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Analytical study of $ \mathcal{ABC} $-fractional pantograph implicit differential equation with respect to another function

  • Received: 12 May 2023 Revised: 01 July 2023 Accepted: 10 July 2023 Published: 31 July 2023
  • MSC : 34A08, 34B15

  • This article aims to establish sufficient conditions for qualitative properties of the solutions for a new class of a pantograph implicit system in the framework of Atangana-Baleanu-Caputo ($ \mathcal{ABC} $) fractional derivatives with respect to another function under integral boundary conditions. The Schaefer and Banach fixed point theorems (FPTs) are utilized to investigate the existence and uniqueness results for this pantograph implicit system. Moreover, some stability types such as the Ulam-Hyers $ (\mathbb{UH}) $, generalized $ \mathbb{UH} $, Ulam-Hyers-Rassias $ (\mathbb{UHR}) $ and generalized $ \mathbb{UHR} $ are discussed. Finally, interpretation mathematical examples are given in order to guarantee the validity of the main findings. Moreover, the fractional operator used in this study is more generalized and supports our results to be more extensive and covers several new and existing problems in the literature.

    Citation: Sabri T. M. Thabet, Miguel Vivas-Cortez, Imed Kedim. Analytical study of $ \mathcal{ABC} $-fractional pantograph implicit differential equation with respect to another function[J]. AIMS Mathematics, 2023, 8(10): 23635-23654. doi: 10.3934/math.20231202

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  • This article aims to establish sufficient conditions for qualitative properties of the solutions for a new class of a pantograph implicit system in the framework of Atangana-Baleanu-Caputo ($ \mathcal{ABC} $) fractional derivatives with respect to another function under integral boundary conditions. The Schaefer and Banach fixed point theorems (FPTs) are utilized to investigate the existence and uniqueness results for this pantograph implicit system. Moreover, some stability types such as the Ulam-Hyers $ (\mathbb{UH}) $, generalized $ \mathbb{UH} $, Ulam-Hyers-Rassias $ (\mathbb{UHR}) $ and generalized $ \mathbb{UHR} $ are discussed. Finally, interpretation mathematical examples are given in order to guarantee the validity of the main findings. Moreover, the fractional operator used in this study is more generalized and supports our results to be more extensive and covers several new and existing problems in the literature.



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    [1] M. I. Abbas, M. Ghaderi, S. Rezapour, S. T. M. Thabet, On a coupled system of fractional differential equations via the generalized proportional fractional derivatives, J. Funct. Space., 2022 (2022), 4779213. https://doi.org/10.1155/2022/4779213 doi: 10.1155/2022/4779213
    [2] M. S. Abdo, T. Abdeljawad, K. D. Kucche, M. A. Alqudah, S. M. Ali, M. B. Jeelani, On nonlinear pantograph fractional differential equations with Atangana-Baleanu-Caputo derivative, Adv. Diff. Equ., 2021 (2021), 65. https://doi.org/10.1186/s13662-021-03229-8 doi: 10.1186/s13662-021-03229-8
    [3] S. T. M. Thabet, M. B. Dhakne, On boundary value problems of higher order abstract fractional integro-differential equations, Int. J. Nonlinear Anal. Appl., 7 (2016), 165–184.
    [4] M. S. Abdo, T. Abdeljawad, K. Shah, F. Jarad, Study of impulsive problems under Mittag-Leffler power law, Heliyon, 2020, 6e05109. https://doi.org/10.1016/j.heliyon.2020.e05109 doi: 10.1016/j.heliyon.2020.e05109
    [5] R. Hilfer, Applications of fractional calculus in physics, World Scientific, Singapore, 2000.
