A New Extension of CJ Metric Spaces—Partially Controlled J Metric Spaces
Abstract
:1. Introduction
2. Preliminaries
- (P1)
- if and only if
- (P2)
- (P3)
- (P4)
- (i)
- implies for any .
- (ii)
- There are some , where for each and
- (i)
- implies for all .
- (ii)
- There exists a function , and
3. Main Result
- (i)
- if .
- (ii)
- .
- (iii)
- There exists a continuous function , and , such that for all we have
- (1)
- Let be a metric space. A sequence converges to an element if and only if= = =
- (2)
- Let be a metric space. A sequence is called Cauchy if and only ifexists and it is finite.
- (3)
- A metric space is denoted as complete if each Cauchy sequence in Υ is convergent.
- (1)
- , which means that, which implies that .
- (2)
- Due to , then
- (3)
- Let
- (1)
- is a symmetric and complete partially controlled J metric space.
- (2)
- , because
4. Application of Theorem 1 to Polynomial Equations
5. Application to Fractional Differential Equation
6. Conclusions
- What will happen if is not necessarily held, which is a metric similar to this space?
- Could this metric-like space be a generalization to the metric and could we prove the existence and the uniqueness of the given contractions?
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Subhi Aiadi, S.; Mior Othman, W.A.; Wong, K.B.; Mlaiki, N. A New Extension of CJ Metric Spaces—Partially Controlled J Metric Spaces. Mathematics 2023, 11, 2973. https://doi.org/10.3390/math11132973
Subhi Aiadi S, Mior Othman WA, Wong KB, Mlaiki N. A New Extension of CJ Metric Spaces—Partially Controlled J Metric Spaces. Mathematics. 2023; 11(13):2973. https://doi.org/10.3390/math11132973
Chicago/Turabian StyleSubhi Aiadi, Suhad, Wan Ainun Mior Othman, Kok Bin Wong, and Nabil Mlaiki. 2023. "A New Extension of CJ Metric Spaces—Partially Controlled J Metric Spaces" Mathematics 11, no. 13: 2973. https://doi.org/10.3390/math11132973
APA StyleSubhi Aiadi, S., Mior Othman, W. A., Wong, K. B., & Mlaiki, N. (2023). A New Extension of CJ Metric Spaces—Partially Controlled J Metric Spaces. Mathematics, 11(13), 2973. https://doi.org/10.3390/math11132973