Existence and iterative approximation of hybrid best proximity points for set-valued mappings. (English) Zbl 07383314
Summary: In this paper, hybrid best proximity points for a class of proximally mixed monotone set-valued mappings are discussed. In order to get rid of the fetter of continuity, we introduce the proximally ordered contraction instead of various proximal distance contractions widely applied in recent literatures and present some new existence and iterative approximation theorems of best proximity points for such mappings. Some examples are given to illustrate the advantages of the obtained results.
Keywords:
hybrid best proximity point; partially ordered set; proximally mixed monotone mapping; proximally ordered contractionReferences:
| [1] | Suzuki, T.; Kikkawa, M.; Vetro, C., The existence of best proximity points in metric spaces with the property UC, Nonlin. Anal, 71, 7-8, 2918-2926 (2009) · Zbl 1178.54029 · doi:10.1016/j.na.2009.01.173 |
| [2] | Bari, C. D.; Suzuki, T.; Vetro, C., Best proximity points for cyclic Meir-Keeler contractions, Nonlin. Anal, 69, 11, 3790-3794 (2008) · Zbl 1169.54021 · doi:10.1016/j.na.2007.10.014 |
| [3] | Felhi, A., A note on” Convergence and best proximity points for Berinde’s cyclic contraction with proximally complete property, Math. Meth. Appl. Sci., 41, 1, 140-143 (2018) · Zbl 1395.54043 · doi:10.1002/mma.4601 |
| [4] | Abkar, A.; Gabeleh, M., Best proximity points for cyclic mappings in ordered metric spaces, J. Optim. Theory Appl., 150, 1, 188-193 (2011) · Zbl 1232.54035 · doi:10.1007/s10957-011-9810-x |
| [5] | Ilchev, A.; Zlatanov, B., Error estimates for approximation of coupled best proximity points for cyclic contractive maps, Appl. Math. Comp, 290, 412-425 (2016) · Zbl 1410.41010 · doi:10.1016/j.amc.2016.06.022 |
| [6] | Gabeleh, M., Best proximity points: global minimization of multivalued non-self mappings, Optim. Lett., 8, 3, 1101-1112 (2014) · Zbl 1321.90152 · doi:10.1007/s11590-013-0628-3 |
| [7] | Khammahawong, K.; Kumam, P.; Lee, D.; Cho, Y.-J., Best proximity points for multi-valued Suzuki-F-proximal contractions, J. Fixed Point Theory Appl, 19, 4, 2847-2871 (2017) · Zbl 1489.54156 · doi:10.1007/s11784-017-0457-6 |
| [8] | Hussain, N.; Latif, A.; Salimi, P., Best proximity point results for modified Suzuki-proximal contractions, Fixed Point Theory Appl, 2014, 1, 10 (2014) · Zbl 1310.41029 · doi:10.1186/1687-1812-2014-10 |
| [9] | Basha, S. S., Best proximity point theorems on partially ordered sets, Optim. Lett., 7, 5, 1035-1043 (2013) · Zbl 1267.90104 · doi:10.1007/s11590-012-0489-1 |
| [10] | Basha, S. S.; Shahzad, N.; Vetro, C., Best proximity point theorems for proximal cyclic contractions, J. Fixed Point Theory Appl, 19, 4, 2647-2661 (2017) · Zbl 1490.54040 · doi:10.1007/s11784-017-0447-8 |
| [11] | Mongkolkeha, C.; Kumam, P., Some common best proximity points for proximity commuting mappings, Optim. Lett., 7, 8, 1825-1836 (2013) · Zbl 1287.54041 · doi:10.1007/s11590-012-0525-1 |
| [12] | Liu, H.; Fan, X.; Yan, L.; Wang, Z., On the existence of coupled best proximity point and best proximity point for Suzuki type alpha(+)-theta-proximal multivalued mappings, J. Nonlin. Sci. Appl, 10, 4, 1801-1819 (2017) · Zbl 1412.47067 · doi:10.22436/jnsa.010.04.41 |
| [13] | Amini-Harandi, A., Common best proximity points theorems in metric spaces, Optim. Lett., 8, 2, 581-589 (2014) · Zbl 1310.90120 · doi:10.1007/s11590-012-0600-7 |
| [14] | Kumam, P.; Pragadeeswarar, V.; Marudai, M.; Sitthithakerngkiet, K., Coupled best proximity points in ordered metric spaces, Fixed Point Theory Appl, 2014, 1, 107 (2014) · Zbl 1429.54056 · doi:10.1155/2014/274062 |
| [15] | Anastassiou, G. A.; Cho, Y.-J.; Saadati, R., Common best proximity points for proximity commuting mappings in non-Archimedean pm-spaces, J. Comp. Anal. Appl, 20, 6, 1021-1030 (2016) · Zbl 1335.90069 |
