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Modified mann iterations for nonexpansive semigroups in Banach space

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Abstract

Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E*, and C be a nonempty closed convex subset of E. Let {T(t): t ≥ 0} be a nonexpansive semigroup on C such that F:= ∩t≥0 Fix(T(t)) ≠ ∅, and f: C → C be a fixed contractive mapping. If {α n }, {β n }, {a n }, {b n }, {t n } satisfy certain appropriate conditions, then we suggest and analyze the two modified iterative processes as:

$$ \left\{ {\begin{array}{*{20}c} {y_n = \alpha _n x_n + (1 - \alpha _n )T(t_n )x_n ,} \\ {x_n = \beta _n f(x_n ) + (1 - \beta _n )y_n .} \\ \end{array} } \right. $$
$$ \left\{ {\begin{array}{*{20}c} {u_0 \in C,} \\ {v_n = a_n u_n + (1 + a_n )T(t_n )u_n ,} \\ {u_{n + 1} = b_n f(u_n ) + (1 - b_n )v_n .} \\ \end{array} } \right. $$

We prove that the approximate solutions obtained from these methods converge strongly to q ∈ ∩t≥0 Fix(T(t)), which is a unique solution in F to the following variational inequality:

$$ \left\langle {(I - f)q,j(q - u)} \right\rangle \leqslant 0 \forall u \in F. $$

Our results extend and improve the corresponding ones of Suzuki [Proc. Amer. Math. Soc., 131, 2133–2136 (2002)], and Kim and XU [Nonlear Analysis, 61, 51–60 (2005)] and Chen and He [Appl. Math. Lett., 20, 751–757 (2007)].

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Authors and Affiliations

  1. Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300160, P. R. China

    Ru Dong Chen & Hui Min He

  2. Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan

    Muhammad Aslam Noor

Authors
  1. Ru Dong Chen
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  2. Hui Min He
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  3. Muhammad Aslam Noor
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Correspondence to Ru Dong Chen.

Additional information

Supported by the National Natural Science Foundation of China (Grant No. 10771050); the third author is supported by the Higher Education Commission, Pakistan, through Research Grant No. I-29/HEC/HRD/2005/90

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Chen, R.D., He, H.M. & Noor, M.A. Modified mann iterations for nonexpansive semigroups in Banach space. Acta. Math. Sin.-English Ser. 26, 193–202 (2010). https://doi.org/10.1007/s10114-010-7446-7

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  • DOI: https://doi.org/10.1007/s10114-010-7446-7

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