Title: Iterative Solutions of Nonlinear Integral Equations of Hammerstein Type
Author(s): Abebe R. Tufa, H. Zegeye, M. Thuto
Pages: 129-141
Cite as:
Abebe R. Tufa, H. Zegeye, M. Thuto, Iterative Solutions of Nonlinear Integral Equations of Hammerstein Type, Int. J. Anal. Appl., 9 (2) (2015), 129-141.

Abstract


Let H be a real Hilbert space. Let F,K : H → H be Lipschitz monotone mappings with Lipschtiz constants L1and L2, respectively. Suppose that the Hammerstein type equation u + KFu = 0 has a solution in H. It is our purpose in this paper to construct a new explicit iterative sequence and prove strong convergence of the sequence to a solution of the generalized Hammerstein type equation. The results obtained in this paper improve and extend known results in the literature.


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References


  1. H. Brezis and F. Browder, Nonlinear integral equations and systems of Hammerstein type, Advances in Math., 18 (1975), 115-147.

  2. H. Brezis and F. Browder, Existence theorems for nonlinear integral equations of Hammerstein type, Bull. Amer. Math. Soc. 81 (1975), 73-78.

  3. F. E. Browder, D. G. de Figueiredo and P. Gupta, Maximal monotone operators and a nonlinear integral equations of Hammerstein type, Bull. Amer. Math. Soc. 76 (1970), 700- 705.

  4. Felix E. Browder and Chaitan P. Gupta, Monotone operators and nonlinear integral equations of Hammerstein type, Bull. Amer. Math. Soc. 75 (1969), 1347-1353.

  5. C.E. Chidume and N. Djitte, Approximation of solutions of nonlinear integral equations of Hammerstein type, ISRN Mathematical Analysis, 2012(2012), Article ID 169751, 12 pages.

  6. C.E. Chidume and H. Zegeye, Approximation of solutions of nonlinear equations of Hammerstein type in Hilbert space, Proc. Amer. Math. Soc. 133(2004), 851-858.

  7. C. L. Dolph, Nonlinear integral equations of the Hammerstein type,Trans. Amer. Math. Soc. 66 (1949), 289-307.

  8. A. Hammerstein, Nichtlineare integralgleichungen nebst anwendungen, Acta Math. Soc. 54 (1930), 117-176.

  9. Kacurovskii, On monotone operators and convex functionals, Uspekhi Mat. Nauk, 15 (1960), 213-215.

  10. P. E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal. 16 (2008), 899-912.

  11. G. J. Minty, Monotone operators in Hilbert spaces. Duke Math. J., 29 (1962), 341-346.

  12. D. Pascali and Sburlan, Nonlinear mappings of monotone type, editura academiai, Bucuresti, Romania (1978).

  13. W. Takahashi, Nonlinear Functional Analysis. Fixed Point Theory and Its Applications,Yokohama Publishers, Yokohama, Japan (2000).

  14. M. M. Vainberg and R. I. Kacurovskii, On the variational theory of nonlinear operators and equations, Dokl. Akad. Nauk 129(1959), 1199-1202.

  15. H.K.Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc., 65(2002),109-113.

  16. E. H. Zarantonello, Solving functional equations by contractive averaging, Mathematics Research Center Rep, #160, Mathematics Research Centre,Univesity of Wisconsin, Madison, 1960.

  17. H. Zegeye and David M. Malonza, Habrid approximation of solutions of integral equations of the Hammerstein type, Arab.J.Math., 2(2013),221-232.

  18. H. Zegeye and N. Shahzad, Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive mappings, Comput. Math. Appl. 62(2011), 4007-4014.

  19. H. Zegeye and N. Shahzad, Approximating common solution of variational inequality problems for two monotone mappings in Banach spaces, Optim. Lett. 5(2011), 691-704.

  20. E. Zeidler, Nonlinear Functional Analysis and Its Applications, Part II: Monotone Operators, Springer-verlag, Berlin(1985).