Double-Composed Metric Spaces
Abstract
:1. Introduction
2. Preliminaries
- if and only if for all ;
- for all ;
- for all .
- if and only if for all ;
- for all ;
- for all .
- if and only if for all
- for all
- for all .
- if and only if for all
- for all
- for all .
- 1.
- That a Cauchy sequence if exists and is finite;
- 2.
- That converges to x if ;
- 3.
- That is complete if every Cauchy sequence in is convergent to some point in .
3. Main Results
- 1.
- α and β are continuous and non-decreasing functions with , and β is sub-additive.
- 2.
- where and denote the composite functions.
- 1.
- where , and and denote the composite functions.
- 2.
- α and β are continuous and non-decreasing functions with , and β is sub-additive with for every and
4. Applications
5. Conclusions
- i.
- Establish the new fixed-point results in a double-composed metric space for the Chatterjee contraction, the Reich contraction, the Hardy–Rogers contraction, and other rational-type contraction mappings.
- ii.
- Establish some non-trivial applications of Theorems 3 and 4. In particular, provide a non-trivial application of Theorem 3 to the theory of non-linear integral equations.
- iii.
- Finally, we propose an important direction for further research in the framework of a double-composed metric space. When there is no unique fixed point, one method for generalizing fixed-point results is to analyze the geometric properties of the set of fixed points. The fixed-circle problem (see [18]) and the fixed-figure problem (see [19,20]) have been introduced accordingly.
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Ayoob, I.; Chuan, N.Z.; Mlaiki, N. Double-Composed Metric Spaces. Mathematics 2023, 11, 1866. https://doi.org/10.3390/math11081866
Ayoob I, Chuan NZ, Mlaiki N. Double-Composed Metric Spaces. Mathematics. 2023; 11(8):1866. https://doi.org/10.3390/math11081866
Chicago/Turabian StyleAyoob, Irshad, Ng Zhen Chuan, and Nabil Mlaiki. 2023. "Double-Composed Metric Spaces" Mathematics 11, no. 8: 1866. https://doi.org/10.3390/math11081866
APA StyleAyoob, I., Chuan, N. Z., & Mlaiki, N. (2023). Double-Composed Metric Spaces. Mathematics, 11(8), 1866. https://doi.org/10.3390/math11081866