Some New Results for (α, β)-Admissible Mappings in 𝔽-Metric Spaces with Applications to Integral Equations
Abstract
:1. Introduction and Preliminaries
- if and only if ;
- ;
- For every , and for every with , we have
- (1)
- A sequence is called -convergent to if
- (2)
- A sequence is called -Cauchy if
- (3)
- The space is said to be -complete if every -Cauchy sequence in Γ is -convergent to some element in
- (i)
- for some implies
- (ii)
- for some implies
2. Main Results
- (i)
- There exists such that and
- (ii)
- Υ is continuous or;
- (iii)
- If is a sequence in Γ such that and for all , then .
- (i)
- There exists such that and
- (ii)
- is continuous or;
- (iii)
- If is a sequence in such that and for all , then .
- (i)
- There exists such that and
- (ii)
- is continuous or;
- (iii)
- If is a sequence in such that and for all , then .
- (i)
- There exists such that and
- (ii)
- is continuous or;
- (iii)
- If is a sequence in such that and for all , then .
- (i)
- There exists such that and ;
- (ii)
- Υ and S are two (-admissible mappings;
- (iii)
- If is a sequence in Γ such that as and for all , then .
- (i)
- There exists such that and ;
- (ii)
- Υ is an -admissible mapping;
- (iii)
- If is a sequence in Γ such that as and for all , then .
3. Application to Integral Equation
- (1)
- For all we have ;
- (2)
- There exist such that if and for some then for every we obtain
- (3)
- There exists such that and ;
- (4)
- for some implies and for some implies ;
- (5)
- If is a sequence in Γ such that as and for all then
4. Conclusions
Author Contributions
Funding
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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i | Exact Solution | Approximation Solution | Approximation Solution |
---|---|---|---|
(m = 64) | (m = 128) | ||
0.0 | 0 | 0.010417 | 0.005208 |
0.1 | 0.199667 | 0.197570 | 0.192399 |
0.2 | 0.397339 | 0.382942 | 0.398412 |
0.3 | 0.591040 | 0.605205 | 0.589930 |
0.4 | 0.778837 | 0.781174 | 0.785758 |
0.5 | 0.958851 | 0.967335 | 0.963098 |
0.6 | 1.129285 | 1.126666 | 1.122812 |
0.7 | 1.288435 | 1.276056 | 1.289847 |
0.8 | 1.434712 | 1.446451 | 1.433200 |
0.9 | 1.566654 | 1.569934 | 1.572171 |
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Faraji, H.; Mirkov, N.; Mitrović, Z.D.; Ramaswamy, R.; Abdelnaby, O.A.A.; Radenović, S. Some New Results for (α, β)-Admissible Mappings in 𝔽-Metric Spaces with Applications to Integral Equations. Symmetry 2022, 14, 2429. https://doi.org/10.3390/sym14112429
Faraji H, Mirkov N, Mitrović ZD, Ramaswamy R, Abdelnaby OAA, Radenović S. Some New Results for (α, β)-Admissible Mappings in 𝔽-Metric Spaces with Applications to Integral Equations. Symmetry. 2022; 14(11):2429. https://doi.org/10.3390/sym14112429
Chicago/Turabian StyleFaraji, Hamid, Nikola Mirkov, Zoran D. Mitrović, Rajagopalan Ramaswamy, Ola A. Ashour Abdelnaby, and Stojan Radenović. 2022. "Some New Results for (α, β)-Admissible Mappings in 𝔽-Metric Spaces with Applications to Integral Equations" Symmetry 14, no. 11: 2429. https://doi.org/10.3390/sym14112429
APA StyleFaraji, H., Mirkov, N., Mitrović, Z. D., Ramaswamy, R., Abdelnaby, O. A. A., & Radenović, S. (2022). Some New Results for (α, β)-Admissible Mappings in 𝔽-Metric Spaces with Applications to Integral Equations. Symmetry, 14(11), 2429. https://doi.org/10.3390/sym14112429