Compactness of soft cone metric space and fixed point theorems related to diametrically contractive mapping

In this article, we describe the concepts such as sequentially soft closeness, sequential compactness, totally boundedness and sequentially continuity in any soft cone metric space and prove their some properties. Also, we examine soft closed set, soft closure, compactness and continuity in an elementary soft topological cone metric space. Unlike classical cone metric space, sequential compactness and compactness are not the same here. Because the compactness is an elementary soft topological property and cannot be defined for every soft cone metric space. However, in the restricted soft cone metric spaces, they are the same. Additionally, we prove some fixed point theorems related to diametrically contractive mapping in a complete soft cone metric space.

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[1] Abbas M, Ali B, Vetro C. A Suzuki type fixed point theorem for a generalized multivalued mapping on partial Hausdorff metric spaces. Topology and its Applications 2013; 160 (3): 553-563. doi: 10.1016/j.topol.2013.01.006
[2] Abbas M, Jungck G. Common fixed point results for noncommuting mappings without continuity in cone metric spaces. Journal of Mathematical Analysis and Applications 2008; 341 (1): 416-420. doi: 10.1016/j.jmaa.2007.09.070
[3] Abbas M, Murtaza G, Romaguera S. On the fixed point theory of soft metric spaces. Fixed Point Theory and Applications 2016; 2016 (17). doi: 10.1186/s13663-016-0502-y
[4] Altıntaş İ, Şimşek D, Taşköprü K. Topology of soft cone metric spaces. AIP Conference Proceedings 2017; 1880 (1): 030006. doi: 10.1063/1.5000605
[5] Arshad M, Azam A, Vetro P. Some common fixed point results in cone metric spaces. Fixed Point Theory and Applications 2009. doi: 10.1155/2009/493965
[6] Aygünoğlu A, Aygün H. Some notes on soft topological spaces. Neural Computing and Applications 2012; 21: 113-119. doi: 10.1007/s00521-011-0722-3
[7] Azam A, Arshad M, Beg I. Common fixed points of two maps in cone metric spaces. Rendiconti del Circolo Matematico di Palermo 2009; 57: 433-441. doi: 10.1007/s12215-008-0032-5
[8] Azam A, Arshad M, Beg I. Existence of fixed points in complete cone metric spaces. International Journal of Modern Mathematics 2010; 5 (1): 91-99.
[9] Azam A, Beg I, Arshad M. Fixed point in topological vector space-valued cone metric spaces. Fixed Point Theory and Applications 2010. doi: 10.1155/2010/604084
[10] Azam A, Mehmood N. Multivalued fixed point theorems in tvs-cone metric spaces. Fixed Point Theory and Applications 2013. doi: 10.1186/1687-1812-2013-184
[11] Azam A, Mehmood N, Ahmad J, Radenović S. Multivalued fixed point theorems in cone b-metric spaces. Journal of Inequalities and Applications volume 2013. doi: 10.1186/1029-242X-2013-582
[12] Beg I, Azam A, Arshad M. Common fixed points for maps on topological vector space valued cone metric spaces. International Journal of Mathematics and Mathematical Sciences 2009; 2009. doi: 10.1155/2009/560264
[13] Chen D, Tsang ECC, Yeung DS, Wang X. The parametrization reduction of soft sets and its applications. Computers & Mathematics with Applications 2005; 49 (5-6): 757-763. doi: 10.1016/j.camwa.2004.10.036
[14] Cho SH, Bae JS. Fixed point theorems for multivalued maps in cone metric spaces. Fixed Point Theory and Applications 2011. doi: 10.1186/1687-1812-2011-87
[15] Çaksu Güler A, Dalan Yıldırım E, Bedre Özbaır O. A fixed point theorem on soft G-metric spaces. Journal of Nonlinear Sciences and Applications 2016; 9 (3): 885-894. doi: 10.22436/jnsa.009.03.18
[16] Das S, Majumdar P, Samanta SK. On soft linear spaces and soft normed linear spaces. Annals of Fuzzy Mathematics and Informatics 2015; 9 (1): 91-109.
[17] Das S, Samanta SK. Soft real sets, soft real numbers and their properties. Journal of Fuzzy Mathematics 2012; 20: 551-576.
[18] Das S, Samanta SK. On soft complex sets and soft complex numbers. Journal of Fuzzy Mathematics 2013; 21: 195-216.
[19] Das S, Samanta SK. Soft linear operators in soft normed linear spaces. Annals of Fuzzy Mathematics and Informatics 2013; 6 (2): 295-314.
[20] Das S, Samanta SK. On soft metric spaces. Journal of Fuzzy Mathematics 2013; 21: 707-734.
