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

Akuchu Besheng George; Celestin Akwumbuom Nse

doi:10.4236/am.2014.515212

Home > Journals > Physics & Mathematics > AM

Applied Mathematics > Vol.5 No.15, August 2014

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

Akuchu Besheng George, Celestin Akwumbuom Nse

Department of Mathematics, Federal University of Technology, Owerri, Nigeria.
Department of Mathematics, University of Nigeria, Nsukka, Nigeria.

DOI: 10.4236/am.2014.515212 PDF HTML XML 2,274 Downloads 2,672 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 x_n+ 1 = (1 - a_n) x_n + a_nT x_n, where a_n ∈ (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.

Keywords

Demicontractive Maps, Mann Iterative Sequence, Strong Convergence, Monotonicity, Hilbert Spaces

Share and Cite:

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.

Conflicts of Interest

The authors declare no conflicts of interest.

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.