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Common fixed point and invariant approximation results in certain metrizable topological vector spaces

Fixed Point Theory and Applications volume 2006, Article number: 23582 (2006) Cite this article

Abstract

We obtain common fixed point results for generalized -nonexpansive -subweakly commuting maps on nonstarshaped domain. As applications, we establish noncommutative versions of various best approximation results for this class of maps in certain metrizable topological vector spaces.

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References

  1. Al-Thagafi MA: Common fixed points and best approximation. Journal of Approximation Theory 1996,85(3):318–323. 10.1006/jath.1996.0045

    Article  MathSciNet  MATH  Google Scholar 

  2. Berinde V: A common fixed point theorem for quasi contractive type mappings. Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Mathematica 2003, 46: 101–110 (2004).

    MathSciNet  MATH  Google Scholar 

  3. Dotson WG Jr.: Fixed point theorems for non-expansive mappings on star-shaped subsets of Banach spaces. Journal of the London Mathematical Society. Second Series 1971/1972, 4: 408–410.

    MathSciNet  MATH  Google Scholar 

  4. Guseman LF Jr., Peters BC Jr.: Nonexpansive mappings on compact subsets of metric linear spaces. Proceedings of the American Mathematical Society 1975, 47: 383–386. 10.1090/S0002-9939-1975-0353072-2

    Article  MathSciNet  Google Scholar 

  5. Habiniak L: Fixed point theorems and invariant approximations. Journal of Approximation Theory 1989,56(3):241–244. 10.1016/0021-9045(89)90113-5

    Article  MathSciNet  MATH  Google Scholar 

  6. Hussain N, Khan AR: Common fixed-point results in best approximation theory. Applied Mathematics Letters. An International Journal of Rapid Publication 2003,16(4):575–580. 10.1016/S0893-9659(03)00039-9

    MathSciNet  MATH  Google Scholar 

  7. Hussain N, O'Regan D, Agarwal RP: Common fixed point and invariant approximation results on non-starshaped domain. Georgian Mathematical Journal 2005,12(4):559–669.

    MathSciNet  MATH  Google Scholar 

  8. Jungck G: Common fixed points for commuting and compatible maps on compacta. Proceedings of the American Mathematical Society 1988,103(3):977–983. 10.1090/S0002-9939-1988-0947693-2

    Article  MathSciNet  MATH  Google Scholar 

  9. Khan LA, Khan AR: An extension of Brosowski-Meinardus theorem on invariant approximation. Approximation Theory and its Applications. New Series 1995,11(4):1–5.

    MathSciNet  MATH  Google Scholar 

  10. Meinardus G: Invarianz bei linearen Approximationen. Archive for Rational Mechanics and Analysis 1963, 14: 301–303.

    MathSciNet  MATH  Google Scholar 

  11. Naimpally SA, Singh KL, Whitfield JHM: Fixed points and nonexpansive retracts in locally convex spaces. Polska Akademia Nauk. Fundamenta Mathematicae 1984,120(1):63–75.

    MathSciNet  MATH  Google Scholar 

  12. Rudin W: Functional Analysis, International Series in Pure and Applied Mathematics. 2nd edition. McGraw-Hill, New York; 1991.

    Google Scholar 

  13. Sahab SA, Khan MS, Sessa S: A result in best approximation theory. Journal of Approximation Theory 1988,55(3):349–351. 10.1016/0021-9045(88)90101-3

    Article  MathSciNet  MATH  Google Scholar 

  14. Shahzad N: A result on best approximation. Tamkang Journal of Mathematics 1998,29(3):223–226.

    MathSciNet  MATH  Google Scholar 

  15. Shahzad N: Correction to: "A result on best approximation". Tamkang Journal of Mathematics 1999.,30(2): 165

    MathSciNet  Google Scholar 

  16. Shahzad N: Noncommuting maps and best approximations. Radovi Matematički 2000/2001,10(1):77–83.

    MathSciNet  MATH  Google Scholar 

  17. Shahzad N: Invariant approximations and -subweakly commuting maps. Journal of Mathematical Analysis and Applications 2001,257(1):39–45. 10.1006/jmaa.2000.7274

    Article  MathSciNet  MATH  Google Scholar 

  18. Shahzad N: Invariant approximations, generalized -contractions, and -subweakly commuting maps. Fixed Point Theory and Applications 2005,2005(1):79–86. 10.1155/FPTA.2005.79

    Article  MathSciNet  MATH  Google Scholar 

  19. Singh SP: An application of a fixed-point theorem to approximation theory. Journal of Approximation Theory 1979,25(1):89–90. 10.1016/0021-9045(79)90036-4

    Article  MathSciNet  MATH  Google Scholar 

  20. Talman LA: A fixed point criterion for compact -spaces. Proceedings of the American Mathematical Society 1975, 51: 91–93.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Centre for Advanced Studies in Pure Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan

    Nawab Hussain

  2. Department of Mathematics, Faculty of Science, King Abdul Aziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

    Nawab Hussain

  3. Department of Mathematics and Computer Science, Faculty of Sciences, North University of Baia Mare, Victoriei Nr. 76, Baia Mare, 430122, Romania

    Vasile Berinde

Authors
  1. Nawab Hussain
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  2. Vasile Berinde
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Correspondence to Vasile Berinde.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Hussain, N., Berinde, V. Common fixed point and invariant approximation results in certain metrizable topological vector spaces. Fixed Point Theory Appl 2006, 23582 (2006). https://doi.org/10.1155/FPTA/2006/23582

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