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

[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.