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Opial-Type Theorems and the Common Fixed Point Problem

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Fixed-Point Algorithms for Inverse Problems in Science and Engineering

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 49))

Abstract

The well-known Opial theorem says that an orbit of a nonexpansive and asymptotically regular operator T having a fixed point and defined on a Hilbert space converges weakly to a fixed point of T. In this paper we consider recurrences generated by a sequence of quasi-nonexpansive operators having a common fixed point or by a sequence of extrapolations of an operator satisfying Opial’s demiclosedness principle and having a fixed point. We give sufficient conditions for the weak convergence of sequences defined by these recurrences to a fixed point of an operator which is closely related to the sequence of operators. These results generalize in a natural way the classical Opial theorem. We give applications of these generalizations to the common fixed point problem.

AMS 2010 Subject Classification: 46B45, 37C25, 65K15, 90C25

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References

  1. Aharoni, R., Berman, A., Censor, Y.: An interior point algorithm for the convex feasibility problem. Advances in Applied Mathematics 4, 479–489 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aharoni, R., Censor, Y.: Block-iterative projection methods for parallel computation of solutions to convex feasibility problems. Linear Algebra and Its Applications 120, 165–175 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  3. Appleby, G., Smolarski, D.C.: A linear acceleration row action method for projecting onto subspaces. Electronic Transactions on Numerical Analysis 20, 253–275 (2005)

    MathSciNet  MATH  Google Scholar 

  4. Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Review 38, 367–426 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bauschke, H.H., Combettes, P.L.: A weak to strong convergence principle for Fejér-monotone methods in Hilbert spaces. Mathematics of Operation Research 26, 248–264 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  6. Berinde, V.: Iterative Approximation of Fixed Points. Springer-Verlag, Berlin (2007)

    MATH  Google Scholar 

  7. Browder, F.E.: Convergence Theorems for Sequences of Nonlinear Operators in Banach Spaces. Math. Zeitschr. 100, 201–225 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  8. Butnariu, D., Censor, Y.: On the behavior of a block-iterative projection method for solving convex feasibility problems. Intern. J. Computer Math. 34, 79–94 (1990)

    Article  MATH  Google Scholar 

  9. Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems 20, 103–120 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cegielski, A.: Relaxation Methods in Convex Optimization Problems. Monographs, Vol. 67, Institute of Mathematics, Higher College of Engineering, Zielona Góra (1993) (in Polish)

    Google Scholar 

  11. Cegielski, A.: A generalization of the Opial’s Theorem. Control and Cybernetics 36, 601–610 (2007)

    MathSciNet  MATH  Google Scholar 

  12. Cegielski, A.: Generalized relaxations of nonexpansive operators and convex feasibility problems. Contemporary Mathematics 513, 111–123 (2010)

    MathSciNet  Google Scholar 

  13. Censor, Y.: Row-action methods for huge and sparse systems and their applications. SIAM Review 23, 444–466 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  14. Censor, Y., Segal, A.: On the string averaging method for sparse common fixed point problems. International Transactions in Operational Research 16, 481–494 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Censor, Y., Segal, A.: The split common fixed point problem for directed operators. Journal of Convex Analysis 16, 587–600 (2009)

    MathSciNet  MATH  Google Scholar 

  16. Censor, Y., Segal, A.: On string-averaging for sparse problems and on the split common fixed point problem. Contemporary Mathematics 513, 125–142 (2010)

    MathSciNet  Google Scholar 

  17. Censor, Y., Zenios, S.A.: Parallel Optimization: Theory, Algorithms and Applications. Oxford University Press, New York, NY, USA (1997)

    MATH  Google Scholar 

  18. Cimmino, G.: Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari. La Ricerca Scientifica XVI Series II, Anno IX 1, 326–333 (1938)

    Google Scholar 

  19. Combettes, P.L.: Convex set theoretic image recovery by extrapolated iterations of parallel subgradient projections. IEEE Transactions on Image Processing 6, 493–506 (1997)

    Article  Google Scholar 

  20. Combettes, P.L.: Hilbertian convex feasibility problems: Convergence of projection methods. Appl. Math. Optim. 35, 311–330 (1997)