    [6] S. T. M. Thabet, M. B. Dhakne, M. A. Salman, R. Gubran, Generalized fractional Sturm-Liouville and Langevin equations involving Caputo derivative with nonlocal conditions, Progr. Fract. Differ. Appl., 6 (2020), 225–237. https://doi.org/10.18576/pfda/060306 doi: 10.18576/pfda/060306
    [7] O. Nikan, S. M. Molavi-Arabshai, H. Jafari, Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves, Discrete Cont. Dyn.-S, 14 (2021), 3685–3701. https://doi.org/10.3934/dcdss.2020466 doi: 10.3934/dcdss.2020466
    [8] N. H. Can, O. Nikan, M. N. Rasoulizadeh, H. Jafari, Y. S. Gasimov, Numerical computation of the time non-linear fractional generalized equal width model arising in shallow water channel, Therm. Sci., 24 (2020), 49–58. https://doi.org/10.2298/TSCI20S1049C doi: 10.2298/TSCI20S1049C
    [9] W. Qiu, D. Xu, H. Chen, J. Guo, An alternating direction implicit Galerkin finite element method for the distributed-order time-fractional mobile-immobile equation in two dimensions, Comput. Math. Appl., 80 (2020), 3156–3172. https://doi.org/10.1016/j.camwa.2020.11.003 doi: 10.1016/j.camwa.2020.11.003
    [10] X. Yang, W. Qiu, H. Zhang, L. Tang, An efficient alternating direction implicit finite difference scheme for the three-dimensional time-fractional telegraph equation, Comput. Math. Appl., 102 (2021), 233–247. https://doi.org/10.1016/j.camwa.2021.10.021 doi: 10.1016/j.camwa.2021.10.021
    [11] W. Qiu, D. Xu, J. Guo, J. Zhou, A time two-grid algorithm based on finite difference method for the two-dimensional nonlinear time-fractional mobile/immobile transport model, Numer. Algorithms, 85 (2020), 39–58. https://doi.org/10.1007/s11075-019-00801-y doi: 10.1007/s11075-019-00801-y
    [12] S. T. M. Thabet, M. M. Matar, M. A. Salman, M. E. Samei, M. Vivas-Cortez, I. Kedim, On coupled snap system with integral boundary conditions in the $\mathbb{G}$-Caputo sense, AIMS Math., 8 (2023), 12576–12605. https://doi.org/10.3934/math.2023632 doi: 10.3934/math.2023632
    [13] F. Jarad, T. Abdeljawad, Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative, Chaos Soliton. Fract., 117 (2018), 16–20.
    [14] S. T. M. Thabet, B. Ahmad, R. P. Agarwal, On abstract Hilfer fractional integrodifferential equations with boundary conditions, Arab J. Math. Sci., 26 (2020), 107–125. https://doi.org/10.1016/j.ajmsc.2019.03.001 doi: 10.1016/j.ajmsc.2019.03.001
    [15] S. Rezapour, S. T. M. Thabet, M. M. Matar, J. Alzabut, S. Etemad, Some existence and stability criteria to a generalized FBVP having fractional composite p-Laplacian operator, J. Funct. Space., 2021 (2021), 9554076. https://doi.org/10.1155/2021/9554076 doi: 10.1155/2021/9554076
    [16] S. T. M. Thabet, M. B. Dhakne, On positive solutions of higher order nonlinear fractional integro-differential equations with boundary conditions, Malaya J. Mat., 7 (2019), 20–26. https://doi.org/10.26637/MJM0701/0005 doi: 10.26637/MJM0701/0005
    [17] M. S. Abdo, Boundary value problem for fractional neutral differential equations with infinite delay, Abh. J. Basic Appl. Sci., 1 (2022), 1–18. https://doi.org/10.59846/ajbas.v1i1.357 doi: 10.59846/ajbas.v1i1.357
    [18] A. S. Rafeeq, Periodic solution of Caputo-Fabrizio fractional integro-differential equation with periodic and integral boundary conditions, Eur. J. Pure Apppl. Math., 15 (2022), 144–157. https://doi.org/10.29020/nybg.ejpam.v15i1.4247 doi: 10.29020/nybg.ejpam.v15i1.4247
    [19] S. T. M. Thabet, I. Kedim, Study of nonlocal multiorder implicit differential equation involving Hilfer fractional derivative on unbounded domains, J. Math., 2023 (2023), 1–14. https://doi.org/10.1155/2023/8668325 doi: 10.1155/2023/8668325
    [20] A. Boutiara, S. Etemad, S. T. M. Thabet, S. K. Ntouyas, S. Rezapour, J. Tariboon, A mathematical theoretical study of a coupled fully hybrid $(\kappa, \phi)$-fractional order system of BVPs in generalized Banach spaces, Symmetry, 15 (2023), 1041. https://doi.org/10.3390/sym15051041 doi: 10.3390/sym15051041
    [21] S. T. M. Thabet, M. B. Dhakne, On nonlinear fractional integro-differential equations with two boundary conditions, Adv. Stud. Contemp. Math., 26 (2016), 513–526.
