Stability and existence results for a system of fractional differential equations via Atangana-Baleanu derivative with \(\phi_{p}\)-Laplacian operator

Volume 27, Issue 2, pp 184--195 http://dx.doi.org/10.22436/jmcs.027.02.08

Publication Date: April 13, 2022 Submission Date: July 26, 2021 Revision Date: October 02, 2021 Accteptance Date: February 15, 2022

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Authors

T. Q. S. Abdullah - School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, P. R. China. - Department of Mathematics, Faculty of Applied Sciences, Thamar University, Dhamar, Yemen. H. Xiao - School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, P. R. China. G. Huang - School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, P. R. China. W. Al-Sadi - School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, P. R. China.

Abstract

This study focused on the existence and uniqueness(EU) and stability of the solution for a system of fractional differential equations(FDEs) via Atangana-Baleanu derivative in the sense of Caputo (ABC) with \(\phi_{p}\)-Laplacian operator. Green function \( \mathcal{G}^{\eth}(t,s)\), \(m<\eth<m+1\), \(m\geq4\) used for converting the suggested problem to an integral equation. Guo-Krasnoselskii theorem used for proving the EU of solution for the suggested problem. The stability of the solution was derived by Hyers-Ulam stability method(HUS). One illustrative example is used for manifesting the results.

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Keywords

• ABC-Derivative
• Riemann-Liuoville fractional derivative
• Green function
• existence and uniqueness
• Hyers-Ulam stability

MSC

•  34B82
•  26A33
•  45N05

References

• [1] T. Abdeljawad, A Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel, J. Inequal. Appl., 2017 (2017), 11 pages
  • View Article
  • MathSciNet


• [2] B. Ahmed, J. J. Nieto, Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions, Bound. Value Probl., 2011 (2011), 9 pages
  • View Article
  • MathSciNet


• [3] M. Almushaira, F. Liu, Fourth-order time-stepping compact finite difference method for multi-dimensional space-fractional coupled nonlinear Schrodinger equations, SN Partial Differ. Equ. Appl., 1 (2020), 29 pages
  • View Article
  • Google Scholar


• [4] M. Almushaira, F. Liu, A Fast Conservative Scheme for the Space Fractional Nonlinear Schrödinger Equation with Wave Operator, J. Math. Study, 54 (2021), 407--426
  • MathSciNet
  • Google Scholar


• [5] W. Al-Sadi, A. Alkhazan, T. Q. S. Abdullah, M. Al-Soswa, Stability and existence the solution for a coupled system of hybrid fractional differential equation with uniqueness, Arab. J. Basic Appl. Sci., 28 (2021), 340--350
  • View Article
  • Google Scholar


• [6] W. Al-Sadi, M. Hussein, T. Q. S. Abdullah, Existence and stability criterion for the results of fractional order $\Phi$p-Laplacian operator boundary value problem, Comput. Methods Differ. Equ., 9 (2021), 1042--1058
  • View Article
  • Google Scholar


• [7] W. Al-Sadi, H. Zhenyou, A. Alkhazzan, Existence and stability of a positive solution for nonlinear hybrid fractional differential equations with singularity, J. Taibah Univ. Sci., 13 (2019), 951--960
  • View Article
  • Google Scholar


• [8] A. Atangana, Modelling the spread of COVID-19 with new fractal-fractional operators: Can the lockdown save mankind before vaccination?, Chaos Solitons Fractals, 136 (2020), 38 pages
  • View Article
  • MathSciNet


• [9] A. Atangana, D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: theory and application to heat transfer model, arXiv, 2016 (2016), 8 pages
  • View Article
  • Google Scholar


• [10] K. Balachandran, J. Y. Park, J. J. Trujillo, Controllability of nonlinear fractional dynamical systems, Nonlinear Anal., 75 (2012), 1919--1926
  • View Article
  • MathSciNet


• [11] D. Baleanu, B. Shiri, Collocation methods for fractional differential equations involving non-singular kernel, Chaos Solitons Fractals, 116 (2018), 136--145
  • View Article
  • MathSciNet


• [12] A. Boukhouima, K. Hattaf, E. M. Lotfi, M. Mahrouf, D. F. M. Torres, N. Yousfi, Lyapunov functions for fractional-order systems in biology: Methods and applications, Chaos Solitons Fractals, 140 (2020), 9 pages
  • View Article
  • MathSciNet


