Analysis study of hybrid Caputo-Atangana-Baleanu fractional pantograph system under integral boundary conditions
Abstract
This manuscript investigates the qualitative analysis of a new hybrid fractional pantograph system involving AtanganaBaleanu-Caputo derivatives, complemented by hybrid integral boundary conditions. Dhage’s fixed point theorem is employed to investigate the existence theorem of the solutions, while uniqueness is proven by using Perov’s approach and Lipschitz’s matrix. The Hyers-Ulam (HU) stability is also demonstrated using the Lipschitz’s matrix and techniques from nonlinear analysis. Finally, illustrative example is enhanced to examine the effectiveness of the obtained results.
Keyword : Caputo-Atangana-Baleanu operator, coupled hybrid fractional differential system, pantograph problem, Dhage and Perov techniques, Lipschitzian’s matrix
How to Cite
Thabet, S. T. M., Kedim, I., Samei, M. E., & Abdeljawad, T. (2025). Analysis study of hybrid Caputo-Atangana-Baleanu fractional pantograph system under integral boundary conditions. Mathematical Modelling and Analysis, 30(2), 386–404. https://doi.org/10.3846/mma.2025.22328
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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S.M. Ali, M.S. Abdo, B. Sontakke, K. Shah and T. Abdeljawad. New results on a coupled system for second-order pantograph equations with ABC−fractional derivatives. AIMS Mathematics, 7(10):19520–19538, 2022. https://doi.org/10.3934/math.20221071
S. Aljoudi. Existence and uniqueness results for coupled system of fractional differential equations with exponential kernel derivatives. AIMS Mathematics, 8(1):590–606, 2022. https://doi.org/10.3934/math.2023027
H. Alrabaiah, G. Ali, A. Ali, K. Shah and T. Abdeljawad. On existence and stability results for pantograph fractional boundary value problems. Fractals, 30(8):2240231, 2022. https://doi.org/10.1142/S0218348X22402319
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A. Boutiara, M. Benbachir, M.K.A. Kaabar, F. Martínez, M.E. Samei and M. Kaplan. Explicit iteration and unbounded solutions for fractional q-difference equations with boundary conditions on an infinite interval. Journal of Inequalities and Applications, 2022:29, 2022. https://doi.org/10.1186/s13660-022-02764-6
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B.C. Dhage. On a fixed point theorem in Banach algebras with applications. Applied Mathematics Letters, 18(3):273–280, 2005. https://doi.org/10.1016/j.aml.2003.10.014
M. Houas, K. Kaushik, A. Kumar, A. Khan and T. Abdeljawad. Existence and stability results of pantograph equation with three sequential fractional derivatives. AIMS Mathematics, 8(3):5216–5232, 2022. https://doi.org/10.3934/math.2023262
K.D. Kucche and S.T. Sutar. Analysis of nonlinear fractional differential equations involving Atangana-Baleanu-Caputo derivative. Chaos, Solitons & Fractals, 143:110556, 2021. https://doi.org/10.1016/j.chaos.2020.110556
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G. Mani, A.J. Gnanaprakasam, L. Guran, R. George and Z.D. Mitrović. Some results in fuzzy b-metric space with b-triangular property and applications to fredholm integral equations and dynamic programming. Mathematics, 11(19):4101, 2023. https://doi.org/10.3390/math11194101
G. Mani, S. Haque, A.J. Gnanaprakasam, O. Ege and N. Mlaiki. The study of bicomplex-valued controlled metric spaces with applications to fractional differential equations. Mathematics, 11(12):2742, 2023. https://doi.org/10.3390/math11122742
A.I. Perov. On the Cauchy problem for a system of ordinary differential equations. Priblijen. Metod Res. Dif. Urav. Kiev, 2:115–134, 1964. (in Russian)
I. Podlubny. Fractional Differential Equations. Academic Press, San Diego, 1999.
B. Precup and A. Viorel. Existence results for systems of nonlinear evolution equations. International Journal of Pure and Applied Mathematics, 47(2):199– 206, 2008. https://doi.org/10.1186/s13660-017-1400-5
A.S. Rafeeq, S.T.M. Thabet, M.O. Mohammed, I. Kedim and M. Vivas-Cortez. On Caputo-Hadamard fractional pantograph problem of two different orders with dirichlet boundary conditions. Alexandria Engineering Journal, 86:386– 398, 2024. https://doi.org/10.1016/j.aej.2023.11.081
I.A. Rus. Generalized Contractions and Applications. Cluj University Press, Cluj, 2001.