    [22] A. Ali, I. Mahariq, K. Shah, T. Abdeljawad, B. Al-Sheikh, Stability analysis of initial value problem of pantograph-type implicit fractional differential equations with impulsive conditions, Adv. Diff. Equ., 2021 (2021), 55. https://doi.org/10.1186/s13662-021-03218-x doi: 10.1186/s13662-021-03218-x
    [23] S. T. M. Thabet, S. Al-Sádi, I. Kedim, A. S. Rafeeq, S. Rezapour, Analysis study on multi-order $\varrho$-Hilfer fractional pantograph implicit differential equation on unbounded domains, AIMS Math., 8 (2023), 18455–18473. https://doi.org/10.3934/math.2023938 doi: 10.3934/math.2023938
    [24] M. Houas, K. Kaushik, A. Kumar, A. Khan, T. Abdeljawad, Existence and stability results of pantograph equation with three sequential fractional derivatives, AIMS Math., 8 (2022), 5216–5232. https://doi.org/10.3934/math.2023262 doi: 10.3934/math.2023262
    [25] Z. H. Yu, Variational iteration method for solving the multi-pantograph delay equation, Phys. Lett. A, 372 (2008), 6475–6479. https://doi.org/10.1016/j.physleta.2008.09.013 doi: 10.1016/j.physleta.2008.09.013
    [26] K. Balachandran, S. Kiruthika, J. J. Trujillo, Existence of solution of nonlinear fractional pantograph equations, Acta Math. Sci., 33 (2013), 712–720. https://doi.org/10.1016/S0252-9602(13)60032-6 doi: 10.1016/S0252-9602(13)60032-6
    [27] E. Tohidi, A. H. Bhrawy, K. Erfani, A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation, Appl. Math. Model., 37 (2013), 4283–4294. https://doi.org/10.1016/j.apm.2012.09.032 doi: 10.1016/j.apm.2012.09.032
    [28] A. Atangana, D. Baleanu, New fractional derivatives with non-local and non-singular kernel: Theory and application to heat transfer model, Therm. Sci., 20 (2016), 763–769. https://doi.org/10.2298/TSCI160111018A doi: 10.2298/TSCI160111018A
    [29] T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, J. Inequal. Appl., 2017 (2017), 130. https://doi.org/10.1186/s13660-017-1400-5 doi: 10.1186/s13660-017-1400-5
    [30] T. Abdeljawad, D. Baleanu, Integration by parts and its applications of a new nonlocal fractional derivative with Mittag-Leffler nonsingular kernel, J. Nonlinear Sci. Appl., 9 (2017), 1098–1107.
    [31] M. I. Ayari, S. T. M. Thabet, Qualitative properties and approximate solutions of thermostat fractional dynamics system via a nonsingular kernel operator, Arab J. Math. Sci., 2023. https://doi.org/10.1108/AJMS-06-2022-0147 doi: 10.1108/AJMS-06-2022-0147
    [32] M. Khan, Z. Ahmad, F. Ali, N. Khan, I. Khan, K. S. Nisar, Dynamics of two-step reversible enzymatic reaction under fractional derivative with Mittag-Leffler Kernel, Plos One, 18 (2023), e0277806. https://doi.org/10.1371/journal.pone.0277806 doi: 10.1371/journal.pone.0277806
    [33] S. Rashid, Z. Hammouch, R. Ashraf, Y. M. Chu, New computation of unified bounds via a more general fractional operator using generalized Mittag-Leffler function in the kernel, Comput. Model. Eng. Sci., 126 (2021), 359–378. https://doi.org/10.32604/cmes.2021.011782 doi: 10.32604/cmes.2021.011782
    [34] Y. M. Chu, M. F. Khan, S. Ullah, S. A. Shah, M. Farooq, M. B. Mamat, Mathematical assessment of a fractional-order vector-host disease model with the Caputo-Fabrizio derivative, Math. Method. Appl. Sci., 46 (2023), 232–247. https://doi.org/10.1002/mma.8507 doi: 10.1002/mma.8507
    [35] S. Rashid, Z. Hammouch, D. Baleanu, Y. M. Chu, New generalizations in the sense of the weighted non-singular fractional integral operator, Fractals, 28 (2020), 2040003. https://doi.org/10.1142/S0218348X20400034 doi: 10.1142/S0218348X20400034
