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Regularization Method for the Approximate Split Equality Problem in Infinite‐Dimensional Hilbert Spaces

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  • Rudong Chen
  • Junlei Li
  • Yijie Ren

Abstract

We studied the approximate split equality problem (ASEP) in the framework of infinite‐dimensional Hilbert spaces. Let H1, H2, and H3 be infinite‐dimensional real Hilbert spaces, let C ⊂ H1 and Q ⊂ H2 be two nonempty closed convex sets, and let A : H1 → H3 and B : H2 → H3 be two bounded linear operators. The ASEP in infinite‐dimensional Hilbert spaces is to minimize the function fx,y=(12)/Ax-By22 over x ∈ C and y ∈ Q. Recently, Moudafi and Byrne had proposed several algorithms for solving the split equality problem and proved their convergence. Note that their algorithms have only weak convergence in infinite‐dimensional Hilbert spaces. In this paper, we used the regularization method to establish a single‐step iterative for solving the ASEP in infinite‐dimensional Hilbert spaces and showed that the sequence generated by such algorithm strongly converges to the minimum‐norm solution of the ASEP. Note that, by taking B = I in the ASEP, we recover the approximate split feasibility problem (ASFP).

Suggested Citation

  • Rudong Chen & Junlei Li & Yijie Ren, 2013. "Regularization Method for the Approximate Split Equality Problem in Infinite‐Dimensional Hilbert Spaces," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:813635
    DOI: 10.1155/2013/813635
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    References listed on IDEAS

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    1. Charles L. Byrne and Abdellatif Moudafi, 2013. "Extensions of the CQ Algorithm for the Split Feasibility and Split Equality Problems," Documents de Travail 2013-01, CEREGMIA, Université des Antilles et de la Guyane.
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    Cited by:

    1. 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).

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