| [16] | Abkar, A.; Gabeleh, M., The existence of best proximity points for multivalued non-self-mappings, RACSAM, 107, 2, 319-325 (2013) · Zbl 1287.54036 · doi:10.1007/s13398-012-0074-6 |
| [17] | Plebaniak, R., On best proximity points for set-valued contractions of Nadler type with respect to b-generalized pseudodistances in b-metric spaces, Fixed Point Theory Appl, 2014, 1, 9 (2014) · Zbl 1333.54048 · doi:10.1186/1687-1812-2014-39 |
| [18] | Gabeleh, M., Proximal weakly contractive and proximal nonexpansive non-self-mappings in metric and Banach spaces, J. Optim. Theory Appl., 158, 2, 615-625 (2013) · Zbl 1272.90111 · doi:10.1007/s10957-012-0246-8 |
| [19] | Choudhury, B. S.; Jleli, M.; Maity, P., Best proximity points of discontinuous operator in partially ordered metric spaces, J. Nonlin. Sci. Appl, 10, 1, 308-315 (2017) · Zbl 1412.54039 · doi:10.22436/jnsa.010.01.29 |
| [20] | Ninsri, A.; Sintunavarat, W., Toward a generalized contractive condition in partial metric spaces with the existence results of fixed points and best proximity points, J. Fixed Point Theoty Appl., 20, 1, 15 (2018) · Zbl 1489.54186 · doi:10.1007/s11784-018-0499-4 |
| [21] | Hong, S.-H., Fixed points for mixed monotone multivalued operators in Banach spaces with applications, J. Math. Anal. Appl, 337, 151-162 (2008) |
| [22] | Hong, S.-H.; Guan, D.; Wang, L., Hybrid fixed points of multivalued operators in metric spaces with applications, Nonlin. Anal, 70, 11, 4106-4117 (2009) · Zbl 1206.54045 · doi:10.1016/j.na.2008.08.020 |
| [23] | Hong, S.-H., Fixed points of multivalued operators in ordered metric spaces with applications, Nonlin. Anal, 72, 11, 3929-3942 (2010) · Zbl 1184.54041 · doi:10.1016/j.na.2010.01.013 |
| [24] | Ahmad, J.; Al-Mazrooei, A. E.; Cho, Y. J.; Yang, Y.-O., Fixed point results for generalized Θ-contractions, J. Nonlin. Sci. Appl, 10, 5, 2350-2358 (2017) · Zbl 1412.46088 · doi:10.22436/jnsa.010.05.07 |
| [25] | Kadelburg, Z.; Nashine, H.-K.; Radenovic, S., Coupled fixed points in partial metric spaces, J. Adv. Math. Stud, 2013, 1, 1-172 (2013) · Zbl 1268.54028 · doi:10.1155/2013/726387 |
| [26] | Nashine, H.-K.; Kadelburg, Z.; Radenovic, S., Fixed point theorems via various cyclic contractive conditions in partial metric spaces, Publ. Inst. Math, 107, 69-93 (2013) · Zbl 1313.54097 |
| [27] | De la Sen, M., Some results on fixed and best proximity points of multivalued cyclic self-mappings with a partial order, Abstr. Appl. Anal, 2013, 1-11 (2013) · Zbl 1273.54050 · doi:10.1155/2013/968492 |
| [28] | Nieto, J. J.; Lopez, R. R., Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22, 3, 223-229 (2005) · Zbl 1095.47013 · doi:10.1007/s11083-005-9018-5 |
| [29] | Tan, K. K.; Xu, H. K., Appoximating fixed points of nonexpansive mapping by the Ishikawa iteration process, J. Math. Anal. Appl, 178, 2, 301-308 (1993) · Zbl 0895.47048 · doi:10.1006/jmaa.1993.1309 |
| [30] | Deng, L., Convergence of the Ishikawa iteration process for nonexpansive mappings, J. Math. Anal. Appl, 199, 3, 769-775 (1996) · Zbl 0856.47041 · doi:10.1006/jmaa.1996.0174 |
| [31] | Jung, J.-S.; Sahu, D. R., Dual convergences of iteration processes for nonexpansive mapping in Banach spaces, Czechoslovak Math. J, 53, 2, 397-404 (2003) · Zbl 1030.47037 · doi:10.1023/A:1026291421505 |
| [32] | Ishikawa, S., Fixed point by a new iteration method, Proc. Am. Math. Soc, 44, 1, 147-150 (1974) · Zbl 0286.47036 · doi:10.1090/S0002-9939-1974-0336469-5 |
| [33] | Nadler, N. S. Jr., Multi-valued contraction mappings, Pacific J. Math, 30, 2, 475-488 (1969) · Zbl 0187.45002 · doi:10.2140/pjm.1969.30.475 |
| [34] | Dhage, B. C., Hybrid fixed point theory for strictly monotone increasing multivalued mappings with applications, Comput. Math. Appl, 53, 5, 803-824 (2007) · Zbl 1144.47041 · doi:10.1016/j.camwa.2006.10.020 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
