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

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.


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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

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