A Proposal for Revisiting Banach and Caristi Type Theorems in b-Metric Spaces
Abstract
:1. Introduction and Preliminaries
- (b1)
- ;
- (b2)
- if and only if ;
- (b3)
- (Expanded triangle inequality).
2. Main Result
- (i)
- φ is bounded from below ,
- (ii)
- implies .
- 1.
- From Example 1, it follows that Theorem 1 (over metric spaces) is not a consequence of the Banach contraction principle.
- 2.
- Question for further study: It is natural to ask if the Banach contraction principle is a consequence of Theorem 1 (over metric spaces).
Author Contributions
Funding
Conflicts of Interest
References
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Karapınar, E.; Khojasteh, F.; Mitrović, Z.D. A Proposal for Revisiting Banach and Caristi Type Theorems in b-Metric Spaces. Mathematics 2019, 7, 308. https://doi.org/10.3390/math7040308
Karapınar E, Khojasteh F, Mitrović ZD. A Proposal for Revisiting Banach and Caristi Type Theorems in b-Metric Spaces. Mathematics. 2019; 7(4):308. https://doi.org/10.3390/math7040308
Chicago/Turabian StyleKarapınar, Erdal, Farshid Khojasteh, and Zoran D. Mitrović. 2019. "A Proposal for Revisiting Banach and Caristi Type Theorems in b-Metric Spaces" Mathematics 7, no. 4: 308. https://doi.org/10.3390/math7040308
APA StyleKarapınar, E., Khojasteh, F., & Mitrović, Z. D. (2019). A Proposal for Revisiting Banach and Caristi Type Theorems in b-Metric Spaces. Mathematics, 7(4), 308. https://doi.org/10.3390/math7040308