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Strong Convergence for Hybrid Implicit S‐Iteration Scheme of Nonexpansive and Strongly Pseudocontractive Mappings

Author

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  • Shin Min Kang
  • Arif Rafiq
  • Faisal Ali
  • Young Chel Kwun

Abstract

Let K be a nonempty closed convex subset of a real Banach space E, let S : K → K be nonexpansive, and let T : K → K be Lipschitz strongly pseudocontractive mappings such that p ∈ F(S)∩F(T) = {x ∈ K : Sx = Tx = x} and ∥x − Sy∥ ≤ ∥Sx − Sy∥ and ∥x − Ty∥ ≤ ∥Tx − Ty∥ for all x, y ∈ K. Let {βn} be a sequence in [0, 1] satisfying (i) ∑n=1∞βn=∞; (ii) limn→∞⁡βn = 0. For arbitrary x0 ∈ K, let {xn} be a sequence iteratively defined by xn = Syn, yn = (1 − βn)xn−1 + βnTxn, n ≥ 1. Then the sequence {xn} converges strongly to a common fixed point p of S and T.

Suggested Citation

  • Shin Min Kang & Arif Rafiq & Faisal Ali & Young Chel Kwun, 2014. "Strong Convergence for Hybrid Implicit S‐Iteration Scheme of Nonexpansive and Strongly Pseudocontractive Mappings," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:735673
    DOI: 10.1155/2014/735673
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    1. Shin Min Kang & Arif Rafiq & Young Chel Kwun, 2013. "Strong Convergence for Hybrid S‐Iteration Scheme," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).
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