Convergence Theorems for Generalized Viscosity Explicit Methods for Nonexpansive Mappings in Banach Spaces and Some Applications
Abstract
:1. Introduction
2. Preliminaries
- (1)
- E is uniformly convex.
- (2)
- There is a strictly-increasing, continuous, and convex function such that and:
- (i)
- ;
- (ii)
- or .
3. The Main Results
- (C1)
- and ;
- (C2)
- .
- (1)
- (2)
- (3)
- We give new control conditions and techniques to prove our results.
- (4)
- (5)
- Our results are applicable for the family of nonexpansive mappings, for example -mapping, a countable family of nonexpansive mappings, and nonexpansive semigroups.
Open Problem
4. Convergence Theorems for a Strict Pseudo-Contraction Mapping
5. Some Applications
5.1. Periodic Solution of a Nonlinear Evolution Equation
- (1)
- and ;
- (2)
- and ;
- (3)
- There exists a mild solution u of the Equation (31) on for each initial value ;
- (4)
- There exists some such that for all with and .
5.2. Nonlinear Fredholm Integral Equation
6. Numerical Examples
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Number of Iterates | ||
---|---|---|
1 | (1.0000000, −2.0000000, 3.0000000, 0, 0, 0, …) | 2.7144176 |
2 | (0.2327849, −0.4480184, 0.7292703, 0, 0, 0, …) | 0.6771823 |
3 | (0.0526107, −0.0931765, 0.1750471, 0, 0, 0, …) | 0.1675113 |
4 | (0.0118323, −0.0180499, 0.0418631, 0, 0, 0, …) | 0.0410437 |
5 | (0.0026730, −0.0031216, 0.0099981, 0, 0, 0, …) | 0.0099602 |
6 | (0.0006105, −0.0004172, 0.0023865, 0, 0, 0, …) | 0.0023955 |
7 | (0.0001418, −9.9203 × , 0.0005694, 0, 0, 0, …) | 5.7233 × |
8 | (3.3678 × , 2.2004 × , 0.0001358, 0, 0, 0, …) | 1.3688 × |
9 | (8.2042 × , 1.1855 × , 3.2365 × , 0, 0, 0, …) | 3.3056 × |
10 | (2.0542 × , 4.6141 × , 7.7051 × , 0, 0, 0, …) | 8.2639 × |
⋮ | ⋮ | ⋮ |
15 | (2.8321 × , 1.3627 × , 5.6702 × , 0, 0, 0, …) | 1.3986 × |
Number of Iterates | ||
---|---|---|
1 | (1.0000000, −2.0000000, 3.0000000, 0, 0, 0, …) | 2.7144176 |
2 | (−0.0470355, 0.3985926, 0.3853261, 0, 0, 0, …) | 0.4938371 |
3 | (0.0337876, 0.3481921, 0.0732708, 0, 0, 0, …) | 0.3493757 |
4 | (0.0344241, 0.1863456, 0.0570267, 0, 0, 0, …) | 0.1867381 |
5 | (0.0196135, 0.0844269, −0.0054563, 0, 0, 0, …) | 0.0847707 |
6 | (0.0091690, 0.0345417, −0.0045213, 0, 0, 0, …) | 0.0347302 |
7 | (0.0038295, 0.0130688, −0.0024315, 0, 0, 0, …) | 0.0131499 |
8 | (0.0014729, 0.0046081, −0.0011026, 0, 0, 0, …) | 0.0046371 |
9 | (5.2742 × , 0.0015100, −4.5084 × , 0, 0, 0, …) | 0.0015180 |
10 | (1.7573 × , 4.5282 × , −1.7034 × , 0, 0, 0, …) | 4.5360 × |
⋮ | ⋮ | ⋮ |
15 | (−1.7307 × , −1.6353 × , −3.0799 × , 0, 0, 0, …) | 8.1981 × |
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Sunthrayuth, P.; Pakkaranang, N.; Kumam, P.; Thounthong, P.; Cholamjiak, P. Convergence Theorems for Generalized Viscosity Explicit Methods for Nonexpansive Mappings in Banach Spaces and Some Applications. Mathematics 2019, 7, 161. https://doi.org/10.3390/math7020161
Sunthrayuth P, Pakkaranang N, Kumam P, Thounthong P, Cholamjiak P. Convergence Theorems for Generalized Viscosity Explicit Methods for Nonexpansive Mappings in Banach Spaces and Some Applications. Mathematics. 2019; 7(2):161. https://doi.org/10.3390/math7020161
Chicago/Turabian StyleSunthrayuth, Pongsakorn, Nuttapol Pakkaranang, Poom Kumam, Phatiphat Thounthong, and Prasit Cholamjiak. 2019. "Convergence Theorems for Generalized Viscosity Explicit Methods for Nonexpansive Mappings in Banach Spaces and Some Applications" Mathematics 7, no. 2: 161. https://doi.org/10.3390/math7020161
APA StyleSunthrayuth, P., Pakkaranang, N., Kumam, P., Thounthong, P., & Cholamjiak, P. (2019). Convergence Theorems for Generalized Viscosity Explicit Methods for Nonexpansive Mappings in Banach Spaces and Some Applications. Mathematics, 7(2), 161. https://doi.org/10.3390/math7020161