Fixed point theorems via comparable mappings in ordered metric spaces. (English) Zbl 1491.54133
Summary: In this paper, we introduce two new types of comparable multivalued mappings and prove a variant of well known classical Mizoguchi and Takahashi fixed point theorem for these mappings in partially ordered metric spaces. The method we use in the proof of our results is technically connected with the proof given by T. Suzuki [J. Math. Anal. Appl. 340, No. 1, 752–755 (2008; Zbl 1137.54026)]. We also provide examples to vindicate our claims and usability of the present results.
MSC:
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 47H10 | Fixed-point theorems |
| 54E40 | Special maps on metric spaces |
| 54F05 | Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces |
Citations:
Zbl 1137.54026References:
| [1] | Abbas, M., A.R. Khan, and T. Nazir. 2012. Common fixed point of multivalued mappings in ordered generalized metric spaces. Filomat 26 (5): 1045-1053. · Zbl 1289.54109 · doi:10.2298/FIL1205045A |
| [2] | Alam, A., and M. Imdad. 2017. Comparable linear contractions in ordered metric spaces. Fixed Point Theory 18 (2): 415-432. · Zbl 1395.54036 · doi:10.24193/fpt-ro.2017.2.33 |
| [3] | Alam, A., and M. Imdad. 2016. Monotone generalized contractions in ordered metric spaces. Bulletin of the Korean Mathematical Society 53 (1): 61-81. · Zbl 1337.54033 · doi:10.4134/BKMS.2016.53.1.061 |
| [4] | Alam, A., A.R. Khan, and M. Imdad. 2014. Some coincidence theorems for generalized nonlinear contractions in ordered metric spaces with applications. Fixed Point Theory and Applications 2014: 216. · Zbl 1505.54060 · doi:10.1186/1687-1812-2014-216 |
| [5] | Almezel, S., Q.H. Ansari, and M.A. Khamsi. 2014. Topics in fixed point theory. Switzerland: Springer. · Zbl 1278.47002 · doi:10.1007/978-3-319-01586-6 |
| [6] | Assad, N.A., and W.A. Kirk. 1972. Fixed point theorems for set-valued mappings of contractive type. Pacific Journal of Mathematics 43: 553-562. · Zbl 0239.54032 · doi:10.2140/pjm.1972.43.553 |
| [7] | Beg, I., and A.R. Butt. 2010. Common fixed point for generalized set valued contractions satisfying an implicit relation in partially ordered metric spaces. Mathematical Communications 15: 65-75. · Zbl 1195.54068 |
| [8] | Beg, I., and A.R. Butt. 2009. Fixed point for set valued mappings satisfying an implicit relation in partially ordered metric spaces. Nonlinear Analysis 71: 3699-3704. · Zbl 1176.54028 · doi:10.1016/j.na.2009.02.027 |
| [9] | Bhaskar, T.G., and V. Laskhmikantham. 2006. Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Analysis 65 (7): 1379-1393. · Zbl 1106.47047 · doi:10.1016/j.na.2005.10.017 |
| [10] | Ciric, L.B. 2006. Common fixed point theorems for multi-valued mappings. Demonstratio Mathematica 39 (2): 419-428. · Zbl 1112.47047 · doi:10.1515/dema-2006-0220 |
| [11] | Ciric, L.B. 2009. Multi-valued nonlinear contraction mappings. Nonlinear Analysis 71: 2716-2723. · Zbl 1179.54053 · doi:10.1016/j.na.2009.01.116 |
| [12] | Daffer, P.Z., H. Kaneko, and W. Li. 1996. On a conjecture of S. Reich. Proceedings of The American Mathematical Society 124 (10): 3159-3162. · Zbl 0866.47040 · doi:10.1090/S0002-9939-96-03659-3 |
| [13] | Dimri, R.C., and G. Prasad. 2017. Coincidence theorems for comparable generalized non-linear contractions in ordered partial metric spaces. Communications of the Korean Mathematical Society 32 (2): 375-387. · Zbl 1371.54178 · doi:10.4134/CKMS.c160127 |
| [14] | Feng, Y., and S. Liu. 2006. Fixed point theorems for multi-valued contractive mappings and multivalued Caristi type mappings. Journal of Mathematical Analysis and Applications 317: 103-112. · Zbl 1094.47049 · doi:10.1016/j.jmaa.2005.12.004 |
| [15] | Klim, D., and D. Wardowski. 2007. Fixed point theorems for set-valued contractions in complete metric spaces. Journal of Mathematical Analysis and Applications 334: 132-139. · Zbl 1133.54025 · doi:10.1016/j.jmaa.2006.12.012 |
| [16] | Kutbi, M.A., A. Alam, and M. Imdad. 2015. Sharpening some core theorems of Nieto and Rodriguez-Lopez with application to boundary value problem. Fixed Point Theory and Applications 2015: 198. · Zbl 1470.54078 · doi:10.1186/s13663-015-0446-7 |
| [17] | Mizoguchi, N., and W. Takahashi. 1989. Fixed point theorems for multivalued mappings on complete metric spaces. Journal of Mathematical Analysis and Applications 141: 177-188. · Zbl 0688.54028 · doi:10.1016/0022-247X(89)90214-X |
| [18] | Nadler, S. 1969. Multi-valued contraction mappings. Pacific Journal of Mathematics 20 (2): 475-488. · Zbl 0187.45002 · doi:10.2140/pjm.1969.30.475 |
| [19] | Nieto, J.J., and R. Rodriguez-Lopez. 2005. Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22: 223-239. · Zbl 1095.47013 · doi:10.1007/s11083-005-9018-5 |
| [20] | Ran, A.C.M., and M.C.B. Reurings. 2003. A fixed point theorem in partially ordered sets and some applications to matrix equations. Proceedings of the American Mathematical Society 132 (5): 1435-1443. · Zbl 1060.47056 · doi:10.1090/S0002-9939-03-07220-4 |
| [21] | Reich, S. 1972. Fixed points of contractive functions. Bollettino Unione Matenatica Italiana 5: 26-42. · Zbl 0249.54026 |
| [22] | Suzuki, T. 2008. Mizoguchi-Takahashi’s fixed point theorem is a real generalization of Nadler’s. Journal of Mathematical Analysis and Applications 340: 752-755. · Zbl 1137.54026 · doi:10.1016/j.jmaa.2007.08.022 |
| [23] | Tiammee, J., and S. Suantai. 2014. Fixed point theorems for monotone multi-valued mappings in partially ordered metric spaces. Fixed Point Theory and Applications 2014: 110. · Zbl 1345.54076 · doi:10.1186/1687-1812-2014-110 |
| [24] | Turinici, M. 1986. Fixed points for monotone iteratively local contractions. Demonstratio Mathematica 19 (1): 171-180. · Zbl 0651.54020 |
| [25] | Turinici, M. 2011. Ran and Reurings theorems in ordered metric spaces. Journal of the Indian Mathematical Society 78: 207-214. · Zbl 1251.54057 |
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.
