Research article

On a novel impulsive boundary value pantograph problem under Caputo proportional fractional derivative operator with respect to another function

  • Received: 14 October 2021 Revised: 27 January 2022 Accepted: 06 February 2022 Published: 17 February 2022
  • MSC : 26A33, 34A37, 34A08, 34D20, 38B82

  • In this manuscript, we study the existence and Ulam's stability results for impulsive multi-order Caputo proportional fractional pantograph differential equations equipped with boundary and integral conditions with respect to another function. The uniqueness result is proved via Banach's fixed point theorem, and the existence results are based on Schaefer's fixed point theorem. In addition, the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of the proposed problem are obtained by applying the nonlinear functional analysis technique. Finally, numerical examples are provided to supplement the applicability of the acquired theoretical results.

    Citation: Songkran Pleumpreedaporn, Chanidaporn Pleumpreedaporn, Weerawat Sudsutad, Jutarat Kongson, Chatthai Thaiprayoon, Jehad Alzabut. On a novel impulsive boundary value pantograph problem under Caputo proportional fractional derivative operator with respect to another function[J]. AIMS Mathematics, 2022, 7(5): 7817-7846. doi: 10.3934/math.2022438

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  • In this manuscript, we study the existence and Ulam's stability results for impulsive multi-order Caputo proportional fractional pantograph differential equations equipped with boundary and integral conditions with respect to another function. The uniqueness result is proved via Banach's fixed point theorem, and the existence results are based on Schaefer's fixed point theorem. In addition, the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of the proposed problem are obtained by applying the nonlinear functional analysis technique. Finally, numerical examples are provided to supplement the applicability of the acquired theoretical results.



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    [1] I. Podlubny, Fractional differential equation, Mathematics in Science and Engineering, Vol. 198, New York: Academic Press, 1999.
    [2] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam: Elsevier, 2006.
    [3] R. Caponetto, G. Dongola, L. Fortuna, I. Petras, Fractional order systems: Modeling and control applications, Singapore: World Scientific, 2010.
    [4] F. Mainardi, Fractional calculus and waves in linear viscoelasticity, London: Imperiall College Press, 2010.
    [5] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional calculus: Models and numerical methods, Singapore: World Scientific, 2012.
    [6] J. R. Ockendon, A. B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. Roy. Soc. Lond. A, 322 (1971), 447–468. https://doi.org/10.1098/rspa.1971.0078 doi: 10.1098/rspa.1971.0078
    [7] A. Iserles, On the generalized pantograph functional-differential equation, Eur. J. Appl. Math., 1 (1993), 1–38. https://doi.org/10.1017/S0956792500000966 doi: 10.1017/S0956792500000966
    [8] A. Iserles, Y. K. Liu, On pantograph integro-differential equations, J. Integral Equ. Appl., 6 (1994), 213–237.
    [9] G. Derfel, A. Iserles, The pantograph equation in the complex plane, J. Math. Anal. Appl., 213 (1997), 117–132. https://doi.org/10.1006/jmaa.1997.5483 doi: 10.1006/jmaa.1997.5483
    [10] M. Z. Liu, D. S. Li, Properties of analytic solution and numerical solution of multi-pantograph equation, Appl. Math. Comput., 155 (2004), 853–871. https://doi.org/10.1016/j.amc.2003.07.017 doi: 10.1016/j.amc.2003.07.017
    [11] D. Li, M. Z. Liu, Runge-Kutta methods for the multi-pantograph delay equation, Appl. Math. Comput., 163 (2005), 383–395. https://doi.org/10.1016/j.amc.2004.02.013 doi: 10.1016/j.amc.2004.02.013
    [12] M. Sezer, S. Yalçinbaş, N. Şahin, Approximate solution of multi-pantograph equation with variable coefficients, J. Comput. Appl. Math., 214 (2008), 406–416. https://doi.org/10.1016/j.cam.2007.03.024 doi: 10.1016/j.cam.2007.03.024
    [13] L. Bogachev, G. Derfel, S. Molchanov, J. Ochendon, On bounded solutions of the balanced generalized pantograph equation, In: P. L. Chow, B. S. Mordukhovich, G. Yin, Topics in stochastic analysis and nonparametric estimation, New York: Springer, 2008. https://doi.org/10.1007/978-0-387-75111-5_3
    [14] 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
    [15] S. K. Vanani, J. S. Hafshejani, F. Soleymani, M. Khan, On the numerical solution of generalized pantograph equation, World Appl. Sci. J., 13 (2011), 2531–2535.
