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Solving Split Common Fixed‐Point Problem of Firmly Quasi‐Nonexpansive Mappings without Prior Knowledge of Operators Norms

Author

Listed:
  • Jing Zhao
  • Hang Zhang

Abstract

Very recently, Moudafi introduced alternating CQ‐algorithms and simultaneous iterative algorithms for the split common fixed‐point problem concerned two bounded linear operators. However, to employ Moudafi’s algorithms, one needs to know a prior norm (or at least an estimate of the norm) of the bounded linear operators. To estimate the norm of an operator is very difficult, if it is not an impossible task. It is the purpose of this paper to introduce a viscosity iterative algorithm with a way of selecting the stepsizes such that the implementation of the algorithm does not need any prior information about the operator norms. We prove the strong convergence of the proposed algorithms for split common fixed‐point problem governed by the firmly quasi‐nonexpansive operators. As a consequence, we obtain strong convergence theorems for split feasibility problem and split common null point problems of maximal monotone operators. Our results improve and extend the corresponding results announced by many others.

Suggested Citation

  • Jing Zhao & Hang Zhang, 2014. "Solving Split Common Fixed‐Point Problem of Firmly Quasi‐Nonexpansive Mappings without Prior Knowledge of Operators Norms," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:389689
    DOI: 10.1155/2014/389689
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    References listed on IDEAS

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    3. Rudong Chen & Junlei Li & Yijie Ren, 2013. "Regularization Method for the Approximate Split Equality Problem in Infinite-Dimensional Hilbert Spaces," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-5, May.
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    5. Songnian He & Caiping Yang, 2013. "Solving the Variational Inequality Problem Defined on Intersection of Finite Level Sets," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-8, May.
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