In the setting of uniformly convex Banach spaces equipped with a partially ordered relation, we survey the existence of fixed points for monotone orbitally nonexpansive mappings. In this way, we extend and improve the main results of Alfuraidan and Khamsi [M. R. Alfuraidan, M. A. Khamsi, Proc. Amer. Math. Soc., 146, (2018), 2451-2456]. Examples are given to show the usability of our main conclusions. We also study the existence of an optimal solution for cyclic contractions in such spaces.

Type of Study: Research paper |
Subject:
General

1. A. Abkar, M. Gabeleh, Best Proximity Points for Cyclic Mappings in Ordered Metric Spaces, J. Optim. Theory Appl., 150, (2011), 188-193. [DOI:10.1007/s10957-011-9810-x]

2. A. Abkar, M. Gabeleh, Generalized Cyclic Contractions in Partially Ordered Metric Spaces, Optim. Lett., 6, (2012), 1819-1830. [DOI:10.1007/s11590-011-0379-y]

3. K. Aoyama, F. Kohsaka, Fixed Point Theorem for α-nonexpansive Mappings in Banach Spaces, Nonlinear Anal., 74, (2011), 4387-4391. [DOI:10.1016/j.na.2011.03.057]

4. M. Bachar, M. A. Khamsi, Fixed Points of Monotone Mappings and Application to Integral Equations, Fixed Point Theory Appl., 2015(110), (2015), 1-7. [DOI:10.1186/s13663-015-0362-x]

5. L. P. Belluce, W. A. Kirk, E. F. Steiner, Normal Structure in Banach Spaces, Pacific J. Math., 26, (1968), 433-440. [DOI:10.2140/pjm.1968.26.433]

6. J. Bogin, A Generalization of a Fixed Point Theorem of Goebel, Kirk and Shimi, Canadian Math. Bull., 19, (1976), 7-12. [DOI:10.4153/CMB-1976-002-7]

7. A. A. Eldred, P. Veeramani, Existence and Convergence of Best Proximity Points, J. Math. Anal. Appl., 323, (2006), 1001-1006. [DOI:10.1016/j.jmaa.2005.10.081]

8. J. Garcia-Falset, E. Llorens-Fuster, T. Suzuki, Fixed Point Theory for a Class of Generalized Nonexpansive Mappings, J. Math. Anal. Appl., 375, (2011), 185-195. [DOI:10.1016/j.jmaa.2010.08.069]

9. K. Goebel, W. A. Kirk, A Fixed Point Theorem for Asymptotically Nonexpansive Mappings, Proc. Amer. Math. Soc., (1972), 7, 171-174. [DOI:10.1090/S0002-9939-1972-0298500-3]

10. K. Goebel, W. A. Kirk, T. N. Shimi, A Fixed Point Theorem in Uniformly Convex Spaces, Boll. :union:e Mat. Ital., 7, (1973), 67-75.

11. W. A. Kirk, A Fixed Point Theorem for Mappings which Do not Increase Distances, Amer. Math. Monthly, 72, (1965), 1004-1373. [DOI:10.2307/2313345]

12. E. Llorens-Fuster, Orbitally Nonexpansive Mappings, Bull. Australian Math. Soc., 93, (2016), 497-503 [DOI:10.1017/S0004972715001318]

13. J. J. Nieto, R. Rodriguez-Lopez, Contractive Mapping Theorems in Partially Ordered Sets and Applications to Ordinary Differential Equations, Order, 22, (2005), 223-239. [DOI:10.1007/s11083-005-9018-5]

14. A. M. Rashed, M. A. Khamsi, A Fixed Point Theorem for Monotone Asymptotic Nonexpansive Mappings, Proc. Amer. Math. Soc., 146, (2018), 2451-2456. [DOI:10.1090/proc/13385]

15. S. Sadiq Basha, N. Shahzad, Common Best Proximity Point Theorems: Global Minimization of Some Real-valued Multi-objective Functions, Journal of Fixed Point Theory and Applications, 18, (2016), 587-600. [DOI:10.1007/s11784-016-0295-y]

16. T. Suzuki, Fixed Point Theorems and Convergence Theorems for Some Generalized Nonexpansive Mappings, Journal of Mathematical Analysis and Applications, 340, (2008), 1088-1095. [DOI:10.1016/j.jmaa.2007.09.023]

Rights and permissions | |

This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. |