In this paper, we investigate and obtain a new discrete $ q $-fractional version of the Gronwall inequality. As applications, we consider the existence and uniqueness of the solution of $ q $-fractional damped difference systems with time delay. Moreover, we formulate the novel sufficient conditions such that the $ q $-fractional damped difference delayed systems is finite time stable. Our result extend the main results of the paper by Abdeljawad et al. [A generalized $ q $-fractional Gronwall inequality and its applications to nonlinear delay $ q $-fractional difference systems, J.Inequal. Appl. 2016,240].
Citation: Jingfeng Wang, Chuanzhi Bai. Finite-time stability of $ q $-fractional damped difference systems with time delay[J]. AIMS Mathematics, 2021, 6(11): 12011-12027. doi: 10.3934/math.2021696
In this paper, we investigate and obtain a new discrete $ q $-fractional version of the Gronwall inequality. As applications, we consider the existence and uniqueness of the solution of $ q $-fractional damped difference systems with time delay. Moreover, we formulate the novel sufficient conditions such that the $ q $-fractional damped difference delayed systems is finite time stable. Our result extend the main results of the paper by Abdeljawad et al. [A generalized $ q $-fractional Gronwall inequality and its applications to nonlinear delay $ q $-fractional difference systems, J.Inequal. Appl. 2016,240].
| [1] | F. M. Atici, P. W. Eloe, Initial value problems in discrete fractional calculus, Proc. Am. Math. Soc., 137 (2009), 981–989. |
| [2] |
G. A. Anastassiou, Nabla discrete fractional calculus and nabla inequalities, Math. Comput. Model., 51 (2010), 562–571. doi: 10.1016/j.mcm.2009.11.006
|
| [3] |
C. S. Goodrich, Continuity of solutions to discrete fractional initial value problems, Comput. Math. Appl., 59 (2010), 3489–3499. doi: 10.1016/j.camwa.2010.03.040
|
| [4] |
N. R. O. Bastos, R. A. C. Ferreira, D. F. M. Torres, Necessary optimality conditions for fractional difference problems of the calculus of variations, Discrete Contin. Dyn. Syst., Ser. A, 29 (2011), 417–437. doi: 10.3934/dcds.2011.29.417
|
| [5] |
R. Floreanini, L. Vinet, Quantum symmetries of q-difference equations, J. Math. Phys., 36 (1995), 3134–3156. doi: 10.1063/1.531017
|
| [6] | M. Marin, On existence and uniqueness in thermoelasticity of micropolar bodies, C. R. Acad. Sci. Paris, Ser. Ⅱ, 321 (1995), 475–480. |
| [7] |
R. Finkelstein, E. Marcus, Transformation theory of the q-oscillator, J. Math. Phys., 36 (1995), 2652–2672. doi: 10.1063/1.531057
|
| [8] |
M. Marin, Lagrange identity method for microstretch thermoelastic materials, J. Math. Anal. Appl., 363 (2010), 275–286. doi: 10.1016/j.jmaa.2009.08.045
|
| [9] | T. Ernst, A Comprehensive Treatment of q-Calculus, Birkh$\ddot{a}$user, Basel, 2012. |
| [10] |
F. Jarad, T. Abdeljawad, D. Baleanu, Stability of q-fractional non-autonomous systems, Nonlinear Anal., Real World Appl., 14 (2013), 780–784. doi: 10.1016/j.nonrwa.2012.08.001
|
| [11] |
T. Abdeljawad, D. Baleanu, Caputo $q$-fractional initial value problems and a q-analogue Mittag-Leffler function, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), 4682–4688. doi: 10.1016/j.cnsns.2011.01.026
|
| [12] | Z. S. I. Mansour, Linear sequential $q$-difference equations of fractional order, Fract. Calc. Appl. Anal., 12 (2009), 159–178. |
| [13] |
X. Li, Z. Han, X. Li, Boundary value problems of fractional q-difference Schr$\ddot{o}$dinger equations, Appl. Math. Lett., 46 (2015), 100–105. doi: 10.1016/j.aml.2015.02.013
|
| [14] |
J. Mao, Z. Zhao, C. Wang, The unique iterative positive solution of fractional boundary value problem with q-difference, Appl. Math. Lett., 100 (2020), 106002. doi: 10.1016/j.aml.2019.106002
|
| [15] |
Y. Liang, H. Yang, H. Li, Existence of positive solutions for the fractional $q$-difference boundary value problem, Adv. Differ. Equ., 2020 (2020), 1–11. doi: 10.1186/s13662-019-2438-0
