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                

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.

References

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