network Theory, vol. 2 of Studies in Automation and Control, Elsevier Scientific Publishing, Amsterdam, The Netherlands, 1979.
  • C. E. Chidume and H. Zegeye, “Approximation of solutions of nonlinear equations of Hammerstein type in Hilbert space,” Proceedings of the American Mathematical Society, vol. 133, no. 3, pp. 851–858, 2005.
  • C. E. Chidume and E. U. Ofoedu, “Solution of nonlinear integral equations of Hammerstein type,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 13, pp. 4293–4299, 2011.
  • C. E. Chidume and N. Djitté, “Approximation of solutions of nonlinear integral equations of Hammerstein type,” ISRN Mathematical Analysis, vol. 2012, Article ID 169751, 12 pages, 2012.
  • K. Aoyama, Y. Kimura, W. Takahashi, and M. Toyoda, “Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 8, pp. 2350–2360, 2007.
  • W. Takahashi, Nonlinear Functional Analysis. Fixed Point Theory and Its Applications, Yokohama Publishers, Yokohama, Japan, 2000.
  • N. Shioji and W. Takahashi, “Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces,” Proceedings of the American Mathematical Society, vol. 125, no. 12, pp. 3641–3645, 1997.
  • W. Takahashi and J.-C. Yao, “Fixed point theorems and ergodic theorems for nonlinear mappings in Hilbert spaces,” Taiwanese Journal of Mathematics, vol. 15, no. 2, pp. 457–472, 2011.
  • T. Kawasaki and W. Takahashi, “Fixed point and nonlinear ergodic theorems for new nonlinear mappings in Hilbert spaces,” Journal of Nonlinear and Convex Analysis, vol. 13, pp. 529–540, 2012.
  • C. Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, vol. 1965 of Lecture Notes in Mathematics, Springer, London, UK, 2009.
  • N. Djitte and M. Sene, “Approximation of solutions of nonlinear integral equations of Hammerstein type with Lipschitz and bounded nonlinear operators,” ISRN Applied Mathematics, vol. 2012, Article ID 963802, 15 pages, 2012.