K. Shah, T. Abdeljawad and B. Abdalla. On a coupled system under coupled integral boundary conditions involving non-singular differential operator. AIMS Mathematics, 8(4):9890–9910, 2023. https://doi.org/10.3934/math.2023500
K. Shah, R. Amin, G. Ali, N. Mlaiki and T. Abdeljawad. Algorithm for the solution of nonlinear variable-order pantograph fractional integrodifferential equations using haar method. Fractals, 30(8):2240225, 2022. https://doi.org/10.1142/S0218348X22402253
S.T. Sutar and K.D. Kucche. On nonlinear hybrid fractional differential equations with Atangana-Baleanu-Caputo derivative. Chaos, Solitons & Fractals, 143:110557, 2021. https://doi.org/10.1016/j.chaos.2020.110557
S.T.M. Thabet, M. Vivas-Cortez and I. Kedim. Analytical study of ABC-fractional pantograph implicit differential equation with respect to another function. AIMS Mathematics, 8(10):23635–23654, 2023. https://doi.org/10.3934/math.20231202
C. Urs. Coupled fixed point theorems and applications to periodic boundary value problems. Miskolc Mathematical Notes, 14(1):323–333, 2013. https://doi.org/10.18514/MMN.2013.598
R.S. Varga. Matrix Iterative Analysis, Second Revised and Expanded Edition. Springer Series in Computational Mathematics 27, Springer-Verlag, Berlin, 2000. https://doi.org/10.1007/978-3-642-05156-2
L. Zhao and Y. Jiang. Existence and stability for a coupled hybrid system of fractional differential equations with Atangana-Baleanu-Caputo derivative. Journal of Mathematics, 2022:12 pages, 2022. https://doi.org/10.1155/2022/4741224
Y. Zhou. Basic Theory of Fractional Differential Equations, vol. 6. World Scientific, Singapore, 2014. https://doi.org/10.1142/9069
T. Abdeljawad, S.T.M. Thabet, I. Kedim and M. Vivas-Cortez. On a new structure of multi-term Hilfer fractional impulsive neutral Levin-Nohel integrodifferential system with variable time delay. AIMS Mathematics, 9(3):7372–7395, 2024. https://doi.org/10.3934/math.2024357
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. Advances in Difference Equations, 2021:55, 2021. https://doi.org/10.1186/s13662-021-03218-x
S.M. Ali, M.S. Abdo, B. Sontakke, K. Shah and T. Abdeljawad. New results on a coupled system for second-order pantograph equations with ABC−fractional derivatives. AIMS Mathematics, 7(10):19520–19538, 2022. https://doi.org/10.3934/math.20221071
S. Aljoudi. Existence and uniqueness results for coupled system of fractional differential equations with exponential kernel derivatives. AIMS Mathematics, 8(1):590–606, 2022. https://doi.org/10.3934/math.2023027
H. Alrabaiah, G. Ali, A. Ali, K. Shah and T. Abdeljawad. On existence and stability results for pantograph fractional boundary value problems. Fractals, 30(8):2240231, 2022. https://doi.org/10.1142/S0218348X22402319
A. Atangana and D. Baleanu. New fractional derivatives with non-local and non-singular kernel: theory and application to heat transfer model. Thermal Science, 20(2):763–769, 2016. https://doi.org/10.2298/TSCI160111018A
A. Boutiara, J. Alzabut, M. Ghaderi and S. Rezapour. On a coupled system of fractional (p,q)-differential equation with Lipschitzian matrix in generalized metric space. AIMS Mathematics, 8(1):1566–1591, 2022. https://doi.org/10.3934/math.2023079
A. Boutiara, M. Benbachir, M.K.A. Kaabar, F. Martínez, M.E. Samei and M. Kaplan. Explicit iteration and unbounded solutions for fractional q-difference equations with boundary conditions on an infinite interval. Journal of Inequalities and Applications, 2022:29, 2022. https://doi.org/10.1186/s13660-022-02764-6
A. Boutiara, M. Benbachir and M. Lotayif. Existence solutions for a nonlinear Langevin fractional q-difference system in Banach space. Progress in Fractional Differentiation & Applications, 8(4):485–494, 2022. https://doi.org/10.18576/pfda/080403
A. Boutiara, S. Etemad, J. Alzabut, A. Hussain, M. Subramanian and S. Rezapour. On a nonlinear sequential four-point fractional q-difference equation involving q-integral operators in boundary conditions along with stability criteria. Advances in Difference Equations, 2021:367, 2021. https://doi.org/10.1186/s13662-021-03525-3
