A Quicker Iteration Method for Approximating the Fixed Point of Generalized α-Reich-Suzuki Nonexpansive Mappings with Applications
Abstract
:1. Introduction
2. A Short Survey on the Applications of Fixed Point Theory
- Economics: Kenneth Arrow utilized fixed point theorems to argue that there is no perfect voting system, which is a foundational conclusion in social choice theory [41].
- Computer Science: the halting problem: Alan Turing’s fixed point theorem proved the undecidability of the halting problem, which is a key notion in computer science [42].
- Topology: the Brouwer fixed point theorem is a key conclusion in topology that has implications in domains such as game theory and economics [43].
- Physics: fixed point theory is utilized in the study of quantum mechanics, specifically in understanding the behavior of quantum systems in regard to fixed points [44].
- Image Processing: image registration: image registration, a technique that aligns two or more pictures for diverse applications such as medical imaging, uses fixed point algorithms [45].
- Game Theory: Nash equilibrium: fixed point theorems were used by John Nash to demonstrate the existence of Nash equilibria in non-co-operative games [46].
- Finance: fixed point approaches are used in the formulation of the Black–Scholes equation, which is a key model in financial mathematics [47].
- Social Sciences: sociology and network theory: fixed point theory is used to investigate network architecture, social dynamics, and social system stability [47].
- Optimization: fixed point iterations are often employed in optimization methods such as the Gauss–Seidel method for solving linear equations [48].
3. Preliminaries
4. Comparison Analysis and Stability
- Conversely, let . Therefore,
5. Convergence Analysis
- is bounded. Let . Now, by combining the above inequality and Theorem 6, =q. Additionally,
6. Application to Fractional 1-Fredholm Integro Differential Equation
- If we take , then (15) can be written as
7. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Iteration | Maan | Ishikawa | F | D | |
---|---|---|---|---|---|
1 | 3.81359 | 3.91521 | 4.96625 | 4.96876 | 4.99062 |
2 | 4.22201 | 4.28724 | 4.97828 | 4.97994 | 4.99398 |
3 | 4.49145 | 4.53352 | 4.98597 | 4.98707 | 4.99612 |
4 | 4.66824 | 4.69545 | 4.99092 | 4.99163 | 4.99749 |
5 | 4.78385 | 4.80148 | 4.99411 | 4.99458 | 4.99837 |
6 | 4.85929 | 4.87073 | 4.99618 | 4.99648 | 4.99895 |
7 | 4.90845 | 4.91587 | 4.99752 | 4.99772 | 4.99932 |
8 | 4.94045 | 4.94528 | 4.99839 | 4.99852 | 4.99956 |
9 | 4.96128 | 4.96441 | 4.99895 | 4.99904 | 4.99971 |
10 | 4.97482 | 4.97686 | 4.99932 | 4.99937 | 4.99981 |
11 | 4.98363 | 4.98496 | 4.99956 | 4.99959 | 4.99988 |
12 | 4.98936 | 4.99022 | 4.99971 | 4.99974 | 4.99992 |
13 | 4.99308 | 4.99364 | 4.99981 | 4.99983 | 4.99995 |
14 | 4.99550 | 4.99587 | 4.99988 | 4.99989 | 4.99997 |
15 | 4.99708 | 4.99731 | 4.99992 | 4.99993 | 4.99998 |
16 | 4.99810 | 4.99825 | 4.99995 | 4.99995 | 4.99999 |
17 | 4.99877 | 4.99886 | 4.99997 | 4.99997 | 4.99999 |
18 | 4.99920 | 4.99926 | 4.99998 | 4.99998 | 4.99999 |
19 | 4.99948 | 4.99952 | 4.99999 | 4.99999 | 5.00000 |
20 | 4.99966 | 4.99969 | 4.99999 | 4.99999 | 5.00000 |
21 | 4.99978 | 4.99980 | 4.99999 | 4.99999 | 5.00000 |
22 | 4.99986 | 4.99987 | 5.00000 | 5.00000 | 5.00000 |
23 | 4.99991 | 4.99991 | 5.00000 | 5.00000 | 5.00000 |