    [16] 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
    [17] C. M. Pappalardo, M. C. De Simone, D. Guida, Multibody modeling and nonlinear control of the pantograph/catenary system, Arch. Appl. Mech., 89 (2019), 1589–1626. https://doi.org/10.1007/s00419-019-01530-3 doi: 10.1007/s00419-019-01530-3
    [18] M. Chamekh, T. M. Elzaki, N. Brik, Semi-analytical solution for some proportional delay differential equations, SN Appl. Sci., 1 (2019), 1–6. https://doi.org/10.1007/s42452-018-0130-8 doi: 10.1007/s42452-018-0130-8
    [19] D. F. Li, C. J. Zhang, Long time numerical behaviors of fractional pantograph equations, Math. Comput. Simulat., 172 (2020), 244–257. https://doi.org/10.1016/j.matcom.2019.12.004 doi: 10.1016/j.matcom.2019.12.004
    [20] K. Balachandran, S. Kiruthika, J. J. Trujillo, Existence of solutions 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
    [21] V. Lakshmikantham, D. D. Bainov, P. S. Semeonov, Theory of impulsive differential equations, Singapore: Worlds Scientific, 1989.
    [22] A. M. Samoilenko, N. A. Perestyuk, Impulsive differential equations, Singapore: World Scientific, 1995. https://doi.org/10.1142/2892
    [23] M. Benchohra, J. Henderson, S. Ntouyas, Impulsive differential equations and inclusions, New York: Hindawi Publishing Corporation, 2006.
    [24] J. R. Wang, Y. Zhou, M. Fečkan, On recent developments in the theory of boundary value problems for impulsive fractional differential equations, Comput. Math. Appl., 64 (2012), 3008–3020. https://doi.org/10.1016/j.camwa.2011.12.064 doi: 10.1016/j.camwa.2011.12.064
    [25] S. M. Ulam, A collection of mathematical problems, New York: Interscience Publishers, 1960.
    [26] D. H. Hyers, On the stability of the linear functional equations, Proc. Natl. Acad. Sci., 27 (1941), 222–224. https://doi.org/10.1073/pnas.27.4.222 doi: 10.1073/pnas.27.4.222
    [27] T. M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300. https://doi.org/10.1090/s0002-9939-1978-0507327-1 doi: 10.1090/s0002-9939-1978-0507327-1
    [28] S. M. Jung, Hyers-Ulam-Rassias stability of functional equations in nonlinear analysis, New York: Springer, 2011. https://doi.org/10.1007/978-1-4419-9637-4
    [29] M. Benchohra, B. A. Slimani, Existence and uniqueness of solutions to impulsive fractional diferential equations, Electron. J. Differ. Equ., 2009 (2009), 1–11.
    [30] M. Benchohra, D. Seba, Impulsive fractional differential equations in Banach spaces, Electron. J. Qual. Theory Differ. Equ., 2009 (2009), 1–14.
    [31] J. R. Wang, Y. Zhou, W. Wei, Study in fractional differential equations by means of topological degree methods, Numer. Funct. Anal. Optimiz., 33 (2012), 216–238. https://doi.org/10.1080/01630563.2011.631069 doi: 10.1080/01630563.2011.631069
    [32] M. Benchohra, J. E. Lazreg, Existence results for nonlinear implicit fractional differential equations with impulse, Commun. Appl. Anal., 19 (2015), 413–426.