    [36] F. Jin, Z. S. Qian, Y. M. Chu, M. Rahman, On nonlinear evolution model for drinking behavior under Caputo-Fabrizio derivative, J. Appl. Anal. Comput., 12 (2022), 790–806. https://doi.org/10.11948/20210357 doi: 10.11948/20210357
    [37] M. D. Ikram, M. A. Imran, Y. M. Chu, A. Akgül, MHD flow of a Newtonian fluid in symmetric channel with ABC fractional model containing hybrid nanoparticles, Comb. Chem. High T. Scr., 25 (2022), 1087–1102. https://doi.org/10.2174/1386207324666210412122544 doi: 10.2174/1386207324666210412122544
    [38] A. Fernandez, D. Baleanu, Differintegration with respect to functions in fractional models involving Mittag-Leffler functions, SSRN Electron. J., 2018. https://doi.org/10.2139/ssrn.3275746 doi: 10.2139/ssrn.3275746
    [39] T. Abdeljawad, S. T. M. Thabet, I. Kedim, M. I. Ayari, A. Khan, A higher-order extension of Atangana-Baleanu fractional operators with respect to another function and a Gronwall-type inequality, Bound. Value Probl., 2023 (2023), 49. https://doi.org/10.1186/s13661-023-01736-z doi: 10.1186/s13661-023-01736-z
    [40] G. Ali, K. Shah, T. Abdeljawad, H. Khan, G. U. Rahman, A. Khan, On existence and stability results to a class of boundary value problems under Mittag-Leffler power law, Adv. Diff. Equ., 2020 (2020), 407. https://doi.org/10.1186/s13662-020-02866-9 doi: 10.1186/s13662-020-02866-9
    [41] S. Asma, K. Shabbir, T. Shah, T. Abdeljawad, Stability analysis for a class of implicit fractional differential equations involving Atangana-Baleanu fractional derivative, Adv. Diff. Equ., 2021 (2021), 395. https://doi.org/10.1186/s13662-021-03551-1 doi: 10.1186/s13662-021-03551-1
    [42] M. S. Abdo, T. Abdeljawad, S. M. Ali, K. Shah, On fractional boundary value problems involving fractional derivatives with Mittag-Leffler kernel and nonlinear integral conditions, Adv. Diff. Equ., 2021 (2021), 37. https://doi.org/10.1186/s13662-020-03196-6 doi: 10.1186/s13662-020-03196-6
    [43] R. Almeida, A. B. Malinowska, M. T. T. Monteiro, Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications, Math. Method. Appl. Sci., 41 (2018), 336–352. https://doi.org/10.1002/mma.4617 doi: 10.1002/mma.4617
    [44] P. O. Mohammed, T. Abdeljawad, Integral inequalities for a fractional operator of a function with respect to another function with nonsingular kernel, Adv. Diff. Equ., 2020 (2020), 363. https://doi.org/10.1186/s13662-020-02825-4 doi: 10.1186/s13662-020-02825-4
    [45] A. Granas, J. Dugundji, Fixed point theory, New York, Springer, 2003. https://doi.org/10.1007/978-0-387-21593-8
    [46] Y. Zhou, Basic theory of fractional differential equations, World Scientific, Singapore, 2014.
    [47] Z. Ahmad, F. Ali, N. Khan, I. Khan, Dynamics of fractal-fractional model of a new chaotic system of integrated circuit with Mittag-Leffler kernel, Chaos Soliton. Fract., 153 (2021), 111602. https://doi.org/10.1016/j.chaos.2021.111602 doi: 10.1016/j.chaos.2021.111602
    [48] S. M. Ulam, A problem of stability in functional equations, Bull. Am. Math. Soc., 46 (1940), 1–14.
    [49] D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA, 27 (1941), 222–224. https://doi.org/10.1073/pnas.27.4.222 doi: 10.1073/pnas.27.4.222
    [50] T. M. Rassias, On the stability of functional equations in Banach spaces, J. Funct. Anal., 32 (1978), 168–175.
    [51] C. Wang, T. Xu, Hyers-Ulam stability of fractional linear differential equations involving Caputo fractional derivatives, Appl. Math., 60 (2015), 383–393. https://doi.org/10.1007/s10492-015-0102-x doi: 10.1007/s10492-015-0102-x
    [52] T. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math., 62 (2000), 23–130. https://doi.org/10.1023/A:1006499223572 doi: 10.1023/A:1006499223572
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