A Novel Fixed-Point Iterative Process for Multivalued Mappings Applied in Solving a HIV Model of Fractional Order
Abstract
:1. Introduction
2. Preliminaries
- (i)
- , and
- (ii)
- , as
- (1)
- (2)
- (3)
- (4)
3. Convergence Analysis
- (i)
- (ii)
- and
- (iii)
4. Rate of Convergence
- (i)
- (ii)
- (iii)
5. Data-Dependence Results
- (i)
- (ii)
- for all and
- (iii)
- .
6. Stability Results
7. Application to Disease Model
- Let be a constant, then
- Assume that .
8. Numerical Examples
9. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
- Rhoades, B.E. Comments on two fixed point iteration methods. J. Math. Anal. Appl. 1976, 56, 741–750. [Google Scholar] [CrossRef]
- Agarwal, R.P.; O’Regan, D.; Sahu, D.R. Iteration construction of fixed points of nearly asymptotically nonexpansive mappings. J. Nonlinear Convex Anal 2007, 8, 61–79. [Google Scholar]
- Chugh, R.; Kumar, V.; Kumar, S. Strong convergence of a new step iterative scheme in Banach spaces. Am. J. Comput. Math. 2012, 2, 345–357. [Google Scholar] [CrossRef]
- Karahan, I.; Ozdemir, M. A general iterative method for approximation of fixed points and their applications. Adv. Fixed Point Theory 2013, 3, 510–526. [Google Scholar]
- Mann, W.R. Mean Value methods in iteration. Proc. Am. Math. Soc. 1953, 4, 506–510. [Google Scholar] [CrossRef]
- Ishikawa, S. Fixed points by a new iteration method. Proc. Am. Math. Soc. 1974, 44, 147–150. [Google Scholar] [CrossRef]
- Noor, M.A. New approximation scheme for general variation inequalities. J. Math. Anal. Appl. 2000, 251, 217–229. [Google Scholar] [CrossRef]
- Okeke, G.A. Convergence analysis of the Picard-Ishikawa hybrid iteration process with application. Afr. Math. 2019, 30, 817–835. [Google Scholar] [CrossRef]
- Khan, S.H. A Picard-Mann hybrid iterative process. Fixed Point Theory Appl. 2013, 2013, 69. [Google Scholar] [CrossRef]
- Sarivastava, V.; Rai, K.N. A multi-term fractional diffusion equation for oxygen delivery a capillary to tissues. Math. Comput. Model. 2010, 51, 616–624. [Google Scholar] [CrossRef]
- Tan, K.K.; Xu, H.K. Approximating fixed points of nonexpansive mappings by Ishikawa iteration process. J. Math. Anal. Appl. 1993, 178, 301–308. [Google Scholar] [CrossRef]
- Picard, E. Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives. J. Math. Pures Appl. 1890, 6, 145–210. [Google Scholar]
- Kransel, M.A. Two observations about the method of successive approximations. Uspekhi Mat. Nauk. 1957, 10, 131–140. [Google Scholar]
- Ullah, K.; Arshad, M. Numerical reckoning fixed points for Suzuki’s generalized nonexpansive mappings via new iteration process. Filomat 2018, 32, 187–196. [Google Scholar] [CrossRef]
- Berinde, V. Iterative Approximation of Fixed Points; Efemeride: Baia Mare, Romania, 2002. [Google Scholar]
- Alshehry, A.S.; Mukhtar, S.; Khan, H.S.; Shah, R. Fixed-point theory and numerical analysis of an epidemic model with fractional calculus: Exploring dynamical behavior. Open Phys. 2023, 21, 20230121. [Google Scholar] [CrossRef]
- Baleanu, D.; Arshad, S.; Jajarmi, A.; Shokat, W.; Ghassabzade, F.A.; Wali, M. Dynamical behaviors and stability analyzis of a generalied fractional model with a real case study. J. Adv. Res. 2023, 48, 157–173. [Google Scholar] [CrossRef] [PubMed]
- Agarwal, R.; Hristova, S.; O’Regan, D. Basic concepts on Riemann-Liouville fractional differential equations with non-instantaneous impulses. Symmetry 2019, 11, 614. [Google Scholar] [CrossRef]
- Kajouni, A.; Chafiki, A.; Hilal, K.; Oukessou, M. A new conformable fractional derivative and applications. Int. J. Differ. Equ. 2021, 2021, 6245435. [Google Scholar] [CrossRef]
- Vivas-Cortez, M.; Mohammed, P.O.; Guirao, J.L.G.; Yousif, M.A.; Ibrahim, I.S.; Chorfi, N. Improved fractional differences with kernels of delta Mittag-Leffler and exponential functions. Symmetry 2024, 16, 1562. [Google Scholar] [CrossRef]
