Solving an Integral Equation via Fuzzy Triple Controlled Bipolar Metric Spaces
Abstract
:1. Introduction
2. Preliminaries
- (i)
- ;
- (ii)
- iff ;
- (iii)
- ;
- (iv)
- ;
- (v)
- is continuous.
- (FB1)
- for all ;
- (FB2)
- iff for and ;
- (FB3)
- for all ;
- (FB4)
- for alland ;
- (FB5)
- is left continuous;
- (FB6)
- is non-decreasing for all and .
- (FCB1)
- for all ;
- (FCB2)
- iff for and ;
- (FCB3)
- for all ;
- (FCB4)
- for all and ;
- (FCB5)
- is left continuous;
- (FCB6)
- is non-decreasing for all and .
- (i)
- Sequence is named as a bisequence on .
- (ii)
- If both and converge, the sequence is said to be a convergent sequence. If both and converge to some center point, bisequence is said to be a biconvergent sequence.
- (iii)
- A bisequence on fuzzy triple controlled bipolar metric space is said to be a Cauchy bisequence as , for all .
3. Main Results
- (i)
- and ;
- (ii)
- for all and , where .
- (i)
- and ;
- (ii)
- , for all and , where .
- (i)
- and ;
- (ii)
- For and , where is an increasing function such that and for all .
- (i)
- and ;
- (ii)
- For and .
4. Application
- (T1)
- and ;
- (T2)
- There is a continuous function and such that
- (T3)
- .
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Mani, G.; Gnanaprakasam, A.J.; Mitrović, Z.D.; Bota, M.-F. Solving an Integral Equation via Fuzzy Triple Controlled Bipolar Metric Spaces. Mathematics 2021, 9, 3181. https://doi.org/10.3390/math9243181
Mani G, Gnanaprakasam AJ, Mitrović ZD, Bota M-F. Solving an Integral Equation via Fuzzy Triple Controlled Bipolar Metric Spaces. Mathematics. 2021; 9(24):3181. https://doi.org/10.3390/math9243181
Chicago/Turabian StyleMani, Gunaseelan, Arul Joseph Gnanaprakasam, Zoran D. Mitrović, and Monica-Felicia Bota. 2021. "Solving an Integral Equation via Fuzzy Triple Controlled Bipolar Metric Spaces" Mathematics 9, no. 24: 3181. https://doi.org/10.3390/math9243181
APA StyleMani, G., Gnanaprakasam, A. J., Mitrović, Z. D., & Bota, M.-F. (2021). Solving an Integral Equation via Fuzzy Triple Controlled Bipolar Metric Spaces. Mathematics, 9(24), 3181. https://doi.org/10.3390/math9243181