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The Split Common Fixed Point Problem for Directed Operators.

Yair Censor1, Alexander Segal

  • 1Department of Mathematics, University of Haifa Mt. Carmel, Haifa 31905, Israel.

Journal of Convex Analysis
|February 15, 2011
PubMed
Summary
This summary is machine-generated.

We introduce the split common fixed point problem, finding common fixed points for operator families in one space and their transformed images in another. A parallel algorithm is presented for directed operators, unifying prior work.

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Area of Science:

  • Optimization theory
  • Functional analysis
  • Numerical analysis

Background:

  • Fixed point theory is crucial for solving equations in various scientific domains.
  • Existing methods often address single operator problems or specific operator classes.

Purpose of the Study:

  • To introduce and define the novel split common fixed point problem.
  • To develop and analyze a parallel algorithm for solving this problem.
  • To demonstrate the unification and generalization of existing problems and algorithms.

Main Methods:

  • Formulation of the split common fixed point problem for families of operators.
  • Development of a parallel iterative algorithm.
  • Analysis of the algorithm's convergence properties for directed operators.

Main Results:

  • The proposed split common fixed point problem generalizes existing problems.
  • A parallel algorithm is formulated and analyzed for directed operators.
  • The algorithm unifies and extends previously studied algorithms.

Conclusions:

  • The split common fixed point problem offers a new framework for operator problems.
  • The parallel algorithm provides an efficient method for solving this problem.
  • This work integrates and advances research in fixed point theory and optimization.