Inertial Accelerated Algorithm for Fixed Point of Asymptotically Nonexpansive Map** in Real Uniformly Convex Banach Spaces
Abstract
:1. Introduction
- (i)
- Nonexpansive if
- (ii)
- Asymptotically nonexpansive (see [1] ) if there exists a sequence , with such that
- (iii)
- Uniformly L-Lipschitzian if there exists a constant such that, for all ,
- (C1)
- .
- (C2)
- .
- (C3)
- is bounded.
- (D1) is bounded; and
- (D2) is bounded for any
2. Preliminaries
- (i)
- for weak convergence and → for strong convergence.
- (ii)
- to denote the set of w-weak cluster limits of .
- (1)
- Demiclosed at , if for any sequence in C which converges weakly to and , it holds that .
- (2)
- Semicompact, if for any bounded sequence in C such that there exists a subsequence such that .
- (i)
- The sequence converges.
- (ii)
- In particular, if , then .
3. Main Results
- (i)
- Choose sequences , and with which means .
- (ii)
- Let be arbitrary points, for the iterates and for each , choose such that where, for anyThis idea was obtained from the recent inertial extrapolation step introduced in [32].
4. Numerical Examples
- Case I:
- and
- Case II:
- and
- Case III:
- and
- Case IV:
- and
- Case I:
- Case II:
- Case III:
- Case IV:
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Alg. (25) | Alg. (26) | Alg. (27) | ||||
---|---|---|---|---|---|---|
Iter. | CPU (sec) | Iter. | CPU (sec) | Iter. | CPU (sec) | |
Case I | 32 | 0.0063 | 84 | 0.0750 | 69 | 0.0102 |
Case II | 33 | 0.0065 | 74 | 0.0724 | 69 | 0.0120 |
Case III | 38 | 0.0084 | 87 | 0.0847 | 78 | 0.0103 |
Case IV | 48 | 0.0095 | 123 | 0.0781 | 89 | 0.0131 |
Our Alg. | Pan and Wang Alg. | Vaish and Ahmad Alg. | ||||
---|---|---|---|---|---|---|
Iter. | CPU (sec) | Iter. | CPU (sec) | Iter. | CPU (sec) | |
Case I | 35 | 0.0103 | 88 | 0.0382 | 58 | 0.0156 |
Case II | 22 | 0.0067 | 43 | 0.0163 | 40 | 0.0104 |
Case III | 30 | 0.0090 | 70 | 0.0487 | 45 | 0.0175 |
Case IV | 30 | 0.0072 | 74 | 0.0368 | 45 | 0.0175 |
Our Alg. | Pan and Wang Alg. | Vaish and Ahmad Alg. | |
---|---|---|---|
Cameraman image | 2.7928 | 2.6422 | 2.6709 |
Pout image | 4.8237 | 4.4248 | 3.45630 |
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Harbau, M.H.; Ugwunnadi, G.C.; Jolaoso, L.O.; Abdulwahab, A. Inertial Accelerated Algorithm for Fixed Point of Asymptotically Nonexpansive Map** in Real Uniformly Convex Banach Spaces. Axioms 2021, 10, 147. https://doi.org/10.3390/axioms10030147
Harbau MH, Ugwunnadi GC, Jolaoso LO, Abdulwahab A. Inertial Accelerated Algorithm for Fixed Point of Asymptotically Nonexpansive Map** in Real Uniformly Convex Banach Spaces. Axioms. 2021; 10(3):147. https://doi.org/10.3390/axioms10030147
Chicago/Turabian StyleHarbau, Murtala Haruna, Godwin Chidi Ugwunnadi, Lateef Olakunle Jolaoso, and Ahmad Abdulwahab. 2021. "Inertial Accelerated Algorithm for Fixed Point of Asymptotically Nonexpansive Map** in Real Uniformly Convex Banach Spaces" Axioms 10, no. 3: 147. https://doi.org/10.3390/axioms10030147