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Best Proximity Points: Optimal Solutions

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  • S. Sadiq Basha

    (Anna University)

Abstract

This article elicits a best proximity point theorem for non-self-proximal contractions. As a consequence, it ascertains the existence of an optimal approximate solution to some equations for which it is plausible that there is no solution. Moreover, an algorithm is exhibited to determine such an optimal approximate solution designated as a best proximity point. It is interesting to observe that the preceding best proximity point theorem includes the famous Banach contraction principle.

Suggested Citation

  • S. Sadiq Basha, 2011. "Best Proximity Points: Optimal Solutions," Journal of Optimization Theory and Applications, Springer, vol. 151(1), pages 210-216, October.
    • Handle: RePEc:spr:joptap:v:151:y:2011:i:1:d:10.1007_s10957-011-9869-4
    DOI: 10.1007/s10957-011-9869-4
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    References listed on IDEAS

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    1. S. Sadiq Basha, 2011. "Best proximity points: global optimal approximate solutions," Journal of Global Optimization, Springer, vol. 49(1), pages 15-21, January.
    2. N. Shahzad & S. Sadiq Basha & R. Jeyaraj, 2011. "Common Best Proximity Points: Global Optimal Solutions," Journal of Optimization Theory and Applications, Springer, vol. 148(1), pages 69-78, January.
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    Citations

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    Cited by:

    1. Naeem Saleem & Mujahid Abbas & Manuel De la Sen, 2019. "Optimal Approximate Solution of Coincidence Point Equations in Fuzzy Metric Spaces," Mathematics, MDPI, vol. 7(4), pages 1-13, April.
    2. Manuel De la Sen & Mujahid Abbas & Naeem Saleem, 2017. "On Optimal Fuzzy Best Proximity Coincidence Points of Proximal Contractions Involving Cyclic Mappings in Non-Archimedean Fuzzy Metric Spaces," Mathematics, MDPI, vol. 5(2), pages 1-20, April.
    3. A. Abkar & M. Gabeleh, 2012. "Global Optimal Solutions of Noncyclic Mappings in Metric Spaces," Journal of Optimization Theory and Applications, Springer, vol. 153(2), pages 298-305, May.
    4. S. Sadiq Basha, 2014. "Best proximity point theorems: unriddling a special nonlinear programming problem," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(2), pages 543-553, July.
    5. Hind Alamri & Nawab Hussain & Ishak Altun, 2023. "Proximity Point Results for Generalized p -Cyclic Reich Contractions: An Application to Solving Integral Equations," Mathematics, MDPI, vol. 11(23), pages 1-25, November.
    6. Ali Abkar & Narges Moezzifar & Azizollah Azizi, 2016. "Best Proximity Point Theorems in Partially Ordered b -Quasi Metric Spaces," Mathematics, MDPI, vol. 4(4), pages 1-16, November.
    7. Bessem Samet, 2013. "Some Results on Best Proximity Points," Journal of Optimization Theory and Applications, Springer, vol. 159(1), pages 281-291, October.
    8. S. Sadiq Basha & N. Shahzad & R. Jeyaraj, 2013. "Best proximity point theorems: exposition of a significant non-linear programming problem," Journal of Global Optimization, Springer, vol. 56(4), pages 1699-1705, August.
    9. Binayak S. Choudhury & Nikhilesh Metiya & Pranati Maity, 2014. "Best Proximity Point Results in Complex Valued Metric Spaces," International Journal of Analysis, Hindawi, vol. 2014, pages 1-6, August.
    10. Moosa Gabeleh, 2015. "Best Proximity Point Theorems via Proximal Non-self Mappings," Journal of Optimization Theory and Applications, Springer, vol. 164(2), pages 565-576, February.
    11. Moosa Gabeleh, 2013. "Proximal Weakly Contractive and Proximal Nonexpansive Non-self-Mappings in Metric and Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 615-625, August.
    12. V. Sankar Raj & A. Anthony Eldred, 2014. "A Characterization of Strictly Convex Spaces and Applications," Journal of Optimization Theory and Applications, Springer, vol. 160(2), pages 703-710, February.
    13. Slah Sahmim & Abdelbasset Felhi & Hassen Aydi, 2019. "Convergence and Best Proximity Points for Generalized Contraction Pairs," Mathematics, MDPI, vol. 7(2), pages 1-12, February.
    14. Victoria Olisama & Johnson Olaleru & Hudson Akewe, 2017. "Best Proximity Point Results for Some Contractive Mappings in Uniform Spaces," International Journal of Analysis, Hindawi, vol. 2017, pages 1-8, April.

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    1. A. Abkar & M. Gabeleh, 2012. "Global Optimal Solutions of Noncyclic Mappings in Metric Spaces," Journal of Optimization Theory and Applications, Springer, vol. 153(2), pages 298-305, May.
    2. Ali Abkar & Narges Moezzifar & Azizollah Azizi, 2016. "Best Proximity Point Theorems in Partially Ordered b -Quasi Metric Spaces," Mathematics, MDPI, vol. 4(4), pages 1-16, November.
    3. Moosa Gabeleh, 2013. "Proximal Weakly Contractive and Proximal Nonexpansive Non-self-Mappings in Metric and Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 158(2), pages 615-625, August.
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    5. S. Sadiq Basha & N. Shahzad & R. Jeyaraj, 2013. "Best proximity point theorems: exposition of a significant non-linear programming problem," Journal of Global Optimization, Springer, vol. 56(4), pages 1699-1705, August.
    6. Hind Alamri & Nawab Hussain & Ishak Altun, 2023. "Proximity Point Results for Generalized p -Cyclic Reich Contractions: An Application to Solving Integral Equations," Mathematics, MDPI, vol. 11(23), pages 1-25, November.
    7. Moosa Gabeleh, 2015. "Best Proximity Point Theorems via Proximal Non-self Mappings," Journal of Optimization Theory and Applications, Springer, vol. 164(2), pages 565-576, February.
    8. Slah Sahmim & Abdelbasset Felhi & Hassen Aydi, 2019. "Convergence and Best Proximity Points for Generalized Contraction Pairs," Mathematics, MDPI, vol. 7(2), pages 1-12, February.
    9. S. Sadiq Basha, 2014. "Best proximity point theorems: unriddling a special nonlinear programming problem," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(2), pages 543-553, July.
    10. Naeem Saleem & Mujahid Abbas & Manuel De la Sen, 2019. "Optimal Approximate Solution of Coincidence Point Equations in Fuzzy Metric Spaces," Mathematics, MDPI, vol. 7(4), pages 1-13, April.
    11. V. Sankar Raj & A. Anthony Eldred, 2014. "A Characterization of Strictly Convex Spaces and Applications," Journal of Optimization Theory and Applications, Springer, vol. 160(2), pages 703-710, February.

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