    [33] M. Benchohra, S. Bouriah, J. R. Graef, Boundary value problems for nonlinear implicit Caputo-Hadamard-type fractional differential equations with impulses, Mediterr. J. Math., 14 (2017), 1–21. https://doi.org/10.1007/s00009-017-1012-9 doi: 10.1007/s00009-017-1012-9
    [34] 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. Difference Equ., 2021 (2021), 1–17. https://doi.org/10.1186/s13662-021-03218-x doi: 10.1186/s13662-021-03218-x
    [35] Y. K. Chang, A. Anguraj, P. Karthikeyan, Existence results for initial value problems with integral condition for impulsive fractional differential equations, J. Fract. Calc. Appl., 2 (2012), 1–10.
    [36] K. Shah, A. Ali, S. Bushnaq, Hyers-Ulam stability analysis to implicit Cauchy problem of fractional differential equations with impulsive conditions, Math. Method. Appl. Sci., 41 (2018), 8329–8343. https://doi.org/10.1002/mma.5292 doi: 10.1002/mma.5292
    [37] J. Tariboon, S. K. Ntouyas, B. Sutthasin, Impulsive fractional quantum Hahn difference boundary value problems, Adv. Difference Equ., 2019 (2019), 1–18. https://doi.org/10.1186/s13662-019-2156-7 doi: 10.1186/s13662-019-2156-7
    [38] I. Ahmed, P. Kumam, J. Abubakar, P. Borisut, K. Sitthithakerngkiet, Solutions for impulsive fractional pantograph differential equation via generalized anti-periodic boundary condition, Adv. Difference Equ., 2020 (2020), 1–15. https://doi.org/10.1186/s13662-020-02887-4 doi: 10.1186/s13662-020-02887-4
    [39] A. I. N. Malti, M. Benchohra, J. R. Graef, J. E. Lazreg, Impulsive boundary value problems for nonlinear implicit Caputo-exponential type fractional differential equations, Electron. J. Qual. Theory Differ. Equ., 2020 (2020), 1–17. https://doi.org/10.14232/ejqtde.2020.1.78 doi: 10.14232/ejqtde.2020.1.78
    [40] M. S. Abdo, T. Abdeljawad, K. Shah, F. Jarad, Study of impulsive problems under Mittag-Leffler power law, Heliyon, 6 (2020), 1–8. https://doi.org/10.1016/j.heliyon.2020.e05109 doi: 10.1016/j.heliyon.2020.e05109
    [41] M. I. Abbas, On the initial value problems for the Caputo-Fabrizio impulsive fractional differential equations, Asian-Eur. J. Math., 14 (2021), 2150073. https://doi.org/10.1142/s179355712150073x doi: 10.1142/s179355712150073x
    [42] A. Salim, M. Benchohra, E. Karapinar, J. E. Lazreg, Existence and Ulam stability for impulsive generalized Hilfer-type fractional differential equations, Adv. Difference Equ., 2020 (2020), 1–21. https://doi.org/10.1186/s13662-020-03063-4 doi: 10.1186/s13662-020-03063-4
    [43] A. Salim, M. Benchohra, J. E. Lazreg, G. N'Guérékata, Boundary value problem for nonlinear implicit generalized Hilfer-type fractional differential equations with impulses, Abstr. Appl. Anal., 2021 (2021), 1–17. https://doi.org/10.1155/2021/5592010 doi: 10.1155/2021/5592010
    [44] H. Khan, A. Khan, T. Abdeljawad, A. Alkhazzan, Existence results in Banach space for a nonlinear impulsive system, Adv. Difference Equ., 2019 (2019), 1–16. https://doi.org/10.1186/s13662-019-1965-z doi: 10.1186/s13662-019-1965-z
    [45] A. Ali, K. Shah, T. Abdeljawad, H. Khan, A. Khan, Study of fractional order pantograph type impulsive antiperiodic boundary value problem, Adv. Difference Equ., 2020 (2020), 1–32. https://doi.org/10.1186/s13662-020-03032-x doi: 10.1186/s13662-020-03032-x
    [46] H. Khan, Z. A. Khan, H. Tajadodi, A. Khan, Existence and data-dependence theorems for fractional impulsive integro-differential system, Adv. Difference Equ., 2020 (2020), 1–11. https://doi.org/10.1186/s13662-020-02823-6 doi: 10.1186/s13662-020-02823-6
    [47] F. Jarad, M. A. Alqudah, T. Abdeljawad, On more general forms of proportional fractional operators, Open Math., 18 (2020), 167–176. https://doi.org/10.1515/math-2020-0014 doi: 10.1515/math-2020-0014
    [48] F. Jarad, T. Abdeljawad, S. Rashid, Z. Hammouch, More properties of the proportional fractional integrals and derivatives of a function with respect to another function, Adv. Difference Equ., 2020 (2020), 1–16. https://doi.org/10.1186/s13662-020-02767-x doi: 10.1186/s13662-020-02767-x
    [49] U. N. Katugampola, New fractional integral unifying six existing fractional integrals, arXiv Preprint, 2016. Available from: https://arXiv.org/abs/1612.08596.
