A New Fixed Point Approach for Approximating Solutions of Fractional Differential Equations in Banach Spaces

Nawaz, Bashir; Labsir, Abdallah; Ullah, Kifayat; Ahmad, Junaid

doi:10.28924/2291-8639-22-2024-124

Research Article

A New Fixed Point Approach for Approximating Solutions of Fractional Differential Equations in Banach Spaces

Bashir Nawaz
Abdallah Labsir*
Kifayat Ullah
Junaid Ahmad

* Corresponding author

Abstract

This contribution targets the solution of fractional differential equations (FDEs) via novel iterative approach and in a new class of nonlinear mappings. Our approach is based on the class of (α, β, γ)-nonexpansive mappings and three-step M-iterative scheme. Under various assumptions, we first carry out some weak and strong convergence results in a setting of a Banach spaces. After this, we carry out an application of one our main result to find approximate solution for a broad class of FDEs. Eventually, we we construct a new example of (α, β, γ)-nonexpansive mappings and show that this new mapping is not continuous on its whole domain and hence it is not nonexpansive. Using this example, we perform a numerical simulation of various iterative scheme including our M-iterative scheme. It has been observed the numerical effectiveness of the M-iterative scheme is high as compared to the other iterative schemes. Accordingly, our main outcome is new/extends some known results of the literature.

How to Cite

B. Nawaz, A. Labsir, K. Ullah, J. Ahmad, A New Fixed Point Approach for Approximating Solutions of Fractional Differential Equations in Banach Spaces, Int. J. Anal. Appl. 22 (2024), 124. https://doi.org/10.28924/2291-8639-22-2024-124

BibTeX

@article{Nawaz2024,
  author  = {Nawaz, Bashir and Labsir, Abdallah and Ullah, Kifayat and Ahmad, Junaid},
  title   = {A New Fixed Point Approach for Approximating Solutions of Fractional Differential Equations in Banach Spaces},
  journal = {Int. J. Anal. Appl.},
  volume  = {22},
  year    = {2024},
  pages   = {124},
  doi     = {10.28924/2291-8639-22-2024-124}
}