B.C. Dhage. On a fixed point theorem in Banach algebras with applications. Applied Mathematics Letters, 18(3):273–280, 2005. https://doi.org/10.1016/j.aml.2003.10.014
M. Houas, K. Kaushik, A. Kumar, A. Khan and T. Abdeljawad. Existence and stability results of pantograph equation with three sequential fractional derivatives. AIMS Mathematics, 8(3):5216–5232, 2022. https://doi.org/10.3934/math.2023262
K.D. Kucche and S.T. Sutar. Analysis of nonlinear fractional differential equations involving Atangana-Baleanu-Caputo derivative. Chaos, Solitons & Fractals, 143:110556, 2021. https://doi.org/10.1016/j.chaos.2020.110556
G. Mani, A.J. Gnanaprakasam, O. Ege, A. Aloqaily and N. Mlaiki. Fixed point results in c∗-algebra-valued partial b-metric spaces with related application. Mathematics, 11(5):1158, 2023. https://doi.org/10.3390/math11051158
G. Mani, A.J. Gnanaprakasam, L. Guran, R. George and Z.D. Mitrović. Some results in fuzzy b-metric space with b-triangular property and applications to fredholm integral equations and dynamic programming. Mathematics, 11(19):4101, 2023. https://doi.org/10.3390/math11194101
G. Mani, S. Haque, A.J. Gnanaprakasam, O. Ege and N. Mlaiki. The study of bicomplex-valued controlled metric spaces with applications to fractional differential equations. Mathematics, 11(12):2742, 2023. https://doi.org/10.3390/math11122742
A.I. Perov. On the Cauchy problem for a system of ordinary differential equations. Priblijen. Metod Res. Dif. Urav. Kiev, 2:115–134, 1964. (in Russian)
I. Podlubny. Fractional Differential Equations. Academic Press, San Diego, 1999.
B. Precup and A. Viorel. Existence results for systems of nonlinear evolution equations. International Journal of Pure and Applied Mathematics, 47(2):199– 206, 2008. https://doi.org/10.1186/s13660-017-1400-5
A.S. Rafeeq, S.T.M. Thabet, M.O. Mohammed, I. Kedim and M. Vivas-Cortez. On Caputo-Hadamard fractional pantograph problem of two different orders with dirichlet boundary conditions. Alexandria Engineering Journal, 86:386– 398, 2024. https://doi.org/10.1016/j.aej.2023.11.081
I.A. Rus. Generalized Contractions and Applications. Cluj University Press, Cluj, 2001.
K. Shah, T. Abdeljawad and B. Abdalla. On a coupled system under coupled integral boundary conditions involving non-singular differential operator. AIMS Mathematics, 8(4):9890–9910, 2023. https://doi.org/10.3934/math.2023500
K. Shah, R. Amin, G. Ali, N. Mlaiki and T. Abdeljawad. Algorithm for the solution of nonlinear variable-order pantograph fractional integrodifferential equations using haar method. Fractals, 30(8):2240225, 2022. https://doi.org/10.1142/S0218348X22402253
S.T. Sutar and K.D. Kucche. On nonlinear hybrid fractional differential equations with Atangana-Baleanu-Caputo derivative. Chaos, Solitons & Fractals, 143:110557, 2021. https://doi.org/10.1016/j.chaos.2020.110557
S.T.M. Thabet, M. Vivas-Cortez and I. Kedim. Analytical study of ABC-fractional pantograph implicit differential equation with respect to another function. AIMS Mathematics, 8(10):23635–23654, 2023. https://doi.org/10.3934/math.20231202
C. Urs. Coupled fixed point theorems and applications to periodic boundary value problems. Miskolc Mathematical Notes, 14(1):323–333, 2013. https://doi.org/10.18514/MMN.2013.598
R.S. Varga. Matrix Iterative Analysis, Second Revised and Expanded Edition. Springer Series in Computational Mathematics 27, Springer-Verlag, Berlin, 2000. https://doi.org/10.1007/978-3-642-05156-2
L. Zhao and Y. Jiang. Existence and stability for a coupled hybrid system of fractional differential equations with Atangana-Baleanu-Caputo derivative. Journal of Mathematics, 2022:12 pages, 2022. https://doi.org/10.1155/2022/4741224
Y. Zhou. Basic Theory of Fractional Differential Equations, vol. 6. World Scientific, Singapore, 2014. https://doi.org/10.1142/9069