Some Iterative Approximation Results of F Iteration Process in Banach Spaces
Abstract
:1. Introduction
2. Preliminaries
- ()
- the asymptotic radius r of the sequence at the point s is the real number ;
- ()
- the asymptotic radius of the sequence in the connection with D is given by
- ()
- the asymptotic center A of the sequence in the connection with D is given by .
- (i)
- For every choice of and , follow that .
- (ii)
- The set is always closed. However if X is strictly convex and the domain D is convex, then the set also enjoys convexity.
- (iii)
- For every choice of in D, it follows that
- (iv)
- If we assume that X has Opial property, and is a weakly convergent to some element u with , then u is the point of of the set .
3. Approximation Results
4. Example
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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k | F | M | Thakur | Abbas | Agarwal | Noor | Ishikawa | Mann |
---|---|---|---|---|---|---|---|---|
1 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
2 | 3.0813 | 3.1625 | 3.1931 | 3.2456 | 3.3863 | 3.4851 | 3.5363 | 3.6500 |
3 | 3.0066 | 3.0264 | 3.0373 | 3.0603 | 3.1492 | 3.2353 | 3.2876 | 3.4225 |
4 | 3.0005 | 3.0043 | 3.0072 | 3.0148 | 3.0576 | 3.1141 | 3.1542 | 3.2746 |
5 | 3 | 3.0007 | 3.0014 | 3.0036 | 3.0223 | 3.0554 | 3.0827 | 3.1785 |
6 | 3 | 3.0001 | 3.0003 | 3.0009 | 3.0086 | 3.0269 | 3.0443 | 3.1160 |
7 | 3 | 3 | 3.0001 | 3.0002 | 3.0033 | 3.0130 | 3.0238 | 3.0754 |
8 | 3 | 3 | 3 | 3.0001 | 3.0013 | 3.0063 | 3.0123 | 3.0490 |
9 | 3 | 3 | 3 | 3 | 3.0005 | 3.0031 | 3.0068 | 3.0318 |
10 | 3 | 3 | 3 | 3 | 3.0002 | 3.0015 | 3.0037 | 3.0207 |
11 | 3 | 3 | 3 | 3 | 3.0001 | 3.0007 | 3.0020 | 3.0134 |
12 | 3 | 3 | 3 | 3 | 3 | 3.0003 | 3.0011 | 3.0088 |
13 | 3 | 3 | 3 | 3 | 3 | 3.0002 | 3.0006 | 3.0057 |
14 | 3 | 3 | 3 | 3 | 3 | 3.0001 | 3.0003 | 3.0037 |
15 | 3 | 3 | 3 | 3 | 3 | 3 | 3.0002 | 3.0024 |
16 | 3 | 3 | 3 | 3 | 3 | 3 | 3.0001 | 3.0016 |
17 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3.0010 |
18 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3.0007 |
19 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3.0004 |
20 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3.0003 |
21 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3.0002 |
22 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3.0001 |
23 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
Number of Iterates for Obtaining Fixed Point. | |||
---|---|---|---|
Starting Points | Agarwal | M | F |
28 | 14 | ||
30 | 14 | ||
30 | 15 | ||
31 | 15 | ||
31 | 15 | ||
32 | 15 |
Number of Iterates for Obtaining Fixed Point. | |||
---|---|---|---|
Starting Points | Agarwal | M | F |
24 | 13 | ||
26 | 14 | ||
26 | 15 | ||
27 | 15 | ||
27 | 15 | ||
27 | 15 |
Number of Iterates for Obtaining Fixed Point. | |||
---|---|---|---|
Starting Points | Agarwal | M | F |
28 | 12 | ||
29 | 13 | ||
30 | 13 | ||
31 | 13 | ||
31 | 14 | ||
31 | 14 |
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Ahmad, J.; Ullah, K.; Ahmad, I.; Arshad, M.; Jarasthitikulchai, N.; Sudsutad, W. Some Iterative Approximation Results of F Iteration Process in Banach Spaces. Axioms 2022, 11, 153. https://doi.org/10.3390/axioms11040153
Ahmad J, Ullah K, Ahmad I, Arshad M, Jarasthitikulchai N, Sudsutad W. Some Iterative Approximation Results of F Iteration Process in Banach Spaces. Axioms. 2022; 11(4):153. https://doi.org/10.3390/axioms11040153
Chicago/Turabian StyleAhmad, Junaid, Kifayat Ullah, Imtiaz Ahmad, Muhammad Arshad, Nantapat Jarasthitikulchai, and Weerawat Sudsutad. 2022. "Some Iterative Approximation Results of F Iteration Process in Banach Spaces" Axioms 11, no. 4: 153. https://doi.org/10.3390/axioms11040153