Fixed Point Results for a New Rational Contraction in Double Controlled Metric-like Spaces
Abstract
:1. Introduction
2. Preliminaries
- (d1)
- if and only if;
- (d2)
- , for all;
- (d3)
- , for all.
- (dθ1)
- if and only if;
- (dθ2)
- , for all;
- (dθ3)
- , for all.
- (dχ1)
- if and only if;
- (dχ2)
- ;
- (dχ3)
- .
- (dχ1)
- if and only if;
- (dχ2)
- ;
- (dχ3)
- .
- (dχ1)
- ;
- (dχ2)
- ;
- (dχ3)
- .
- (1)
- ifexists and is finite, then we say thata Cauchy sequence;
- (2)
- if, then we say thatconverges to u;
- (3)
- if every Cauchy sequence in X is convergent to some point in X, then we say thatis complete.
3. Main Results
- Case 1.
- for .
- Case 2.
- for .
- (1).
- Our results is an improvement of the results of Lateef [19]. On the one hand, if , notice that our contraction becomes a Fisher contraction in [19], the conclusion still holds, i.e., f has a fixed point; on the other hand, we get the result in double controlled metric-like spaces, instead of controlled metric spaces. In other words, we extend the result to double controlled metric-like spaces.
- (2).
- Special cases:
- (3).
- By Remark 1, we know that every double controlled metric space is a double controlled metric-like space, and self-distance in the latter does not need to be zero. Thus, our results still hold in double controlled metric spaces.
- ;
- .
- Case 1.
- , ;
- Case 2.
- , ;
- Case 3.
- , ,
- Case 1.
- , ;
- Case 2.
- ,;
- Case 3.
- ,.
4. Application
5. Conclusions
- (i)
- Consider replacing the rational expression in this article with another rational expression;
- (ii)
- (iii)
- The four constants on the right hand side of the rational contraction inequality may be changed to special functions.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Deng, J.; Liu, X.; Sun, Y.; Zhou, M. Fixed Point Results for a New Rational Contraction in Double Controlled Metric-like Spaces. Mathematics 2022, 10, 1439. https://doi.org/10.3390/math10091439
Deng J, Liu X, Sun Y, Zhou M. Fixed Point Results for a New Rational Contraction in Double Controlled Metric-like Spaces. Mathematics. 2022; 10(9):1439. https://doi.org/10.3390/math10091439
Chicago/Turabian StyleDeng, Jia, Xiaolan Liu, Yan Sun, and Mi Zhou. 2022. "Fixed Point Results for a New Rational Contraction in Double Controlled Metric-like Spaces" Mathematics 10, no. 9: 1439. https://doi.org/10.3390/math10091439