Publications de l'Institut Mathematique 2023 Volume 113, Issue 127, Pages: 99-108
https://doi.org/10.2298/PIM2327099D
Full text (
146 KB)
On Ćirić type theorems in b-metric spaces
Dung Nguyen Van (Faculty of Mathematics - Informatics Teacher Education, Dong Thap University, Cao Lanh City, Dong Thap Province, Vietnam), nvdung@dthu.edu.vn
We show that the range of contraction constant in a result of Karapınar et
al. can not be extended to [0, 1). However, by using some additional
conditions, we prove that the range can be extended to [0,1). The first
result gives a negative answer to the open question on a Ćirić type
theorem in b-metric spaces, and the next results are improvements of Ćirić
type theorems in b-metric spaces in the literature.
Keywords: b-metric, fixed point
Show references
M.A. Alghamdi, N. Hussain, P. Salimi, Fixed point and coupled fixed point theorems on b-metric-like spaces, J. Inequal. Appl. 402 (2013), 1-25.
I.A. Bakhtin, The contraction principle in quasimetric spaces, Func. An., Unianowsk, Gos. Ped. Ins. 30 (1989), 26-37, in Russian.
R.K. Bisht, N.K. Singh, V. Rakočević, B. Fisher, On discontinuity at fixed point via power quasi contraction, Publ. Inst. Math., Nouv. Sér. 108(122) (2020), 5-11.
V.W. Bryant, A remark on a fixed-point theorem for iterated mappings, Am. Math. Mon. 75(4) (1968), 399-400.
Lj. Ćirić, On Sehgal’s maps with a contractive iterate at a point, Publ. Inst. Math., Nouv. Sér. 33(47) (1983), 59-62.
Lj. B. Ćirić, A generalization of Banach’s contraction principle, Proc. Am. Math. Soc. 45 (1974), 267-273.
S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Univ. Ostrav. 1 (1993), no. 1, 5-11.
S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Math. Fis. Univ. Modena 46 (1998), 263-276.
N.V. Dung, V.T. L. Hang, On relaxations of contraction constants and Caristi’s theorem in b-metric spaces, J. Fixed Point Theory Appl. 18(2) (2016), 267-284.
N.V. Dung, W. Sintunavarat, Fixed point theory in b-metric spaces, in: D. Gopal, P. Agarwal, P. Kumam (eds.), Metric Structures and Fixed Point Theory, Chapman and Hall/CRC, 2021, pp. 33-66.
L. F. Guseman, Fixed point theorems for mappings with a contractive iterate at a point, Proc. Am. Math. Soc. 26 (1970), 615-618.
E. Karapınar, Z.D. Mitrović, A. Öztürk, S. Radenović, On a theorem of Ćirić in b-metric spaces, Rend. Circ. Mat. Palermo (2) 70 (2021), 217-225.
N. Lu, F. He, W.-S. Du, Fundamental questions and new counterexamples for b-metric spaces and Fatou property, Mathematics 7(11) (2019), 1-15.
N. Lu, F. He, W.-S. Du, On the best areas for Kannan system and Chatterjea system in b-metric spaces, Optimization 70(5-6) (2021), 973-986.
V. M. Sehgal, A fixed point theorem for mappings with a contractive iterate, Proc. Am. Math. Soc. 23(3) (1969), 631-634.