- Okeke, G.A.; Abbas, M.; De la Sen, M. Approximation of the fixed point of multivalued quasi-nonexpansive mappings via a faster iterative process with applications. Discret. Dyn. Nat. Soc. 2020, 2020, 8634050. [Google Scholar] [CrossRef]
- Grosan, T.; Soltuz, S.M. Data dependence for Ishikawa iteration when dealing with contractive-like operators. Fixed Point Theory Appl. 2008, 2008, 242916. [Google Scholar]
- Soltuz, S.M.; Otrocol, D. Classical results via Mann-Ishikawa iteration. Rev. D’analyse Numer. L’approximation 2007, 36, 193–197. [Google Scholar]
- Shukka, D.P.; Tiwari, V.; Singh, R. Noor iterative processes for multivalued mappings in Banach Spaces. Int. J. Math. Anal. 2014, 8, 649–657. [Google Scholar] [CrossRef]
- Granger, T.; Michelitsch, T.M.; Bestehorn, M.; Riascos, A.P.; Collet, B.A. Stochastic compartment model with mortality and its application to epidemic spreading in complex networks. Entropy 2024, 26, 362. [Google Scholar] [CrossRef]
Step | Picard | Mann | Ishikawa | Krasnosel’kii | Ullah and Arshad |
---|---|---|---|---|---|
0 | 0.100000 | 0.1000000 | 0.1000000 | 0.1000000 | 0.1000000 |
1 | 3.02000000 | 1.56000000 | 1.706000000 | 1.56000000 | 3.6624000000 |
2 | 3.604000000 | 2.436000000 | 2.60536000000 | 2.436000000 | 3.7478976000000 |
3 | 3.7208000000 | 2.9616000000 | 3.1090016000000 | 2.9616000000 | 3.7499495424000000 |
4 | 3.74416000000 | 3.2696000000 | 3.391040896000000 | 3.27696000000 | 3.7499987890176000 |
5 | 3.74883200000 | 3.46617600000 | 3.3.5489829017599996 | 3.4661760000 | 3.7499999709364227 |
6 | 3.74976640000 | 3.5797056000 | 3.637430424985599800 | 3.5797056000 | 3.7499999993024744 |
7 | 3.74995328000 | 3.64782336000 | 3.68696103799193600 | 3.64782336000 | 3.74999999983200 |
8 | 3.49990656000 | 3.688694016000 | 3.7146981812755000 | 3.68869401600 | 3.749999999999598 |
9 | 3.7499981312 | 3.713216409600 | 3.7302309815142713 | 3.71316409600 | 3.74999999999902 |
10 | 3.74999962624 | 3.72792984576 | 3.738929349647992 | 3.72792984576 | 3.7500000000000 |
⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |
Step | Noor | Picard–Ishikawa | IA-Iteration |
---|---|---|---|
0 | 0.1000000 | 0.1000000 | 0.1000000 |
1 | 1.720600000 | 3.314120000 | 3.7268400000 |
2 | 2.621653600000 | 3.7042144000000 | 3.75730545600000 |
3 | 3.1226394016000 | 3.74487201280000 | 3.7575613658304 |
4 | 3.4011875072896 | 3.7494256654336000 | 3.7575635154730000 |
5 | 3.5560602540530173 | 3.7499356745285635 | 3.7575635335299733 |
6 | 3.642169501253478 | 3.749992795547199 | 3.757563533681652 |
7 | 3.69004624269693 | 3.7499991931012864 | 3.757563533682926 |
8 | 3.7166657109394947 | 3.749999909627344 | 3.7575635336829367 |
9 | 3.731466135282359 | 3.7499999898782628 | 3.7575635336829367 |
10 | 3.7396951712169915 | 3.7499999988663655 | 3.7575635336829367 |
⋮ | ⋮ | ⋮ |
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Alqahtani, R.T.; Okeke, G.A.; Ugwuogor, C.I. A Novel Fixed-Point Iterative Process for Multivalued Mappings Applied in Solving a HIV Model of Fractional Order. Mathematics 2025, 13, 739. https://doi.org/10.3390/math13050739
Alqahtani RT, Okeke GA, Ugwuogor CI. A Novel Fixed-Point Iterative Process for Multivalued Mappings Applied in Solving a HIV Model of Fractional Order. Mathematics. 2025; 13(5):739. https://doi.org/10.3390/math13050739
Chicago/Turabian StyleAlqahtani, Rubayyi T., Godwin Amechi Okeke, and Cyril Ifeanyichukwu Ugwuogor. 2025. "A Novel Fixed-Point Iterative Process for Multivalued Mappings Applied in Solving a HIV Model of Fractional Order" Mathematics 13, no. 5: 739. https://doi.org/10.3390/math13050739
APA StyleAlqahtani, R. T., Okeke, G. A., & Ugwuogor, C. I. (2025). A Novel Fixed-Point Iterative Process for Multivalued Mappings Applied in Solving a HIV Model of Fractional Order. Mathematics, 13(5), 739. https://doi.org/10.3390/math13050739