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

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AM> Vol.5 No.15, August 2014

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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 2,203 Downloads 2,554 Views Citations

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Akuchu Besheng George, Celestin Akwumbuom Nse

Affiliation(s)

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

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

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

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