• [13] M. Caputo, Linear modles of dissipation whose $Q$ is almost frequenty independent--II, J. R. Astr. Soc., 13 (1967), 529--539
  • View Article
  • MathSciNet
  • MATH


• [14] M. Caputo, M. Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Differ. Appl., 1 (2015), 73--85
  • View Article
  • Google Scholar


• [15] A. M. A. El-Sayed, S. Z. Rida, Y. A. Gaber, Dynamical of curative and preventive treatments in a two-stage plant disease model of fractional order, Chaos Solitons Fractals, 137 (2020), 11 pages
  • View Article
  • MathSciNet


• [16] O. Guner, A. Bekir, The Exp-function method for solving nonlinear space--time fractional differential equations in mathematical physics, J. Assoc. Arab Univ. Basic Appl. Sci., 24 (2017), 277--282
  • View Article
  • Google Scholar


• [17] D. J. Guo, V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston (2014)
  • View Article


• [18] R. Hilfer, Application of fractional Calculus in Physics, World Scientific Publishing Co., River Edge (2000)
  • View Article
  • MathSciNet
  • Google Scholar


• [19] H. Jafari, D. Baleanu, H. Khan, R. A. Khan, A. Khan, Existence criterion for the solutions of fractional order $p$-Laplacian boundary value problems, Bound. Value Probl., 2015 (2015), 10 pages
  • View Article
  • MathSciNet


• [20] F. Jarad, T. Abdeljawad, Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative, Chaos Solitons Fractals, 117 (2018), 16--20
  • View Article
  • MathSciNet


• [21] H. Khan, W. Chen, H. G. Sun, Analysis of positive solution and Hyers-Ulam stability for a class of singular fractional differential equations with $p$-Laplacian in Banach space, Math. Methods Appl. Sci., 41 (2018), 3430--3440
  • View Article
  • MathSciNet


• [22] H. Khan, F. Jarad, T. Abdeljawad, A. Khan, A singular ABC-fractional differential equation with $p$-Laplacian operator, Chaos Solitons Fractals, 129 (2019), 56--61
  • View Article
  • MathSciNet


• [23] A. Khan, H. Khan, J. Gomez-Aguilar, T. Abdeljawad,, Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel, Chaos Solitons Fractals, 127 (2019), 422--427
  • View Article
  • MathSciNet


• [24] H. Khan, Y. J. Li, W. Chen, D. Baleanu, A. Khan, Existence theorems and Hyers-Ulam stability for a coupled system of fractional differential equations with $p$-Laplacian operator, Bound. Value Probl., 2017 (2017), 16 pages
  • View Article
  • MathSciNet


• [25] Y. H. Li, Existence of positive solutions for fractional differential equation involving integral boundary conditions with $p$-Laplacian operator, Adv. Difference Equ., 2017 (2017), 11 pages
  • View Article
  • MathSciNet


• [26] D. F. Luo, Q. x. Zhu, Z. G. Luo, An averaging principle for stochastic fractional differential equations with time-delays, Appl. Math. Lett., 105 (2020), 8 pages
  • View Article
  • MathSciNet


• [27] J.-H. Mai, Devaney's chaos implies existence of $s$--scrambled sets, Proc. Amer. Math. Soc., 132 (2004), 2761--2767
  • View Article
  • MathSciNet
  • MATH


• [28] D. D. Min, L. S. Liu, Y. H. Wu, Uniqueness of positive solutions for the singular fractional differential equations involving integral boundary value conditions, Bound. Value Probl., 2018 (2018), 18 pages
  • View Article
  • MathSciNet


• [29] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Yverdon (1993)
  • View Article
  • MathSciNet
  • MATH


• [30] M. Saqib, I. Khan, S. Shafie, Application of fractional differential equations to heat transfer in hybrid nanofluid: modeling and solution via integral transforms, Adv. Difference Equ., 2019 (2019), 18 pages
  • View Article
  • MathSciNet


• [31] S. Q. Zhang, The existence of a positive solution for a nonlinear fractional differential equation, J. Math. Anal. Appl., 252 (2000), 804--812
  • View Article
  • MathSciNet