EXISTENCE AND STABILITY RESULTS FOR GENERALIZED FRACTIONAL DIFFERENTIAL EQUATIONS

doi:10.1007/s10473-020-0110-3

数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (1): 141-154.doi: 10.1007/s10473-020-0110-3

EXISTENCE AND STABILITY RESULTS FOR GENERALIZED FRACTIONAL DIFFERENTIAL EQUATIONS

A. BEN MAKHLOUF^1,2, D. BOUCENNA³, M. A. HAMMAMI²

1. Department of Mathematics, College of Sciences, Jouf University, Aljouf, Saudi Arabia;
2. Department of Mathematics, Faculty of Sciences of Sfax, Route Soukra, BP 1171, 3000 Sfax, Tunisia 3. Higher School for Professors of Technological Education, Skikda, Algeria

收稿日期:2018-10-31 修回日期:2019-02-04 出版日期:2020-02-25 发布日期:2020-04-14
通讯作者: A. BEN MAKHLOUF E-mail:benmakhloufabdellatif@gmail.com

EXISTENCE AND STABILITY RESULTS FOR GENERALIZED FRACTIONAL DIFFERENTIAL EQUATIONS

A. BEN MAKHLOUF^1,2, D. BOUCENNA³, M. A. HAMMAMI²

1. Department of Mathematics, College of Sciences, Jouf University, Aljouf, Saudi Arabia;
2. Department of Mathematics, Faculty of Sciences of Sfax, Route Soukra, BP 1171, 3000 Sfax, Tunisia 3. Higher School for Professors of Technological Education, Skikda, Algeria

Received:2018-10-31 Revised:2019-02-04 Online:2020-02-25 Published:2020-04-14
Contact: A. BEN MAKHLOUF E-mail:benmakhloufabdellatif@gmail.com

摘要/Abstract

摘要： In this paper, a sufficient conditions to guarantee the existence and stability of solutions for generalized nonlinear fractional differential equations of order α (1 < α < 2) are given. The main results are obtained by using Krasnoselskii's fixed point theorem in a weighted Banach space. Two examples are given to demonstrate the validity of the proposed results.

关键词: nonlinear fractional differential equations, stability analysis, generalized fractional derivative, Krasnoselskii's fixed point theorem

Abstract: In this paper, a sufficient conditions to guarantee the existence and stability of solutions for generalized nonlinear fractional differential equations of order α (1 < α < 2) are given. The main results are obtained by using Krasnoselskii's fixed point theorem in a weighted Banach space. Two examples are given to demonstrate the validity of the proposed results.

Key words: nonlinear fractional differential equations, stability analysis, generalized fractional derivative, Krasnoselskii's fixed point theorem

中图分类号:

26A33

A. BEN MAKHLOUF, D. BOUCENNA, M. A. HAMMAMI. EXISTENCE AND STABILITY RESULTS FOR GENERALIZED FRACTIONAL DIFFERENTIAL EQUATIONS[J]. 数学物理学报(英文版), 2020, 40(1): 141-154.

A. BEN MAKHLOUF, D. BOUCENNA, M. A. HAMMAMI. EXISTENCE AND STABILITY RESULTS FOR GENERALIZED FRACTIONAL DIFFERENTIAL EQUATIONS[J]. Acta mathematica scientia,Series B, 2020, 40(1): 141-154.

