Discrete Optimization for Ordered Weak Proximal Kannan Contractions

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Abstract

Let us consider a non-self mapping T : A → B, where A and B are two nonempty subsets of a partially ordered set that is equipped a metric. A best proximity point x⋆ for such a mapping T is a point such that d(x⋆, T x⋆) = dist(A,B). In this work, we provide different existence results of best proximity points and so, we establish some new fixed point theorems in the setting of partially ordered set.

Abstract

Let us consider a non-self mapping T : A → B, where A and B are two nonempty subsets of a partially ordered set that is equipped a metric. A best proximity point x⋆ for such a mapping T is a point such that d(x⋆, T x⋆) = dist(A,B). In this work, we provide different existence results of best proximity points and so, we establish some new fixed point theorems in the setting of partially ordered set.

References

  • 1. Abkar, A.; Gabeleh, M. - Best proximity points for cyclic mappings in ordered metric spaces, J. Optim. Theory Appl., 151 (2011), 418-424.

  • 2. Abkar, A.; Gabeleh, M. - Global optimal solutions of noncyclic mappings in metric spaces, J. Optim. Theory Appl., 153 (2012), 298-305.

  • 3. Al-Thagafi, M.A.; Shahzad, N. - Convergence and existence results for best proxi- mity points, Nonlinear Anal., 70 (2009), 3665-3671.

  • 4. Amini-Harandi, A. - Best proximity point theorems for cyclic strongly quasi- contraction mappings, J. Global Optim., 56 (2013), 1667-1674.

  • 5. Di Bari, C.; Suzuki, T.; Vetro, C. - Best proximity points for cyclic Meir-Keeler contractions, Nonlinear Anal., 69 (2008), 3790-3794.

  • 6. Eldred, A.A.; Veeramani, P. - Existence and convergence of best proximity points, J. Math. Anal. Appl., 323 (2006), 1001-1006.

  • 7. Espínola, R. - A new approach to relatively nonexpansive mappings, Proc. Amer. Math. Soc., 136 (2008), 1987-1995.

  • 8. Gabeleh, M. - Global optimal solutions of non-self mappings, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 75 (2013), 67-74.

  • 9. Kannan, R. - Some results on fixed points. II, Amer. Math. Monthly, 76 (1969), 405-408.

  • 10. Khamsi, M.A.; Kirk, W.A. - An introduction to metric spaces and fixed point theory, Pure and Applied Mathematics (New York), Wiley-Interscience, New York, 2001.

  • 11. Kikkawa, M.; Suzuki, T. - Some similarity between contractions and Kannan map- pings, Fixed Point Theory Appl., 2008, Art. ID 649749, 8 pp.

  • 12. Nieto, J.J.; Rodr´ıguez-L´opez, R. - Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (2005), 223-239 (2006).

  • 13. Petric, M.A. - Best proximity point theorems for weak cyclic Kannan contractions, Filomat, 25 (2011), 145-154.

  • 14. Sadiq Basha, S. - Discrete optimization in partially ordered sets, J. Global Optim., 54 (2012), 511-517.

  • 15. Sanhan, W.; Mongkolkeha, C.; Kumam, P. - Generalized proximal -contraction mappings and best proximity points, Abstr. Appl. Anal., 2012, Art. ID 896912, 19 pp.

1. Abkar, A.; Gabeleh, M. - Best proximity points for cyclic mappings in ordered metric spaces, J. Optim. Theory Appl., 151 (2011), 418-424.

2. Abkar, A.; Gabeleh, M. - Global optimal solutions of noncyclic mappings in metric spaces, J. Optim. Theory Appl., 153 (2012), 298-305.

3. Al-Thagafi, M.A.; Shahzad, N. - Convergence and existence results for best proxi- mity points, Nonlinear Anal., 70 (2009), 3665-3671.

4. Amini-Harandi, A. - Best proximity point theorems for cyclic strongly quasi- contraction mappings, J. Global Optim., 56 (2013), 1667-1674.

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

6. Eldred, A.A.; Veeramani, P. - Existence and convergence of best proximity points, J. Math. Anal. Appl., 323 (2006), 1001-1006.

7. Espínola, R. - A new approach to relatively nonexpansive mappings, Proc. Amer. Math. Soc., 136 (2008), 1987-1995.

8. Gabeleh, M. - Global optimal solutions of non-self mappings, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys., 75 (2013), 67-74.

9. Kannan, R. - Some results on fixed points. II, Amer. Math. Monthly, 76 (1969), 405-408.

10. Khamsi, M.A.; Kirk, W.A. - An introduction to metric spaces and fixed point theory, Pure and Applied Mathematics (New York), Wiley-Interscience, New York, 2001.

11. Kikkawa, M.; Suzuki, T. - Some similarity between contractions and Kannan map- pings, Fixed Point Theory Appl., 2008, Art. ID 649749, 8 pp.

12. Nieto, J.J.; Rodr´ıguez-L´opez, R. - Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (2005), 223-239 (2006).

13. Petric, M.A. - Best proximity point theorems for weak cyclic Kannan contractions, Filomat, 25 (2011), 145-154.

14. Sadiq Basha, S. - Discrete optimization in partially ordered sets, J. Global Optim., 54 (2012), 511-517.

15. Sanhan, W.; Mongkolkeha, C.; Kumam, P. - Generalized proximal -contraction mappings and best proximity points, Abstr. Appl. Anal., 2012, Art. ID 896912, 19 pp.

Annals of the Alexandru Ioan Cuza University - Mathematics

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