Sequential fractional pantograph differential equations with nonlocal boundary conditions: Uniqueness and Ulam-Hyers-Rassias stability

Yıl 2022, , 29 - 41, 31.03.2022

Houas Mohamed

https://doi.org/10.53006/rna.928654

Cited By: 8

Öz

In the current manuscript, we study the uniqueness and Ulam-stability of solutions for sequential fractional
pantograph differential equations with nonlocal boundary conditions. The uniqueness of solutions is es-
tablished by Banach's fixed point theorem. We also define and study the Ulam-Hyers stability and the
Ulam-Hyers-Rassias stability of mentioned problem. An example is presented to illustrate the main results.

Anahtar Kelimeler

Caputo derivative, fixed point, existence, pantograph equations, Ulam stability

Kaynakça

• [1] M.S. Abdo, T. Abdeljawad, K.D. Kucche, M.A. Alqudah, S.M. Ali and M.B. Jeelani, On nonlinear pantograph fractional di?erential equations with Atangana-Baleanu-Caputo derivative, Adv. Difference . Equ. 2021: 65 (2021), 1-17.
• [2] M. S. Abdo, T. Abdeljawad, K. Shah and S. M. Ali, On nonlinear coupled evolution system with nonlocal subsidiary conditions under fractal-fractional order derivative, Math Meth Appl Sci. 44(8) (2021), 6581-6600.
• [3] A. Ali, I. Mahariq, K. Shah, T. Abdeljawad and B. Al-Sheikh, Stability analysis of initial value problem of pantograph-type implicit fractional differential equations with impulsive conditions, Adv. Di?erence . Equ. 2021: 55 (2021), 1-17.
• [4] B. Azizollah, Q.M. Al-Mdallal, On the existence of positive solutions for a non-autonomous fractional differential equation with integral boundary conditions, CMDE. 9(1) (2021), 36-51.
• [5] K. Balachandran, S. Kiruthika and J.J. Trujillo, Existence of solutions of Nonlinear fractional pantograph equations, Acta Mathematica Scientia. 33B (2013), 1-9.
• [6] W. Benhamida, S. Hamani and J. Henderson, Boundary value problems for Caputo-Hadamard fractional differential equations, Adv.Theory Nonlinear Anal. Appl. 2(3) (2018), 138-145.
• [7] A. Boutiara, M. S. Abdo, M. A. Alqudah and T. Abdeljawad, On a class of Langevin equations in the frame of Caputo function-dependent- kernel fractional derivatives with antiperiodic boundary conditions, AIMS Mathematics. 6(6) (2021), 5518-5534.
• [8] G A. Derfel, A. Iserles, The pantograph equation in the complex plane, J Math Anal Appl. 213, (1997), 117-132.
• [9] M. Houas, Existence of solutions for a coupled system of Caputo-Hadamard type fractional differential equations with Hadamard fractional integral conditions, Adv.Theory Nonlinear Anal. Appl. 5(3) (2021), 316-329.
• [10] A. Iserles, Exact and discretized stability of the pantograph equation, Appl. Numer. Math. 24 (1997), 295-308.
• [11] A Iserles, Y. Liu, On pantograph integro-differential equations, J. Integral Equations Appl. 6 (1994), 213-237.
• [12] A. Khan, H. Khan, J.F.Gómez-Aguilar and T. Abdeljawad, Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Le?er kernel, Chaos Solitons Fractals. 127 (2019), 422-427.
• [13] A.Khan, J.F. Gómez-Aguilar, T. Abdeljawada and H. Khand, Stability and numerical simulation of a fractional order plant-nectar-pollinator model, Alexandria Engineering Journal. 59 (2020), 49-59.
• [14] A. Khan, Y. Li, K. Shah and T. S. Khan, On coupled p−Laplacian fractional differential equations with nonlinear boundary conditions, Complexity. 2017, Article ID 8197610: (2017), 1-9.
• [15] H. Khan, W. Chen, A. Khan, T. S. Khan and Q. M. Al-Madlal, Hyers-Ulam stability and existence criteria for coupled fractional di?erential equations involving p−Laplacian operator, Adv. Difference . Equ. 2018: 45 (2018), 1-16.
• [16] H. Khan, C. Tunc and A. Khan, Green function's properties and existence theorems for nonlinear singular-delay-fractional di?erential equations, Discrete Contin. Dyn. Syst., Ser. S. 13(9) (2020), 2475-2487.
• [17] H. Khan, J.F.Gomez-Aguilar, T. Abdeljawad and A. Khan, Existence results and stability criteria for abc-fuzzy-Volterra integro-differential equation, Fractals. 28(8) (2020), 204004-1-204004-9.
• [18] M. B. A. Khan, T. Abdeljawad, K. Shah, G. Ali, H. Khan and A. Khan, Study of a nonlinear multi-terms boundary value problem of fractional pantograph differential equations, Adv. Difference Equ. 2021: 143 (2021), 1-15.
• [19] A.A. Kilbas, S.A. Marzan. Nonlinear differential equation with the Caputo fraction derivative in the space of continuously di?erentiable functions. Differ. Equ. 41(1), (2005), 84-89.
• [20] M. Klimek, Sequential fractional differential equations with Hadamard derivative, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 4689-4697.
• [21] V. Lakshmikantham, S.Leela, and J. V.Devi. Theory of fractional dynamic systems. Cambridge Scientific Publishers. 2009.
• [22] N. Mahmudov, M. Awadalla and K. Abuassba, Nonlinear sequential fractional differential equations with nonlocal boundary conditions, Adv. Difference Equ. 2017: 319 (2017), 1-.15
• [23] J R. Ockendon, A B. Taylor, The dynamics of a current collection system for an electric locomotive, Proc RSoc London, Ser.A. 322 (1971), 447-468.
• [24] A. Seemab, M. U. Rehman, J. Alzabut, Y. Adjabi and M.S. Abdo, Langevin equation with nonlocal boundary conditions involving a ψ−Caputo fractional operators of di?erent orders, AIMS Mathematics. 6(7) (2021), 6749-6780.
• [25] D. Vivek, E. M. Elsayed and K. Kanagarajan, Existence and Ulam stability results for a class of boundary value problem of neutral pantograph equations with complex order, SeMA Journal. 77(3) (2021), 243-256.
• [26] H. A. Wahashy, M.S. Abdo, A. M. Saeed and S.K. Panchal, Singular fractional differential equations with ψ−Caputo operator and modified Picard's iterative method. Appl. Math. E-Notes. 20 (2020), 215-229.
• [27] Z. Wei, W. Dong, Periodic boundary value problems for Riemann-Liouville sequential fractional differential equations, Electron. J. Qual. Theory Differ. Equ. 87 (2011), 1-13.
• [28] A. Wongcharoen, S.K. Ntouyas and J.Tariboon, Nonlocal boundary value problemsfor Hilfer-type pantograph fractional-differential equations and inclusions, Adv. Difference. Equ. 2020: 279 (2020), 1-21.
• [29] A. Zada, M. Yar and T. Li, Existence and stability analysis of nonlinear sequential coupled system of Caputo fractional differential equations with integral boundary conditions, Ann. Univ. Paedagog. Crac. Stud. Math. 17 (2018), 103-125.