International Journal of Nonlinear Analysis and Applications

On stability results for a nonlinear generalized fractional hybrid pantograph equation involving deformable derivative

Document Type : Research Paper

Authors

1 Acoustics and Civil Engineering Laboratory Djilali Bounaama University-Khemis, Miliana, Algeria

2 Department of Mathematics and Sciences, Prince Sultan University, 11586 Riyadh, Saudi Arabia

3 Department of Industrial Engineering,\ OST\.{I}M Technical University, Ankara 06374, Turkey

4 Department of Mathematics, Sacred Heart College (Autonomous), Tirupattur-635601, Tamil Nadu, India

5 Cyber Security and Digital Industrial Revolution Centre, National Defence University of Malaysia, Kem Sungai Besi, 57000, Kuala Lumpur, Malaysia

Abstract

The pantograph equation is a special type of delay differential equation with applications in quantum mechanics and electrodynamics. A generalized hybrid pantograph equation of fractional order involving deformable derivative is considered in this work to carry out the stability analysis. The existence of solutions is established by employing the measure of noncompactness and Darbo's fixed point theorem while the contraction mapping principle is used for proving the uniqueness of the solution.  The link between the right-hand term of the given equation and the order of the deformable derivative is established. The paper presents the results on Ulam-Hyers stability and the generalized Ulam-Hyers stability of the proposed equation. Numerical simulations are provided to demonstrate the performed theoretical analysis.