References

[1] J. Ahmad, K. Ullah, H.A. Hammad, R. George, A Solution of a Fractional Differential Equation via Novel FixedPoint Approaches in Banach Spaces, AIMS Math. 8 (2023), 12657–12670. https://doi.org/10.3934/math.2023636.
[2] J. Ahmad, K. Ullah, M. Arshad, Approximation of Fixed Points for a Class of Mappings Satisfying Property (CSC) in Banach Spaces, Math. Sci. 15 (2021), 207–213. https://doi.org/10.1007/s40096-021-00407-3.
[3] J. Ahmad, M. Arshad, A Modified Fixed Point Iterative Scheme Based on Green’s Function With Application to BVPs, Asian-Eur. J. Math. 17 (2024), 2450038. https://doi.org/10.1142/s1793557124500384.
[4] M. Abbas, T. Nazir, A New Faster Iteration Process Applied to Constrained Minimization and Feasibility Problems, Math. Vesnik. 66 (2014), 223–234.
[5] D.R. Sahu, D. O’Regan, R.P. Agarwal, Fixed Point Theory for Lipschitzian-type Mappings with Applications, Springer, New York, 2009. https://doi.org/10.1007/978-0-387-75818-3.
[6] R.P. Agarwal, D. O’Regon, D.R. Sahu, Iterative construction of fixed points of nearly asymtotically non-expansive mappings, J. Nonlinear Convex Anal. 8 (2007), 61–79.
[7] K. Aoyama, F. Kohsaka, Fixed Point Theorem for α–Nonexpansive Mappings in Banach Spaces, Nonlinear Anal.: Theory Meth. Appl. 74 (2011), 4387–4391. https://doi.org/10.1016/j.na.2011.03.057.
[8] S. Banach, Sur les Operations dans les Ensembles Abstraits et leur Application aux Equations Integrales, Fund. Math. 3 (1922), 133–181.
[9] F.E. Browder, Nonexpansive Nonlinear Operators in a Banach Space, Proc. Natl. Acad. Sci. U.S.A. 54 (1965), 1041–1044. https://doi.org/10.1073/pnas.54.4.1041.
[10] J.A. Clarkson, Uniformly Convex Spaces, Trans. Amer. Math. Soc. 40 (1936), 396–414. https://doi.org/10.1090/s0002-9947-1936-1501880-4.
[11] D. Göhde, Zum Prinzip der Kontraktiven Abbildung, Math. Nachr. 30 (1965), 251–258. https://doi.org/10.1002/mana.19650300312.
[12] S. Ishikawa, Fixed Points by a New Iteration Method, Proc. Amer. Math. Soc. 44 (1974), 147–150.
[13] E. Karapınar, T. Abdeljawad, F. Jarad, Applying New Fixed Point Theorems on Fractional and Ordinary Differential Equations, Adv. Differ. Equ. 2019 (2019), 421. https://doi.org/10.1186/s13662-019-2354-3.
[14] E. Karapinar, K. Tas, Generalized (C)–Conditions and Related Fixed Point Theorems, Comp. Math. Appl. 61 (2011), 3370–3380. https://doi.org/10.1016/j.camwa.2011.04.035.
[15] S. Khatoon, I. Uddin, D. Baleanu, Approximation of Fixed Point and Its Application to Fractional Differential Equation, J. Appl. Math. Comp. 66 (2020), 507–525. https://doi.org/10.1007/s12190-020-01445-1.
[16] W.A. Kirk, A Fixed Point Theorem for Mappings which do not Increase Distances, Amer. Math. Mon. 72 (1965), 1004–1006. https://doi.org/10.2307/2313345.
[17] W.R. Mann, Mean Value Methods in Iteration, Proc. Amer. Math. Soc. 4 (1953), 506–510. https://doi.org/10.1090/s0002-9939-1953-0054846-3.
[18] M.A. Noor, New Approximation Schemes for General Variational Inequalities, J. Math. Anal. Appl. 251 (2000), 217–229. https://doi.org/10.1006/jmaa.2000.7042.
[19] Z. Opial, Weak and Strong Convergence of the Sequence of Successive Approximations for Nonexpansive Mappings, Bull. Amer. Math. Soc. 73 (1967), 591–597.
[20] E. Picard, Mémoire sur la Théorie des Équations aux Dérivées Partielles et la Méthode des Approximations Successives, J. Math. Pures Appl. 6 (1880), 145–210.
[21] J. Schu, Weak and Strong Convergence to Fixed Points of Asymptotically Nonexpansive Mappings, Bull. Austral. Math. Soc. 43 (1991), 153–159.
[22] H.F. Senter, W.G. Dotson, Approximating Fixed Points of Nonexpansive Mappings, Proc. Amer. Math. Soc. 44 (1974), 375–380.
[23] H.F. Senter, W.G. Doston, Approximating Fixed Points of Nonexpansive Mappings, Proc. Amer. Math. Soc. 44 (1974), 375–380.
[24] T. Suzuki, Fixed Point Theorems and Convergence Theorems for Some Generalized Nonexpansive Mappings, J. Math. Anal. Appl. 340 (2008), 1088–1095. https://doi.org/10.1016/j.jmaa.2007.09.023.
[25] W. Takahashi, Nonlinear Functional Analysis, Yokohoma Publishers, Yokohoma, 2000.
[26] B.S. Thakur, D. Thakur, M. Postolache, A New Iterative Scheme for Numerical Reckoning Fixed Points of Suzuki’s Generalized Nonexpansive Mappings, Appl. Math. Comp. 275 (2016), 147–155. https://doi.org/10.1016/j.amc.2015.11.065.
[27] K. Ullah, J. Ahmad, H.A. Hammad, R. George, Iterative Schemes for Numerical Reckoning of Fixed Points of New Nonexpansive Mappings with an Application, AIMS Math. 8 (2023), 10711–10727. https://doi.org/10.3934/math.2023543.
[28] K. Ullah, M. Arshad, Numerical Reckoning Fixed Points for Suzuki’s Generalized Nonexpansive Mappings via New Iteration Process, Filomat 32 (2018), 187–196. https://doi.org/10.2298/fil1801187u.