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Best proximity points for proximal generalized contractions in metric spaces

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Abstract

In this paper, we give best proximity point theorem for non-self proximal generalized contractions. Moreover, an algorithm is exhibited to determine such an optimal approximate solution designed as a best proximity point. An example is also given to support our main results.

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References

  1. Fan K.: Extensions of two fixed point theorems of F. E. Browder. Math. Z. 112, 234–240 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  2. Prolla J.B.: Fixed point theorems for set valued mappings and existence of best approximations. Numer. Funct. Anal. Optim. 5, 449–455 (1982–1983)

    Article  MathSciNet  Google Scholar 

  3. Reich S.: Approximate selections, best approximations, fixed points and invariant sets. J. Math. Anal. Appl. 62, 104–113 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  4. Amini-Harandi A., Farajzadeh A.P. Farajzadeh: A best approximation theorem in hyperconvex metric spaces. Nonlinear Anal. 70, 2453–2456 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. Sehgal V.M., Singh S.P.: A generalization to multifunctions of Fans best approximation theorem. Proc. Am. Math. Soc. 102, 534–537 (1988)

    MathSciNet  MATH  Google Scholar 

  6. Sehgal V.M., Singh S.P.: A theorem on best approximations. Numer. Funct. Anal. Optim. 10, 181–184 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. Vetrivel V., Veeramani P., Bhattacharyya P.: Some extensions of Fans best approximation theorem. Numer. Funct. Anal. Optim. 13, 397–402 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  8. Sadiq Basha S.: Best proximity points: optimal solutions. J. Optim. Theory Appl. 151, 210–216 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Al-Thagafi M.A., Shahzad N.: Convergence and existence results for best proximity points. Nonlinear Anal. 70(10), 3665–3671 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Eldred A.A., Veeramani P.: Existence and convergence of best proximity points. J. Math. Anal. Appl. 323, 1001–1006 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Di Bari C., Suzuki T., Vetro C.: Best proximity points for cyclic Meir–Keeler contractions. Nonlinear Anal. 69(11), 3790–3794 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Karpagam, S., Agrawal, S.: Best proximity point theorems for p-cyclic Meir–Keeler contractions. Fixed Point Theory Appl., Art. ID 197308 (2009)

  13. Anuradha J., Veeramani P.: Proximal pointwise contraction. Topol. Appl. 156(18), 2942–2948 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Eldred A.A., Kirk W.A., Veeramani P.: Proximinal normal structure and relatively nonexpanisve mappings. Stud. Math. 171(3), 283–293 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Sadiq Basha S.: Best proximity points: global optimal approximate solutions. J. Glob. Optim. 49, 15–21 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Al-Thagafi M.A., Shahzad N.: Best proximity pairs and equilibrium pairs for Kakutani multimaps. Nonlinear Anal. 70(3), 1209–1216 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Al-Thagafi, M.A., Shahzad, N.: Best proximity sets and equilibrium pairs for a finite family of multimaps. Fixed Point Theory Appl., Art. ID 457069 (2008)

  18. Kim W.K., Kum S., Lee K.H.: On general best proximity pairs and equilibrium pairs in free abstract economies. Nonlinear Anal. 68(8), 2216–2227 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Sadiq Basha S., Veeramani P.: Best approximations and best proximity pairs. Acta Sci. Math. 63, 289–300 (1997)

    MathSciNet  Google Scholar 

  20. Sadiq Basha S., Veeramani P.: Best proximity pair theorems for multifunctions with open fibres. J. Approx. Theory 103, 119–129 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  21. Srinivasan P.S.: Best proximity pair theorems. Acta Sci. Math. 67, 421–429 (2011)

    Google Scholar 

  22. Sadiq Basha, S., Shahzad, N., and Jeyaraj, R.: Best proximity points: approximation and optimization. Optim. Lett. doi:10.1007/s11590-011-0404-1

  23. Pardalos P.M., Rassias T.M., Khan A.A.: Nonlinear Analysis and Variational Problems. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  24. Kirk W.A., Reich S., Veeramani P.: Proximinal retracts and best proximity pair theorems. Numer. Funct. Anal. Optim. 24, 851–862 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Geraghty G.: On contractive mappings. Proc. Am. Math. Soc. 40, 604–608 (1973)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Mathematics, University of Shahrekord, 88186-34141, Shahrekord, Iran

    A. Amini-Harandi

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  1. A. Amini-Harandi
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Correspondence to A. Amini-Harandi.

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Amini-Harandi, A. Best proximity points for proximal generalized contractions in metric spaces. Optim Lett 7, 913–921 (2013). https://doi.org/10.1007/s11590-012-0470-z

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  • DOI: https://doi.org/10.1007/s11590-012-0470-z

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