Document Type : Research Paper
Authors
- Nihal Taş ^{} ^{} ^{1}
- Nihal Yilmaz Ozgur ^{} ^{2}
^{1} Department of Mathematics, Bali kesir University, 10145, Bali kesir, Turkey.
^{2} Department of Mathematics, Bali kesir University, 10145 Bali kesir, Turkey.
Abstract
Banach's contraction principle has been improved and extensively studied on several generalized metric spaces. Recently, complex-valued $S$-metric spaces have been introduced and studied for this purpose. In this paper, we investigate some generalized fixed point results on a complete complex valued $S$-metric space. To do this, we prove some common fixed point (resp. fixed point) theorems using different techniques by means of new generalized contractive conditions and the notion of the closed ball. Our results generalize and improve some known fixed point results. We provide some illustrative examples to show the validity of our definitions and fixed
point theorems.
Keywords
[2] A.H. Ansari, O. Ege, and S. Radenovic, Some fixed point results on complex-valued $G_b$-metric spaces, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM, 112 (2018), pp. 463-472.
[3] A.H. Ansari, T. Dosenovic, S. Radenovic, and J.S. Ume, $C$-class functions and common fixed point theorems satisfying $varphi $-weakly contractive conditions, Sahand Commun. Math. Anal.,(13) (2019), pp. 17-30.
[4] A.H. Ansari and A. Razani, Some fixed point theorems for $C$-class functions in $b$-metric spaces, Sahand Commun. Math. Anal., 10 (2018), pp. 85-96.
[5] A. Azam, B. Fisher, and M. Khan, Common fixed point theorems in complex-valued metric spaces, Numer. Funct. Anal. Optim., 32 (2011), pp. 243-253.
[6] N.H. Dien, Some remarks on common fixed point theorems, J. Math. Anal. Appl., 187 (1994), pp. 76-90.
[7] O. Ege, Complex valued rectangular $b$-metric spaces and an application to linear equations, J. Nonlinear Sci. Appl., 8 (2015), pp. 1014-1021.
[8] O. Ege, Complex valued $G_b$-metric spaces, J. Comput. Anal. Appl., 21 (2016), pp. 363-368.
[9] O. Ege, Some fixed point theorems in complex-valued $G_b$-metric spaces, J. Nonlinear Convex Anal., 18 (2017), pp. 1997-2005.
[10] H. Faraji and K. Nourouzi, Fixed and common fixed points for $(psi,varphi)$-weakly contractive mappings in $b$-metric spaces, Sahand Commun. Math. Anal., 7 (2017), pp. 49-62.
[11] C. Kalaivani and G. Kalpana, Fixed point theorems in $C^ast $-algebra-valued $S$-metric spaces with some applications, Sci. Bull., Ser. A, Appl. Math. Phys., Politeh. Univ. Buchar., 80 (2018), pp. 93-102.
[12] Z. Liu, Y. Xu, and Y.J. Cho, On Characterizations of Ffixed and common fixed points, J. Math. Anal. Appl., 222 (1998), pp. 494-504.
[13] N.M. Mlaiki, Common fixed points in complex $S$-metric space, Adv. Fixed Point Theory, 4 (2014), pp. 509-524.
[14] N. Mlaiki, U. Celik, N. Tas, N.Y. Ozgur, and A. Mukheimer, Wardowski type contractions and the fixed-circle problem on $S$-metric spaces, J. Math., 2018 (2018), 9 pp.
[15] N.Y. Ozgur and N. Tas, Some fixed point theorems on $S$-metric spaces, Mat. Vesnik, 69 (2017), pp. 39-52.
[16] N.Y. Ozgur and N. Tas, Some new contractive mappings on $S$-metric spaces and their relationships with the mapping $(S25)$, Math. Sci., 11 (2017), pp. 7-16.
[17] N.Y. Ozgur and N. Tas, Some generalizations of fixed point theorems on $S$-metric spaces, Essays in Mathematics and Its Applications in Honor of Vladimir Arnold, New York, Springer, 2016.
[18] N.Y. Ozgur and N. Tas, Some fixed-circle theorems on metric spaces, Bull. Malays. Math. Sci. Soc., 42 (2019), pp. 1433-1449.
[19] N.Y. Ozgur and N. Tas, Some fixed-circle theorems and discontinuity at fixed circle, AIP Conference Proceedings 1926, 020048 (2018).
[20] N.Y. Ozgur, N. Tas, and U. Celik, New fixed-circle results on $S$-metric spaces, Bull. Math. Anal. Appl., 9 (2017), pp. 10-23.
[21] N.Y. Ozgur and N. Tas, Fixed-circle problem on $S$-metric spaces with a geometric viewpoint, Facta Universitatis. Series: Mathematics and Informatics, 34 (3) (2019), pp. 459-472.
[22] M. Ozturk, Common fixed point theorems satisfying contractive type conditions in complex-valued metric spaces, Abstr. Appl. Anal., 2014 (2014), 7 pp.
[23] M. Ozturk and N. Kaplan, Common fixed points of $f$-contraction mappings in complex-valued metric spaces, Math. Sci., 8 (2014), 7 pp.
[24] M. Ozturk and N. Kaplan, Some common coupled fixed points of mappings satisfying contractive conditions with rational expressions in complex-valued $G_b$-metric spaces, Bangmod Int. J. Math. & Comp. Sci., 1 (2015), pp. 190-204.
[25] N. Priyobarta, Y. Rohen, and N. Mlaiki, Complex valued Sb-metric spaces, J. Math. Anal., 8 (2017), pp. 13-24.
[26] M.M. Rezaee, S. Sedghi, and K. S. Kim, Coupled common fixed point results in ordered $S$-metric spaces, Nonlinear Funct. Anal. Appl., 23 (2018), pp. 595-612.
[27] B.E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc., 226 (1977), pp. 257-290.
[28] S. Sedghi, N. Shobe, and A. Aliouche, A generalization of fixed point theorems in $S$-metric spaces, Mat. Vesnik, 64 (2012), pp. 258-266.
[29] R.J. Shahkoohi and Z. Bagheri, Rational Geraghty contractive mappings and fixed point theorems in ordered $b_2$-metric spaces, Sahand Commun. Math. Anal., 13 (1) (2019), pp. 179-212.
[30] H. Shayanpour and A. Nematizadeh, Common fixed point theory in modified intuitionistic probabilistic metric spaces with common property $(E.A.)$, Sahand Commun. Math. Anal., 13 (1) (2019), pp. 31-50.
[31] N. Tas, N.Y. Ozgur, and N. Mlaiki, New types of $F_c$-contractions and the fixed-circle problem, Mathematics, 6 (2018), 9 pp.
[32] R.K. Verma and H.K. Pathak, Common fixed point theorems using property $(E.A)$ in complex-valued metric spaces, Thai J. Math., 11 (2013), pp. 347-355.