Real Vector Cone Metric Space and Fixed Point Theorems
Fei HE1,2, Jing Hui QIU1
Author information+
1. School of Mathematical Sciences, Soochow University, Suzhou 215006, P. R. China; 2. School of Mathematical Sciences, Inner Mongolia University, hohhot 010021, P. R. China
We introduce real vector cone metric spaces, where cone metric is the mapping on a real vector space without topological structures. We also prove some new fixed point theorems in real vector cone metric spaces. By using nonlinear scalarization functions, we establish the equivalence between these and some other fixed point results in metric and in real vector cone metric spaces. Our results improve and generalize some results from the literature.
Fei HE, Jing Hui QIU.
Real Vector Cone Metric Space and Fixed Point Theorems. Acta Mathematica Sinica, Chinese Series, 2014, 57(1): 171-180 https://doi.org/10.12386/A2014sxxb0017
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}
参考文献
[1] Abbas M., Jungck G., Common fixed point results for noncommuting mappings without continuity in cone metric spaces, J. Math. Anal. Appl., 2008, 341: 416-420. [2] Abbas M., Rhoades B. E., Fixed and periodic point results in cone metric spaces, Appl. Math. Lett., 2008, 21: 511-515. [3] Altun I., Damnjanović B., Djorić D., Fixed point and common fixed point theorems on ordered cone metric spaces, Appl. Math. Lett., 2010, 23: 310-316. [4] Amini-Harandi A., Fakhar M., Fixed point theory in cone metric spaces obtained via the scalarization method, Comput. Math. Appl., 2010, 59: 3529-3534. [5] Ćirić Lj. B., A generalization of Banach's contraction principle, Proc. Amer. Math. Soc., 1974, 45: 267-273. [6] Das K. M., Naik K. V., Common fixed point theorems for commuting maps on metric spaces, Proc. Amer. Math. Soc., 1979, 77: 369-373. [7] Du W. S., A note on cone metric fixed point theory and its equivalence, Nonlinear Anal., 2010, 72: 2259- 2261. [8] Gopfert A., Riahi H., Tammer C., Zalinescu C., variational Methods in Partially Ordered Spaces, Springer- Verlag, New York, 2003. [9] Hernándz E., Jiménez B., Novo V., Benson Proper Efficiency in Set-valued Optimization on Real Linear Spaces, in Recent Advances in Optimization, Lecture Notes in Econom. and Math. Systems 563, Springer- Verlag, Berlin, 2006: 45-59. [10] Huang L. G., Zhang X., Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 2007, 332: 1468-1476. [11] Ilić D., Rakočević V., Common fixed points for maps on cone metric space, J. Math. Anal. Appl., 2008, 341: 876-882. [12] Ilić D., Rakočević V., Quasi-contraction on a cone metric space, Appl. Math. Lett., 2009, 22: 728-731. [13] Kadelburg Z., Pavlović M., Radenović S., Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces, Comput. Math. Appl., 2010, 59: 3148-3159. [14] Kadelburg Z., Radenović S., Rakočvić V., A note on the equivalence of some metric and cone metric fixed point results, Appl. Math. Lett., 2011, 24: 370-374. [15] Kadelburg Z., Radenović S., Rakočvić V., Remarks on quasi-contraction on a cone metric space, Appl. Math. Lett., 2009, 22: 1674-1679. [16] Rezapour Sh., Haghi R. H., Shahzad N., Some notes on fixed points of quasi-contraction maps, Appl. Math. Lett., 2010, 23: 498-502. [17] Pathak H. K., Shahza N., Fixed point results for generalized quasi-contraction mappings in abstract metric spaces, Nonlinear Anal., 2009, 71: 6068-6076. [18] Qiu J. H., Ekeland's variational principle in locally convex spaces and the density of extremal points, Nonlinear Anal., 2009, 71: 4705-4711. [19] Rezapour Sh., Hamlbarani R., Some notes on the paper cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl., 2008, 345: 719-724. [20] Song G. X., Sun X. Y., Zhao Y., Wang G. T., New common fixed point theorems for maps on cone metric spaces, Appl. Math. Lett., 2010, 23: 1033-1037.