参考文献

[1] Almeida R, Malinowska A B, Odzijewicz T. Fractional differential equations with dependence on the Caputo-Katugampola derivative. J Comput Nonlinear Dynam, 2016, 11:061017
[2] Baleanu D, Wu G C, Zeng S D. Chaos analysis and asymptotic stability of generalized Caputo fractional differential equations. Chaos, Solitons and Fractals, 2017, 102:99-105
[3] Ben Makhlouf A. Stability with respect to part of the variables of nonlinear Caputo fractional differential equations. Math Commun, 2018, 23:119-126
[4] Ben Makhlouf A, Nagy A M. Finite-time stability of linear Caputo-Katugampola fractional-order time delay systems. Asian Journal of Control, 2018, https://doi.org/10.1002/asjc.1880
[5] Boroomand A, Menhaj M B. Fractional-order Hopfeld neural networks. Lecture Notes in Computer Science, 2009, 5509:883-890
[6] Boucenna D, Ben Makhlouf A, Naifar O, Guezane-Lakoud A, Hammami M A. Linearized stability analysis of Caputo-Katugampola fractional-order nonlinear systems. J Nonlinear Funct Anal, 2018, 2018:Article 27
[7] Burov S, Barkai E. Fractional Langevin equation:overdamped, underdamped, and critical behaviors. Phys Rev E, 2008, 78(3):031112
[8] Corduneanu C. Integral Equations and Stability of Feedback Systems. New York, London:Academic Press, 1973
[9] Debnath L. Fractional integrals and fractional differential equations in fluid mechanics. Frac Calc Appl Anal, 2003, 6:119155
[10] Duarte-Mermoud M A, Aguila-Camacho N, Gallegos J A, Castro-Linares R. Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun Nonlinear Sci Numer Simul, 2015, 22(1/3):650-659
[11] Ge F, Kou C. Stability analysis by Krasnoselskii's fixed point theorem for nonlinear fractional differential equations. Appl Math Comput, 2015, 257:308316
[12] Hilfer R. Applications of Fractional Calculus in Physics. Singapore:World Science Publishing, 2000
[13] Katugampola U N. Existence and uniqueness results for a class of generalized fractional differential equations. arXiv:1411.5229, 2014
[14] Kilbas A A, Srivastava H M, Trujillo J J. Theory and Application of Fractional Differential Equations. New York:Elsevier, 2006
[15] Kou C, Zhou H, Yan Y. Existence of solutions of initial value problems for nonlinear fractional differential equations on the half-axis. Nonlinear Anal, 2011, 74:5975-5986
[16] Krasnoselskii M A. Some problems of nonlinear analysis. Amer Math Soc Transl, 1958, 10:345-409
[17] Laskin N. Fractional market dynamics. Phys A, 2000, 287(3/4):482-492
[18] Li Y, Chen Y Q, Podlubny I. Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica, 2009, 45(8):1965-1969
[19] Magin R. Fractional calculus in bioengineering. Critical Reviews in Biomedical Engineering, 2004, 32:195-377
[20] Matignon D. Stability result on fractional differential equations with applications to control processing//IMACS SMC Proc, Lille, France, 1996:963-968
[21] Naifar O, Ben Makhlouf A, Hammami M A. Comments on "Mittag-Leffler stability of fractional order nonlinear dynamic systems[Automatica, 2009, 45(8):1965-1969]". Automatica, 2017, 75:329
[22] Naifar O, Ben Makhlouf A, Hammami M A. Comments on Lyapunov stability theorem about fractional system without and with delay. Commun Nonlinear Sci Numer Simul, 2016, 30:360-361
[23] Naifar O, Ben Makhlouf A, Hammami M A, Chen L. Global practical mittag leffler stabilization by output feedback for a class of nonlinear fractional-order systems. Asian Journal of Control, 2018, 20:599-607
[24] Podlubny I. Fractional Differential Equations. New York:Academic Press, 1999
[25] Soczkiewicz E. Application of fractional calculus in the theory of viscoelasticity. Molecular and Quantum Acoustics, 200223:397-404
[26] Sun H, Abdelwahad A, Onaral B. Linear approximation of transfer function with a pole of fractional order. IEEE Trans Automat Contr, 1984, 29(5):441-444
[27] Tripathil D, Pandey S, Das S. Peristaltic flow of viscoelastic fluid with fractional Maxwell model through a channel. Applied Mathematics and Computation, 2010, 215:3645-3654

EXISTENCE AND STABILITY RESULTS FOR GENERALIZED FRACTIONAL DIFFERENTIAL EQUATIONS

EXISTENCE AND STABILITY RESULTS FOR GENERALIZED FRACTIONAL DIFFERENTIAL EQUATIONS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 3

Metrics

本文评价

推荐阅读 0

[1]	A. M. NAGY, N. H. SWEILAM. NUMERICAL SIMULATIONS FOR A VARIABLE ORDER FRACTIONAL CABLE EQUATION[J]. 数学物理学报(英文版), 2018, 38(2): 580-590.
[2]	Rakesh KUMAR, Anuj Kumar SHARMA, Kulbhushan AGNIHOTRI. STABILITY AND BIFURCATION ANALYSIS OF A DELAYED INNOVATION DIFFUSION MODEL[J]. 数学物理学报(英文版), 2018, 38(2): 709-732.
[3]	Dhananjay GOPAL, Mujahid ABBAS, Deepesh Kumar PATEL, Calogero VETRO. FIXED POINTS OF α-TYPE F-CONTRACTIVE MAPPINGS WITH AN APPLICATION TO NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION[J]. 数学物理学报(英文版), 2016, 36(3): 957-970.