    [50] F. Jarad, E. Uğurlu, T. Abdeljawad, D. Baleanu, On a new class of fractional operators, Adv. Difference Equ., 2017 (2017), 1–16. https://doi.org/10.1186/s13662-017-1306-z doi: 10.1186/s13662-017-1306-z
    [51] T. U. Khan, M. Adil Khan, Generalized conformable fractional operators, J. Comput. Appl. Math., 346 (2019), 378–389. https://doi.org/10.1016/j.cam.2018.07.018 doi: 10.1016/j.cam.2018.07.018
    [52] M. I. Abbas, M. A. Ragusa, On the hybrid fractional differential equations with fractional proportional derivatives of a function with respect to a certain function, Symmetry, 13 (2021), 1–16. https://doi.org/10.3390/sym13020264 doi: 10.3390/sym13020264
    [53] S. S. Zhou, S. Rashid, A. Rauf, F. Jarad, Y. S. Hamed, K. M. Abualnaja, Efficient computations for weighted generalized proportional fractional operators with respect to a monotone function, AIMS Math., 6 (2021), 8001–8029. https://doi.org/10.3934/math.2021465 doi: 10.3934/math.2021465
    [54] S. Rashid, F. Jarad, Z. Hammouch, Some new bounds analogous to generalized proportional fractional integral operator with respect to another function, Discrete Contin. Dyn. Syst. Ser. S, 14 (2021), 3703–3718. https://doi.org/10.3934/dcdss.2021020 doi: 10.3934/dcdss.2021020
    [55] T. Abdeljawad, S. Rashid, A. A. El-Deeb, Z. Hammouch, Y. M. Chu, Certain new weighted estimates proposing generalized proportional fractional operator in another sense, Adv. Difference Equ., 2020 (2020), 1–16. https://doi.org/10.1186/s13662-020-02935-z doi: 10.1186/s13662-020-02935-z
    [56] G. Rahman, T. Abdeljawad, F. Jarad, K. S. Nisar, Bounds of generalized proportional fractional integrals in general form via convex functions and their applications, Mathematics, 8 (2020), 1–19. https://doi.org/10.3390/math8010113 doi: 10.3390/math8010113
    [57] C. Tearnbucha, W. Sudsutad, Stability analysis of boundary value problems for Caputo proportional fractional derivative of a function with respect to another function via impulsive Langevin equation, AIMS Math., 6 (2021), 6647–6686. https://doi.org/10.3934/math.2021391 doi: 10.3934/math.2021391
    [58] M. I. Abbas, Non-instantaneous impulsive fractional integro-differential equations with proportional fractional derivatives with respect to another function, Math. Method. Appl. Sci., 44 (2021), 10432–10447. https://doi.org/10.1002/mma.7419 doi: 10.1002/mma.7419
    [59] A. Granas, J. Dugundji, Fixed point theory, New York: Springer, 2003. https://doi.org/10.1007/978-0-387-21593-8
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