|
| [16] | M. H. Annaby, Z. S. Mansour, $q$-fractional Calculus and Equations. Lecture Notes in Mathematics, vol. 2056, Springer, Heidelberg, 2012. |
| [17] |
T. Abdeljawad, J. Alzabut, D. Baleanu, A generalized $q$-fractional Gronwall inequality and its applications to nonlinear delay $q$-fractional difference systems, J. Inequal. Appl., 2016 (2016), 240. doi: 10.1186/s13660-016-1181-2
|
| [18] | T. Abdeljawad, J. Alzabut, The $q$-fractional analogue for Gronwall-type inequality, J. Funct. Spaces, 2013 (2013), 543839. |
| [19] |
F. Du, B. Jia, A generalized fractional $(q, h)$-Gronwall inequality and its applications to nonlinear fractional delay $(q, h)$-difference systems, Math. Methods Appl. Sci., 44 (2021), 10513–10529. doi: 10.1002/mma.7426
|
| [20] |
J. Sheng, W. Jiang, Existence and uniqueness of the solution of fractional damped dynamical systems, Adv. Differ. Equ., 2017 (2017), 16. doi: 10.1186/s13662-016-1049-2
|
| [21] |
M. P. Lazarevic, A. M. Spasic, Finite-time stability analysis of fractional order time-delay systems: Gronwall approach, Math. Comput. Model., 49 (2009), 475–481. doi: 10.1016/j.mcm.2008.09.011
|
| [22] |
V. N. Phat, N. T. Thanh, New criteria for finite-time stability of nonlinear fractional-order delay systems: A Gronwall inequality approach, Appl. Math. Lett., 83 (2018), 169–175. doi: 10.1016/j.aml.2018.03.023
|
| [23] |
R. Wu, Y. Lu, L. Chen, Finite-time stability of fractional delayed neural networks, Neurocomputing, 149 (2015), 700–707. doi: 10.1016/j.neucom.2014.07.060
|
| [24] | M. Li, J. Wang, Exploring delayed Mittag-Leffler type matrix functions to study finite time stability of fractional delay differential equations, Appl. Math. Comput., 24 (2018), 254–265. |
| [25] | F. Du, J. G. Lu, Finite-time stability of neutral fractional order time delay systems with Lipschitz nonlinearities, Appl. Math. Comput., 375 (2020), 125079. |
| [26] |
F. Du, B. Jia, Finite-time stability of a class of nonlinear fractional delay difference systems, Appl. Math. Lett., 98 (2019), 233–239. doi: 10.1016/j.aml.2019.06.017
|
| [27] |
F. Du, B. Jia, Finite time stability of fractional delay difference systems: A discrete delayed Mittag-Leffler matrix function approach, Chaos, Solitons and Fractals, 141 (2020), 110430. doi: 10.1016/j.chaos.2020.110430
|
| [28] |
K. Ma, S. Sun, Finite-time stability of linear fractional time-delay $q$-difference dynamical system, J. Appl. Math. Comput., 57 (2018), 591–604. doi: 10.1007/s12190-017-1123-2
|
| [29] |
P. M. Rajković, S. D. Marinković, M. S. Stanković, Fractional integrals and derivatives in q-calculus, Appl. Anal. Discr. Math., 1 (2007), 311–323. doi: 10.2298/AADM0701072C
|
| [30] |
M. Mansour, An asymptotic expansion of the $q$-gamma function $\Gamma_q (x)$, J. Nonlinear Math. Phys., 13 (2006), 479–483. doi: 10.2991/jnmp.2006.13.4.2
|
| [31] | R. A. Adams, C. Essex, Calculus A Complete Course, Seventh Edition, Pearson Canada, Toronto, 2010. |
| [32] |
N. Phuong, F. Sakar, S. Etemad, S. Rezapour, A novel fractional structure of a multi-order quantum multi-integro-differential problem, Adv. Differ. Equ., 2020 (2020), 633. doi: 10.1186/s13662-020-03092-z
|
| [33] |
R.Butt, T. Abdeljawad, M. Alqudah, M. Rehman, Ulam stability of Caputo q-fractional delay difference equation: $q$-fractional Gronwall inequality approach, J. Inequal. Appl., 2019 (2019), 305. doi: 10.1186/s13660-019-2257-6
|
| [34] |
L. Franco-Péreza, G. Fernández-Anaya, L. A. Quezada-Téllez, On stability of nonlinear nonautonomous discrete fractional Caputo systems, J. Math. Anal. Appl., 487 (2020), 124021. doi: 10.1016/j.jmaa.2020.124021
|
Jingfeng Wang, Chuanzhi Bai. Finite-time stability of $ q $-fractional damped difference systems with time delay[J]. AIMS Mathematics, 2021, 6(11): 12011-12027. doi: 10.3934/math.2021696
DownLoad: