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A Monotonicity Condition for Strong Convergence of the Mann Iterative Sequence for Demicontractive Maps in Hilbert Spaces

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DOI: 10.4236/am.2014.515212    2,138 Downloads   2,442 Views   Citations

ABSTRACT


Let be a real Hilbert space and C be a nonempty closed convex subset of H. Let T : C → C be a demicontractive map satisfying 〈Tx, x〉 ≥ ‖x‖2 for all x ∈ D (T). Then the Mann iterative sequence given by xn + 1 = (1 - an) xn + anT xn, where an ∈ (0, 1) n ≥ 0, converges strongly to an element of F (T):= {x ∈ C : Tx = x}. This strong convergence is obtained without the compactness-type assumptions on C, which many previous results (see e.g. [1]) employed.


Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

George, A. and Nse, C. (2014) A Monotonicity Condition for Strong Convergence of the Mann Iterative Sequence for Demicontractive Maps in Hilbert Spaces. Applied Mathematics, 5, 2195-2198. doi: 10.4236/am.2014.515212.

References

[1] Rafiq, A. (2007) On the Mann Iteration in Hilbert Spaces. Nonlinear Analysis, 66, 2230-2236.
http://dx.doi.org/10.1016/j.na.2006.03.012
[2] Hicks, H.L. and Kubicek, J.D. (1977) On the Mann Iteration in Hilbert Spaces. Journal of Mathematical Analysis and Applications, 59, 498-505.
http://dx.doi.org/10.1016/0022-247X(77)90076-2
[3] Maruster, St. (1973) Sur le Calcul des Zeros d’un Operateur Discontinu par Iteration. Canadian Mathematical Bulletin, 16, 541-544. http://dx.doi.org/10.4153/CMB-1973-088-7
[4] Maruster, L. and Maruster, S. (2011) Strong Convergence of the Mann Iteration for α-Demicontractive Mappings. Mathematical and Computer Modelling, 54, 2486-2492.
http://dx.doi.org/10.1016/j.mcm.2011.06.006
[5] Mann, W. (1953) Mean Value Methods in Iteration. Proceedings of the American Mathematical Society, 4, 506-510.
http://dx.doi.org/10.1090/S0002-9939-1953-0054846-3
[6] Maruster, St. (1977) The Solution by Iteration of Nonlinear Equations in Hilbert Spaces. Proceedings of the American Mathematical Society, 63, 767-773.
http://dx.doi.org/10.1090/S0002-9939-1977-0636944-2
[7] Chidume, C.E. and Maruster, St. (2010) Iterative Methods for the Computation of Fixed Points of Demicontractive Mappings. Journal of Computational and Applied Mathematics, 234, 861-882.
http://dx.doi.org/10.1016/j.cam.2010.01.050
[8] Osilike, M.O. (2000) Strong and Weak Convergence of the Ishikawa Iteration Method for a Class of Nonlinear Equations. Bulletin of the Korean Mathematical Society, 37, 153-169.
[9] Chidume, C.E. (1984) The Solution by Iteration of Nonlinear Equations in Certain Banach Spaces. Journal of the Nigerian Mathematical Society, 3, 57-62.

  
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