Keywords

[1] A. Aghajani, J. Banas and N. Sabzali, Some generalizations of Darbo fixed point theorem and applications, Bull. Belg. Math. Soc. Simon Stevin 20 (2013), 345–358.
[2] P. Ahuja, F. Zulfeqarr, and A. Ujlayan, Deformable fractional derivative and its applications, AIP Conf. Proc. AIP Publishing LLC, 1897 (2017), no. 1.
[3] J. Alzabut, A.G.M. Selvam, R.A. El-Nabulsi, D. Vignesh, and M.E. Samei, Asymptotic stability of nonlinear discrete fractional pantograph equations with non-local initial conditions, Symmetry 13 (2021), 473.
[4] J. Alzabut, A.G. M. Selvam, D. Vignesh, and Y. Gholami, Solvability and stability of nonlinear hybrid
Δ− difference equations of fractional order, Int. J. Nonlinear Sci. Numer. Simul. 2021, 000010151520210005. https://doi.org/10.1515/ijnsns-2021-0005.
[5] R. Arab, Some generalization of Darbo fixed point theorem and its application, Miskolc Math. Notes 18 (2017), no. 2, 595–610.
[6] K. Balachandran, S. Kiruthika, and J.J. Trujillo, Existence of solutions of nonlinear fractional pantograph equations, Acta Math. Sci. 33 (2013), no. 3, 712–720.
[7] J. Banaks and K. Goebel, Measures of noncompactness related to monotonicity, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, Inc, vol. 60, 1980.
[8] J. Banas and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Appl. Math., Marcel Dekker, New York, 1980.
[9] J. Bana´s and L. Olszowy, On a class of measures of noncompactness in Banach algebras and their application to nonlinear integral equations, J. Anal. Appl. 28 (2009), 475–498.
[10] M.A. Darwish and K. Sadarangani, Existence of solutions for hybrid fractional pantograph equations, Appl. Anal. Discrete Math. 9 (2015), 150–167.
[11] B.C. Dhage, Quadratic perturbation of periodic boundary value problems of second order ordinary differential equations, Differ. Equ. Appl. 2 (2010), no. 4, 465–486.
[12] M. Etefa, G.M. N’Guerekata, and M. Benchohra, Existence and uniqueness of solutions to impulsive fractional differential equations via the deformable derivative, Appl. Anal. (2021), 1-12. doi.org/10.1080/00036811.2021.1979224
[13] S. Harikrishnan, E.M. Elsayed, and K. Kanagarajan, Existence theory and Stability analysis of nonlinear neutral pantograph equations via Hilfer-Katugampola fractional derivative, J. Adv. Appl. Comput. Math. 7 (2020), 1–7.
[14] A. Iserles, On the generalized pantograph functional differential equation, Eur. J. Appl. Math. 4 (1993), no. 1, 1–38.
[15] E.T. Karimov, B. Lopez and K. Sadarangani, About the existence of solutions for a hybrid nonlinear generalized fractional pantograph equation, Fractional Differ. Cal. 6 (2016), no. 1, 95–110.
[16] R. Khalil, M.A. Horani, A. Yousef, and M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014), 65–70.
[17] B. Khaminsou, W. Sudsutad, C. Thaiprayoon, J. Alzabut, and S. Pleumpreedaporn, Analysis of impulsive boundary value pantograph problems via Caputo proportional fractional derivative under Mittag-Leffler functions, Fractal Fractional 5 (2021), no. 4, 251. 
[18] A.N. Kolmogorov and S.V. Fomin, Elements of Function Theory and Functional Analysis, Nauka, Moscow, Russia, 1981.
[19] D. Li and M.Z. Liu, Runge-Kutta methods for the multi-pantograph delay equation, Appl. Math. Comput. 163 (2005), 383–395.
[20] M.Z. Liu and D. Li, Properties of analytic solution and numerical solution of multi-pantograph equation, Appl. Math. Comput. 155 (2004), 853–871.
[21] A.M. Mathai and H.J. Haubold, An Introduction to Fractional Calculus, Mathematics Research Developments, Nova Science Publishers, New York, 2017.
[22] M. Mebrat and G.M. N’Gu`er`ekata, A Cauchy problem for some fractional differential equation via deformable derivatives, J. Nonlinear Evol. Equ. Appl. 2020 (2020), no. 4, 55–63.
[23] M. Mebrat and G.M. N’Gu`er`ekata, An existence result for some fractional-integro differential equations in Banach spaces via deformable derivative, J. Math. Ext. 16 (2022), no. 8, 1–19.
[24] J. Ockendon and A.B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. Royal Soc. A: Mathe. Phys. Engin. Sci. 322 (1971), 447–468.
[25] I. Podlubny, Geometric and physical interpretation of fractional integration and differentiation, Fractional Cal. Appl. Anal. 5 (2002), no. 4, 367–386.
[26] H. Rezazadeh, H. Aminikhah, and A. Refahi Sheikhani, Stability Analysis of Hilfer fractional systems, Math. Commun. 21 (2015), no. 1, 45–64.
[27] I.A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpath. J. Math. 26 (2010), 103–107.
[28] A.G.M. Selvam, J. Alzabut, D. Vignesh, J.M. Jonnalagadda, and K. Abodayeh, Existence and stability of nonlinear discrete fractional initial value problems with application to vibrating eardrum, Math. Biosci. Engin. 18 (2021), no. 4, 3907–3921.
[29] S.T.M. Thabet, S. Etemad, and S. Rezapour, On a coupled Caputo conformable system of pantograph problems, Turk. J. Math. 45(2021), 496–519.
[30] C. Thaiprayoon, W. Sudsutad, J. Alzabut, S. Etamed, and S. Rezapour, On the qualitative analysis of the fractional boundary value problem describing thermostat control model via ψ− Hilfer fractional operator, Adv. Differ. Equ. 2021 (2021), 201.
[31] C. Vanterler da, J. Sousa, and E. Capelas de Oliveira, A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties, Int. J. Anal. Appl. 16 (2018), no. 1, 83–96.
[32] D. Vivek, K. Kanagarajan, and S. Harikrishnan, Existence and uniqueness results for nonlinear neutralpantograph equations with generalized fractional derivative, J. Nonlinear Anal. Appl. 2 (2018), 151–157.
[33] A. Wongcharoen, S.K. Ntouyas, and J. Tariboon, Nonlocal boundary value problems for Hilfer type pantograph fractional differential equations and inclusions, Adv. Differ. Equ. 2020 (2020), 279.
[34] F. Zulfeqarr, A. Ujlayan and P. Ahuja, A new fractional derivative and its fractional integral with some applications, arXiv : 1705.00962v1; 2017.
International Journal of Nonlinear Analysis and Applications
Volume 14, Issue 8
August 2023
Pages 1-14
  • Receive Date: 18 March 2022
  • Accept Date: 14 June 2023