    MathSciNet  MATH  Google Scholar 

  21. Combettes, P.L.: Quasi-Fejérian analysis of some optimization algorithms. In: D. Butnariu, Y. Censor and S. Reich (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, Elsevier Science Publishers, Amsterdam, The Netherlands, 115–152 (2001)

    Google Scholar 

  22. Crombez, G.: A geometrical look at iterative methods for operators with fixed points. Numerical Functional Analysis and Optimization 26, 157–175 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. De Pierro, A.R.: Metodos de projeção para a resolução de sistemas gerais de equações algébricas lienaers. Tese de Doutoramento, Instituto de Matemática, Universidade Federal do Rio de Janeiro (IM-UFRJ) (1981)

    Google Scholar 

  24. Dos Santos, L.T.: A parallel subgradient projections method for the convex feasibility problem. J. Comp. and Applied Math 18, 307–320 (1987)

    Article  MATH  Google Scholar 

  25. Flåm, S.D., Zowe, J.: Relaxed outer projections, weighted averages and convex feasibility. BIT 30, 289–300 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  26. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York and Basel (1984)

    MATH  Google Scholar 

  27. Gurin, L.G., Polyak, B.T., Raik, E.V.: The method of projection for finding the common point in convex sets. Zh. Vychisl. Mat. i Mat. Fiz. 7 1211–1228 (1967) (in Russian). English translation in: USSR Comput. Math. Phys. 7, 1–24 (1967)

    Google Scholar 

  28. Iusem, A.N., De Pierro, A.R.: Convergence results for an accelerated nonlinear Cimmino algorithm. Numerische Mathematik 49, 367–378 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  29. Kiwiel, K.C.: Block-iterative surrogate projection methods for convex feasibility problems. Linear Algebra and Applications 215, 225–259 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  30. Opial, Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Amer. Math. Soc. 73, 591–597 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  31. Schott, D.: A general iterative scheme with applications to convex optimization and related fields. Optimization 22, 885–902 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  32. Segal, A.: Directed Operators for Common Fixed Point Problems and Convex Programming Problems. Ph.D. Thesis, University of Haifa, Haifa, Israel (2008)

    Google Scholar 

  33. Zaknoon, M.: Algorithmic Developments for the Convex Feasibility Problem. Ph.D. Thesis, University of Haifa, Haifa, Israel (2003)

    Google Scholar 

  34. Zhao, J., Yang, Q.: Several solution methods for the split feasibility problem. Inverse Problems 21, 1791–1799 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank two anonymous referees for their constructive comments. This work was partially supported by Award Number R01HL070472 from the National Heart, Lung and Blood Institute and by United States-Israel Binational Science Foundation (BSF) grant No. 2009012.

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Authors and Affiliations

  1. Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, ul. Szafrana 4a, Zielona Góra, 65-514, Poland

    Andrzej Cegielski

Authors
  1. Andrzej Cegielski
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  2. Yair Censor
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Editor information

Editors and Affiliations

  1. Okanagan Campus, Department of Mathematics and Statistic, University of British Columbia, Kelowna, V1V 1V7, British Columbia, Canada

    Heinz H. Bauschke

  2. School of Mathematics & Statistics, Division of Information Technology, University of South Australia, Mawson Lakes Blvd., Mawson Lakes, 5095, South Australia, Australia

    Regina S. Burachik

  3. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Place Jussieu 4, Paris, 75005, France

    Patrick L. Combettes

  4. Dept. Physics, Lab. Atomic & Solid State, Cornell University, Clark Hall, Ithaca, 14853-2501, New York, USA

    Veit Elser

  5. Institut für Numerische und Angewandte M, Universität Göttingen, Lotzestr. 16-18, Göttingen, 37073, Germany

    D. Russell Luke

  6. Faculty of Mathematics, Dept. Combinatorics & Optimization, University of Waterloo, Waterloo, N2L 3G1, Ontario, Canada

    Henry Wolkowicz

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Cegielski, A., Censor, Y. (2011). Opial-Type Theorems and the Common Fixed Point Problem. In: Bauschke, H., Burachik, R., Combettes, P., Elser, V., Luke, D., Wolkowicz, H. (eds) Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Optimization and Its Applications(), vol 49. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9569-8_9

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