On some fixed point theorems for ordered vectorial Ćirić-Prešić type contractions. (English) Zbl 1518.54031
Summary: The Banach contraction mapping principle is one of the building blocks of metric fixed point theory. Numerous generalizations of this principle have been made usually by altering the metric space with a more general space and changing the contraction conditions. In this paper, by utilizing these ways, we present a fixed point theorem for ordered vectorial Ćirić-Prešić-type contractions. This result extends many results in the literature such as the ones obtained for Ćirić-Prešić-type contractions on both metric and partially ordered metric spaces. With some remarks, we emphasize that every ordered Prešić and ordered Ćirić-Prešić-type contractions has to be an ordered vectorial Ćirić-Prešić-type contraction nevertheless the converse may not be true in general. In addition, with an example, we support our conclusion and also show that the results existing in the literature are not applicable to this example.
MSC:
| 54H25 | Fixed-point and coincidence theorems (topological aspects) |
| 54E40 | Special maps on metric spaces |
| 54E50 | Complete metric spaces |
| 54F05 | Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces |
| 47H10 | Fixed-point theorems |
Keywords:
Ćirić-Prešić-type operators; fixed point; Riesz space; partially ordered metric spaces; vector metric; ordered vector metric spacesReferences:
| [1] | Aliprantis, CD; Border, KC, Infinite Dimensional Analysis (1999), Berlin: Springer, Berlin · Zbl 0938.46001 · doi:10.1007/978-3-662-03961-8 |
| [2] | Altun, I.; Çevik, C., Some common fixed point theorems in vector metric spaces, Filomat, 25, 1, 105-113 (2011) · Zbl 1289.54111 · doi:10.2298/FIL1101105A |
| [3] | Banach, S., Sur les operations dans les ensembles abstracits et leur application aux equations integrales, Fundamenta Mathematicae, 3, 133-181 (1922) · JFM 48.0201.01 · doi:10.4064/fm-3-1-133-181 |
| [4] | Ćirić, Lj., and B. 1974. A generalization of Banach’s contraction principle. Proceedings of the American Mathematical Society, 45: 267-273. · Zbl 0291.54056 |
| [5] | Cirić, LB; Prešić, SB, On Preši ć type generalization of the Banach contraction mapping principle, Acta Math Univ Comenianae, 76, 143-147 (2007) · Zbl 1164.54030 |
| [6] | Çevik, C.; Altun, I., Vector metric spaces and some properties, Topological Methods in Nonlinear Analysis, 34, 2, 375-382 (2009) · Zbl 1201.54027 · doi:10.12775/TMNA.2009.048 |
| [7] | Çevik, C. 2014. On continuity of functions between vector metric spaces. Journal of Function Spaces, 2014. · Zbl 1322.46006 |
| [8] | Çevik, C.; Altun, I.; Şahin, H.; Özeken, ÇC, Some fixed point theorems for contractive mapping in ordered vector metric spaces, Journal of Nonlinear Science and Applications, 10, 1424-1432 (2017) · Zbl 1412.47103 · doi:10.22436/jnsa.010.04.12 |
| [9] | Luong, NV; Thuan, NX, Some fixed point theorems of Prešić-Cirić type, Acta Universitatis Apulensis, 30, 237-249 (2012) · Zbl 1289.54133 |
| [10] | Malhotra, SK; Shukla, S.; Sen, R., A generalization of Banach contraction principle in ordered cone metric spaces, Journal of Advanced Mathematical Studies, 5, 2, 59-67 (2012) · Zbl 1273.54065 |
| [11] | Nieto, JJ; Rodríguez-López, R., Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22, 3, 223-239 (2005) · Zbl 1095.47013 · doi:10.1007/s11083-005-9018-5 |
| [12] | Nieto, JJ; Rodríguez-López, R., Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Mathematica Sinica (English Series), 23, 2205-2212 (2007) · Zbl 1140.47045 · doi:10.1007/s10114-005-0769-0 |
| [13] | Özeken, ÇC; Çevik, C., Prešić type operators on ordered vector metric spaces, Journal of Science and Art, 21, 4, 935-942 (2021) · doi:10.46939/J.Sci.Arts-21.4-a05 |
| [14] | Özeken, ÇC; Çevik, C., Ordered vectorial quasi and almost contractions on ordered vector metric spaces, Mathematics, 9, 19, 2443-2450 (2021) · doi:10.3390/math9192443 |
| [15] | Prešić, S.B. 1965. Sur une classe din equations aux differences finite et. sur la convergence de certains suites. Publications de l’Institut Mathematique, 5(19): 75-78. · Zbl 0134.04205 |
| [16] | Prešić, SB, Sur la convergence des suites, Comptes Rendus de l’Académie de Paris, 260, 3828-3830 (1965) · Zbl 0125.39805 |
| [17] | Ran, ACM; Reurings, MCB, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proceedings of the American Mathematical Society, 132, 1435-1443 (2003) · Zbl 1060.47056 · doi:10.1090/S0002-9939-03-07220-4 |
| [18] | Rao, K.P.R., Ali M. Mustaq, and D., Fisher, B. 2011. Some presic type generalizations of the Banach contraction principle. Mathematica Moravica, 15 (1): 41-47. · Zbl 1265.54190 |
| [19] | Reich, S., Zaslavski, A. J. 2022. A fixed point result in generalized metric spaces. The Journal of Analysis. · Zbl 1520.54033 |
| [20] | Reich, S.; Zaslavski, AJ, Fixed points and convergence results for a class of contractive mappings, Journal of Nonlinear and Variational Analysis, 5, 5, 665-671 (2021) · Zbl 1519.47060 |
| [21] | Reich, S.; Zaslavski, AJ, Existence of a unique fixed point for nonlinear contractive mappings, Mathematics, 8, 1, 55 (2020) · doi:10.3390/math8010055 |
| [22] | Sahin, H. 2021. Best proximity point theory on vector metric spaces. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 70: 130-142. · Zbl 1491.54147 |
| [23] | Sahin, H. 2022. Existence and uniqueness results for nonlinear fractional differential equations via new Q-function. Advances in Operator Theory, 7. · Zbl 1476.54112 |
| [24] | Shukla, S., Radenović, S. 2013. A generalization of Preši ć type mappings in o-complete ordered partial metric spaces, Chinese Journal of Mathematics, 2013. · Zbl 1297.54104 |
| [25] | Shukla, S.; Radenović, S., Prešić-Maia type theorems in ordered metric spaces, Gulf Journal of Mathematics, 2, 2, 73-82 (2014) · Zbl 1389.54122 · doi:10.56947/gjom.v2i2.197 |
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