[21] Fallahi K, Abbas M, Rad GS. Generalized c-distance on cone b-metric spaces endowed with a graph and fixed point results. Applied General Topology 2017; 18 (2): 391-400. doi: 10.4995/agt.2017.7673
[22] Fallahi K, Rad GS. Fixed point results in cone metric spaces endowed with a graph. Sahand Communications in Mathematical Analysis 2017; 6 (1): 39-47. doi: 10.22130/SCMA.2017.23163
[23] Gündüz Aras Ç, Öztürk TY, Bayramov S. Separation axioms on neutrosophic soft topological spaces. Turkish Journal of Mathematics 2019; 43 (1): 498-510. doi: 10.3906/mat-1805-110
[24] Huang LG, Zhang X. Cone metric spaces and fixed point theorems of contractive mappings. Journal of Mathematical Analysis and Applications 2007; 332 (2): 1468-1476. doi: 10.1016/j.jmaa.2005.03.087
[25] İlkhan M, Zengin Alp P, Kara EE. On the spaces of linear operators acting between asymmetric cone normed spaces. Mediterranean Journal of Mathematics 2018; 15. doi: 10.1007/s00009-018-1182-0
[26] Janković S, Kadelburg Z, Radenović S. On cone metric spaces: a survey. Nonlinear Analysis: Theory, Methods & Applications 2011; 74 (7): 2591-2601. doi: 10.1016/j.na.2010.12.014
[27] Kong Z, Jia W, Zhang G, Wang L. Normal parameter reduction in soft set based on particle swarm optimization algorithm. Applied Mathematical Modelling 2015; 39 (16): 4808-4820. doi: 10.1016/j.apm.2015.03.055
[28] Maji PK, Biswas R, Roy AR. Soft set theory. Computers & Mathematics with Applications 2003; 45 (4-5): 555-562. doi: 10.1016/S0898-1221(03)00016-6
[29] Maji PK, Roy AR, Biswas R. An application of soft sets in a decision making problem. Computers & Mathematics with Applications 2002; 44 (8-9): 1077-1083. doi: 10.1016/S0898-1221(02)00216-X
[30] Majumdar P, Samanta SK. On soft mappings. Computers & Mathematics with Applications 2010; 60 (9): 2666-2672. doi: 10.1016/j.camwa.2010.09.004
[31] Mehmood N, Azam A, Kočinac LDR. Multivalued fixed point results in cone metric spaces. Topology and its Applications 2015; 179: 156-170. doi: 10.1016/j.topol.2014.07.011
[32] Molodtsov D. Soft set theory-first results. Computers & Mathematics with Applications 1999; 37 (4-5): 19-31. doi: 10.1016/S0898-1221(99)00056-5
[33] Özkan A. On near soft sets. Turkish Journal of Mathematics 2019; 43 (2): 1005-1017. doi: 10.3906/mat-1811-57
[34] Öztunç S, Aslan S. Jungck type fixed point results for weakly compatible mappings in a rectangular soft metric space. Journal of Inequalities and Applications 2019; 2019. doi: 10.1186/s13660-019-2096-5
[35] Öztunç S, Mutlu A, Aslan S. Some Kannan type fixed point results in rectangular soft metric space and an application of fixed point for thermal science problem. Thermal Science 2019; 23 (1): 215-225. doi: 10.2298/TSCI181102035O
[36] Öztürk TY. A new approach to soft uniform spaces. Turkish Journal of Mathematics 2016; 40 (5): 1071-1084. doi: 10.3906/mat-1506-98
[37] Pazar Varol B, Aygün H. Fuzzy soft topology. Hacettepe Journal of Mathematics and Statistics 2012; 41 (3): 407-419.
[38] Pei D, Miao D. From soft sets to information systems. In: 2005 IEEE International Conference on Granular Computing Volume 2; Beijing, China; 2005. pp. 617-621.
[39] Rezapour S, Hamlbarani R. Some notes on the paper ”Cone metric spaces and fixed point theorems of contractive mappings”. Journal of Mathematical Analysis and Applications 2008; 345 (2): 719-724. doi: 10.1016/j.jmaa.2008.04.049
[40] Shabir M, Naz M. On soft topological spaces. Computers & Mathematics with Applications 2011; 61 (7): 1786-1799. doi: 10.1016/j.camwa.2011.02.006
[41] Şimşek D, Altıntaş İ, Ersoy S, Abdullayev F, İmaş Kızı M et al. An introduction to soft cone metric spaces and some fixed point theorems. Manas Journal of Engineering 2017; 5 (3): 69-89.
[42] Taşköprü K, Altıntaş İ. A new approach for soft topology and soft function via soft element. Mathematical Methods in the Applied Sciences 2020 (Early View). doi: 10.1002/mma.6354
[43] Türkoğlu D, Abulova M. Cone metric spaces and fixed point theorems in diametrically contractive mapping. Acta Mathematica Sinica, English Series 2010; 26: 489-496. doi: 10.1007/s10114-010-8019-5
[44] Yazar Mİ, Gündüz Aras Ç, Bayramov S. Fixed point theorems of soft contractive mappings. Filomat 2016; 30 (2): 269-279. doi: 10.2298/FIL1602269Y
[45] Zou Y, Xiao Z. Data analysis approaches of soft sets under incomplete information. Knowledge-Based Systems 2008; 21 (8): 941-945. doi: 10.1016/j.knosys.2008.04.004