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-commuting maps and invariant approximations

Abstract

We obtain common fixed point results for generalized -nonexpansive -commuting maps. As applications, various best approximation results for this class of maps are derived in the setup of 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. Al-Thagafi MA, Shahzad N: Noncommuting selfmaps and invariant approximations. Nonlinear Analysis 2006,64(12):2778–2786. 10.1016/j.na.2005.09.015

    Article  MathSciNet  MATH  Google Scholar 

  3. Dotson WJ Jr.: Fixed point theorems for nonexpansive mappings on starshaped subsets of Banach spaces. Journal of the London Mathematical Society 1972, 4: 408–410. 10.1112/jlms/s2-4.3.408

    Article  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: Common fixed point and invariant approximation results. Demonstratio Mathematica 2006, 39: 389–400.

    MathSciNet  MATH  Google Scholar 

  7. Hussain N, Berinde V: Common fixed point and invariant approximation results in certain metrizable topological vector spaces. Fixed Point Theory and Applications 2006, 2006: 1–13.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Hussain N, O'Regan D, Agarwal RP: Common fixed point and invariant approximation results on non-starshaped domains. Georgian Mathematical Journal 2005,12(4):659–669.

    MathSciNet  MATH  Google Scholar 

  10. 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 

  11. Jungck G, Rhoades BE: Fixed points for set valued functions without continuity. Indian Journal of Pure and Applied Mathematics 1998,29(3):227–238.

    MathSciNet  MATH  Google Scholar 

  12. Jungck G, Sessa S: Fixed point theorems in best approximation theory. Mathematica Japonica 1995,42(2):249–252.

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  15. O'Regan D, Shahzad N: Invariant approximations for generalized -contractions. Numerical Functional Analysis and Optimization 2005,26(4–5):565–575. 10.1080/NFA-200067306

    Article  MathSciNet  MATH  Google Scholar 

  16. Rhoades BE, Saliga L: Common fixed points and best approximations. preprint

  17. Rudin W: Functional Analysis, International Series in Pure and Applied Mathematics. McGraw-Hill, New York; 1991:xviii+424.

    MATH  Google Scholar 

  18. 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 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  21. 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 

  22. 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 

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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., Rhoades, B. -commuting maps and invariant approximations. Fixed Point Theory Appl 2006, 24543 (2006). https://doi.org/10.1155/FPTA/2006/24543

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  • DOI: https://doi.org/10.1155/FPTA/2006/24543

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