24 | 4.99994 | 4.99994 | 5.00000 | 5.00000 | 5.00000 |
25 | 4.99996 | 4.99996 | 5.00000 | 5.00000 | 5.00000 |
26 | 4.99997 | 4.99998 | 5.00000 | 5.00000 | 5.00000 |
27 | 4.99998 | 4.99998 | 5.00000 | 5.00000 | 5.00000 |
28 | 4.99999 | 4.99999 | 5.00000 | 5.00000 | 5.00000 |
29 | 4.99999 | 4.99999 | 5.00000 | 5.00000 | 5.00000 |
30 | 5.00000 | 5.00000 | 5.00000 | 5.00000 | 5.00000 |
Iteration | Noor | S | Thukar | M | |
---|---|---|---|---|---|
1 | 3.93161 | 4.85694 | 4.88787 | 4.52880 | 4.99062 |
2 | 4.29749 | 4.90798 | 4.92774 | 4.69565 | 4.99398 |
3 | 4.54002 | 4.94059 | 4.95329 | 4.80296 | 4.99612 |
4 | 4.69962 | 4.96155 | 4.96975 | 4.87225 | 4.99749 |
5 | 4.80416 | 4.97508 | 4.98038 | 4.91709 | 4.99837 |
6 | 4.87246 | 4.98383 | 4.98727 | 4.94617 | 4.99895 |
7 | 4.91699 | 4.9895 | 4.99173 | 4.96503 | 4.99932 |
8 | 4.94600 | 4.99318 | 4.99463 | 4.97728 | 4.99956 |
9 | 4.96488 | 4.99557 | 4.99651 | 4.98524 | 4.99971 |
10 | 4.97717 | 4.99712 | 4.99773 | 4.99041 | 4.99981 |
11 | 4.98515 | 4.99813 | 4.99853 | 4.99376 | 4.99988 |
12 | 4.99035 | 4.99878 | 4.99904 | 4.99595 | 4.99992 |
13 | 4.99373 | 4.99921 | 4.99938 | 4.99737 | 4.99995 |
14 | 4.99592 | 4.99949 | 4.99960 | 4.99829 | 4.99997 |
15 | 4.99735 | 4.99967 | 4.99974 | 4.99889 | 4.99998 |
16 | 4.99828 | 4.99978 | 4.99983 | 4.99928 | 4.99999 |
17 | 4.99888 | 4.99986 | 4.99989 | 4.99953 | 4.99999 |
18 | 4.99927 | 4.99991 | 4.99993 | 4.99969 | 4.99999 |
19 | 4.99953 | 4.99994 | 4.99995 | 4.99980 | 5.00000 |
20 | 4.99969 | 4.99996 | 4.99997 | 4.99987 | 5.00000 |
21 | 4.99980 | 4.99997 | 4.99998 | 4.99992 | 5.00000 |
22 | 4.99987 | 4.99998 | 4.99999 | 4.99995 | 5.00000 |
23 | 4.99992 | 4.99999 | 4.99999 | 4.99996 | 5.00000 |
24 | 4.99995 | 4.99999 | 4.99999 | 4.99998 | 5.00000 |
25 | 4.99996 | 5.00000 | 5.00000 | 4.99999 | 5.00000 |
26 | 4.99998 | 5.00000 | 5.00000 | 4.99999 | 5.00000 |
27 | 4.99998 | 5.00000 | 5.00000 | 4.99999 | 5.00000 |
28 | 4.99999 | 5.00000 | 5.00000 | 5.00000 | 5.00000 |
29 | 4.99999 | 5.00000 | 5.00000 | 5.00000 | 5.00000 |
30 | 5.00000 | 5.00000 | 5.00000 | 5.00000 | 5.00000 |
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Ali, D.; Ali, S.; Pompei-Cosmin, D.; Antoniu, T.; Zaagan, A.A.; Mahnashi, A.M. A Quicker Iteration Method for Approximating the Fixed Point of Generalized α-Reich-Suzuki Nonexpansive Mappings with Applications. Fractal Fract. 2023, 7, 790. https://doi.org/10.3390/fractalfract7110790
Ali D, Ali S, Pompei-Cosmin D, Antoniu T, Zaagan AA, Mahnashi AM. A Quicker Iteration Method for Approximating the Fixed Point of Generalized α-Reich-Suzuki Nonexpansive Mappings with Applications. Fractal and Fractional. 2023; 7(11):790. https://doi.org/10.3390/fractalfract7110790
Chicago/Turabian StyleAli, Danish, Shahbaz Ali, Darab Pompei-Cosmin, Turcu Antoniu, Abdullah A. Zaagan, and Ali M. Mahnashi. 2023. "A Quicker Iteration Method for Approximating the Fixed Point of Generalized α-Reich-Suzuki Nonexpansive Mappings with Applications" Fractal and Fractional 7, no. 11: 790. https://doi.org/10.3390/fractalfract7110790
APA StyleAli, D., Ali, S., Pompei-Cosmin, D., Antoniu, T., Zaagan, A. A., & Mahnashi, A. M. (2023). A Quicker Iteration Method for Approximating the Fixed Point of Generalized α-Reich-Suzuki Nonexpansive Mappings with Applications. Fractal and Fractional, 7(11), 790. https://doi.org/10.3390/fractalfract7110790