Common Fixed Point Theorems Satisfying Implicit Relations on 2-cone Banach Space with an Application

Year 2019, , 9 - 18, 30.04.2019

D. Dhamodharan Nihal Taş R. Krishnakumar

https://doi.org/10.36753/mathenot.559237

Abstract

In this paper, we discuss the existence and uniqueness of common fixed-point theorems satisfying implicit

relations on 2-cone Banach spaces. Modifying obtained new contractive conditions, we also give an

application to the fixed-circle problem.

Keywords

common fixed point, 2-cone Banach space, 2-cone normed space, fixed circle

References

• [1] Abdeljawad, T., Karapınar, E. and Ta¸s, K., Common fixed point theorem in cone Banach space. Hacet. J. Math. Stat. 40 (2011), no. 2, 211-217.
• [2] Gujetiya, R. K., Mali, D. K. and Hakwadiya, M., Common fixed point theorem for compatible mapping on cone Banach space. Int. J. Math. Anal. 8 (2014), no. 35, 1697-1706.
• [3] Huang, L. G. and Zhang, X., Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. Appl. 332 (2007), 1468-1476.
• [4] Karapınar, E., Fixed point theorems in cone Banach spaces. Fixed Point Theory Appl. 2009, Article ID 609281, 9 pages. doi:10.1155/2009/609281.
• [5] Krishnakumar, R. and Dhamodharan, D., Common fixed point of contractive modulus on 2-cone Banach space. Malaya J. Mat. 5 (2017), no. 3, 608-618.
• [6] Mlaiki, N., Çelik, U., Ta¸s, N., Özgür, N. Y. and Mukheimer, A.,Wardowski type contractions and the fixed-circle problem on S-metric spaces. J. Math. 2018, Art. ID 9127486, 9 pp.
• [7] Mlaiki, N., Ta¸s, N. and Özgür, N. Y., On the fixed-circle problem and Khan type contractions. Axioms 7 (2018), 80.
• [8] Özgür, N. Y. and Ta¸s, N., Some fixed-circle theorems on metric spaces. Bull. Malays. Math. Sci. Soc. (2017). https://doi.org/10.1007/s40840-017-0555-z
• [9] Özgür, N. Y. and Ta¸s, N., Fixed-circle problem on S-metric spaces with a geometric viewpoint. arXiv:1704.08838 [10] Özgür, N. Y., Ta¸s, N. and Çelik, U., New fixed-circle results on S-metric spaces. Bull. Math. Anal. Appl. 9 (2017), no. 2, 10-23.
• [11] Özgür, N. Y. and Ta¸s, N., Some fixed-circle theorems and discontinuity at fixed circle. AIP Conference Proceedings, 1926, 020048 (2018).
• [12] Özgür, N. Y., Fixed-disc results via simulation functions. arXiv:1901.02623
• [13] Pant, R. P., Özgür, N. Y. and Ta¸s, N., On discontinuity problem at fixed point. Bull. Malays. Math. Sci. Soc. (2018). https://doi.org/10.1007/s40840-018-0698-6.
• [14] Pitchaimani, M. and Ramesh Kumar, D., Some common fixed point theorems using implicit relation in 2-Banach spaces. Surv. Math. Appl. 10 (2015), 159-168.
• [15] Rada, E. and Tato, A., An extension of a fixed point result in cone Banach space. Pure and Applied Mathematics Journal 4 (2015), no. 3, 70-74.
• [16] ¸Sahiner, A. and Yi ˘ git, T., 2-cone Banach spaces and fixed point theorem. Numerical Analysis and Applied Mathematics ICNAAM 2012 AIP Conf. Proc. 1479 (2012), 975-978. doi: 10.1063/1.4756305
• [17] ¸Sahiner, A., Fixed point theorems in symmetric 2-cone Banach space (lp; k; :; kc p). J. Nonlinear Anal. Optim. 3(2012), no. 2, 115-120.
• [18] Taş, N., Suzuki-Berinde type fixed-point and fixed-circle results on S-metric spaces. J. Linear Topol. Algebra 7 (2018), no. 3, 233-244.
• [19]Taş, N., Various types of fixed-point theorems on S-metric spaces. J. BAUN Inst. Sci. Technol. 20 (2018), no. 2, 211-223.
• [20] Taş, N., Özgür, N. Y. and Mlaiki, N., New types of FC-contractions and the fixed-circle problem. Mathematics 6 (2018), no. 10, 188.
• [21] Taş, N. and Özgür, N. Y., Some fixed-point results on parametric Nb-metric spaces. Commun. Korean Math. Soc. 33 (2018), no. 3, 943-960.