New Answers to the Rhoades’ Open Problem and the Fixed-Circle Problem

Yıl 2020, Cilt: 3 Sayı: 1, 160 - 165, 15.12.2020

Nihal Taş

Öz

Recently, the Rhoades' open problem which is related to the discontinuity at fixed point of a self-mapping and the fixed-circle problem which is related to the geometric meaning of the set of fixed points of a self-mapping have been studied using various approaches. Therefore, in this paper, we give some solutions to the Rhoades' open problem and the fixed-circle problem on metric spaces. To do this, we inspire from the
Meir-Keeler type, Ciric type and Caristi type fixed-point theorems. Also, we use the simulation functions and Wardowski's technique to obtain new fixed-circle results.

Anahtar Kelimeler

Fixec circle, Fixed disc, Fixed point, Metric space

Kaynakça

• 1 S. Banach, Sur les operations dans les ensembles abstrait et leur application aux equations integrals, Fundam. Math. 3 (1922), 133-181.
• 2 R.K. Bisht, R. P. Pant, A remark on discontinuity at fixed point, J. Math. Anal. Appl. 445(2) (2017) 1239-1242.
• 3 J. Caristi, Fixed point theorems for mappings satisfying inwardness conditions, Trans. Am. Math. Soc. 215 (1976), 241-251.
• 4 U. Çelik, N. Özgür, A new solution to the discontinuity problem on metric spaces, Turk. J. Math. 44 (2020), 1115-1126.
• 5 Lj. B. Ciric, A generalization of Banach’s contraction principle, Proc. Am. Math. Soc. 45(2) (1974), 267-273.
• 6 L. J. Cromme, I. Diener, Fixed point theorems for discontinuous mapping, Math. Program. 51(1-3) (1991), 257-267.
• 7 T. Dosenovic, S. Radenovic, S. Sedghi, Generalized metric spaces: survey, TWMS J. Pure Appl. Math. 9(1) (2018), 3-17.
• 8 E. Karapınar, F. Khojasteh, W. Shatanawia, Revisiting Ciric-Type Contraction with Caristi’s Approach, Symmetry 11 (2019), 726.
• 9 F. Khojasteh, S. Shukla, S. Radenovi´c, A new approach to the study of fixed point theory for simulation functions, Filomat 29(6) (2015), 1189-1194.
• 10 A. Meir, E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28 (1969), 326-329.
• 11 N. Y. Özgür, N. Ta¸s, Some fixed-circle theorems and discontinuity at fixed circle, AIP Conference Proceedings 1926 (2018), 020048.
• 12 N. Y. Özgür, N. Ta¸s, Some fixed-circle theorems on metric spaces, Bull. Malays. Math. Sci. Soc. 42(4) (2019), 1433-1449.
• 13 N. Y. Özgür, Fixed-disc results via simulation functions, Turk. J. Math. 43(6) (2019), 2794-2805.
• 14 R. P. Pant, Discontinuity and fixed points, J. Math. Anal. Appl. 240(1) (1999), 284-289.
• 15 A. Pant, R. P. Pant, Fixed points and continuity of contractive maps, Filomat 31(11) (2017), 3501-3506.
• 16 R. P. Pant, N. Y. Özgür, N. Ta¸s, Discontinuity at fixed points with applications, Bull. Belg. Math. Soc. - Simon Stevin 26 (2019), 571-589.
• 17 R. P. Pant, N. Y. Özgür, N. Ta¸s, On discontinuity problem at fixed point, Bull. Malays. Math. Sci. Soc. 43(1) (2020), 499-517.
• 18 M. Rashid, I. Batool, N. Mehmood, Discontinuous mappings at their fixed points and common fixed points with applications, J. Math. Anal. 9(1) (2018), 90-104.
• 19 B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Am. Math. Soc. 226 (1977), 257-290.
• 20 B. E. Rhoades, Contractive definitions and continuity, Contemp. Math. 72 (1988), 233-245.
• 21 N. Ta¸s, N. Y. Özgür, A new contribution to discontinuity at fixed point, Fixed Point Theory 20(2) (2019), 715-728.
• 22 N. Ta¸s, N. Y. Özgür, N. Mlaiki, New types of FC-contractions and the fixed-circle problem, Mathematics 6(10) (2018), 188.
• 23 N. Ta¸s, Bilateral-type solutions to the fixed-circle problem with rectified linear units application, Turk. J. Math. 44 (2020), 1330-1344.
• 24 M. J. Todd, The computation of fixed points and applications, Berlin, Heidelberg, New York: Springer-Verlag, 1976.
• 25 D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl. 2012 (2012), 94.