On non-instantaneous impulsive fractional differential equations and their equivalent integral equations. (English) Zbl 1485.34035
Summary: Real-world processes that display non-local behaviours or interactions, and that are subject to external impulses over non-zero periods, can potentially be modelled using non-instantaneous impulsive fractional differential equations or systems. These have been the subject of many recent papers, which rely on re-formulating fractional differential equations in terms of integral equations, in order to prove results such as existence, uniqueness, and stability. However, specifically in the non-instantaneous impulsive case, some of the existing papers contain invalid re-formulations of the problem, based on a misunderstanding of how fractional operators behave. In this work, we highlight the correct ways of writing non-instantaneous impulsive fractional differential equations as equivalent integral equations, considering several different cases according to the lower limits of the integro-differential operators involved.
MSC:
| 34A08 | Fractional ordinary differential equations |
| 34A37 | Ordinary differential equations with impulses |
| 45D05 | Volterra integral equations |
Keywords:
Caputo fractional derivative; fractional differential equations; fractional integrals; impulsive differential equations; non-instantaneous impulses; Volterra integral equationsReferences:
| [1] | BaleanuD, DiethelmK, ScalasE, TrujilloJJ. Fractional Calculus Models and Numerical Methods. Series on Complexity, Nonlinearity and Chaos. Singapore: World Scientific; 2012. · Zbl 1248.26011 |
| [2] | DiethelmK. The Analysis of Fractional Differential Equations: An Application‐Oriented Exposition using Differential Operators of Caputo Type. Heidelberg: Springer; 2010. · Zbl 1215.34001 |
| [3] | HilferR. Application of Fractional Calculus in Physics. Singapore: World Scientific; 2000. · Zbl 0998.26002 |
| [4] | KilbasAA, SrivastavaHM, TrujilloJJ. Theory and Applications of Fractional Differential Equations: Elsevier Science B.V.; 2006. · Zbl 1092.45003 |
| [5] | PodlubnyI. Fractional Differential Equations. Math. Sci. Eng., vol. 198, 1999. · Zbl 0924.34008 |
| [6] | FernandezA. The Lerch zeta function as a fractional derivative. Banach Center Publ. 2019;118:113‐124. · Zbl 1457.11120 |
| [7] | GuarigliaE. Riemann zeta fractional derivative—functional equation and link with primes. Adv Differ Equ. 2019;2019:261. · Zbl 1459.26011 |
| [8] | KeiperJB. Fractional calculus and its relationship to Riemann’s zeta function. MSc thesis: Ohio State University; 1975. (37 pages). |
| [9] | FernandezA, BouzouinaC. Fractionalisation of complex d‐bar derivatives. Complex Var Elliptic Equ. 2021;66(3):437‐475. · Zbl 1462.30006 |
| [10] | LiC, DaoX, GuoP. Fractional derivatives in complex planes. Nonlin Anal Theor Meth Appl. 2009;71(5‐6):1857‐1869. · Zbl 1173.26305 |
| [11] | AbbasMI, RagusaMA. On the hybrid fractional differential equations with fractional proportional derivatives of a function with respect to a certain function. Symmetry. 2021;13(2):264. |
| [12] | FahadHM, FernandezA. Operational calculus for Caputo fractional calculus with respect to functions and the associated fractional differential equations. App Math Comput. 2021;409:126400. · Zbl 1510.44008 |
| [13] | RestrepoJE, SuraganD. Oscillatory solutions of fractional integro‐differential equations II. Math Meth Appl Sci. 2021;44(8):7262‐7274. · Zbl 1473.45013 |
| [14] | LakshmikanthamV, BainovDD, SimeonovPS. Theory of Impulsive Differential Equations. Singapore: World Scientific; 1989. · Zbl 0719.34002 |
| [15] | O’ReganD. Impulsive Differential Equations. In: D.O’Regan (ed.), ed. Existence Theory for Nonlinear Ordinary Differential Equations. Dordrecht: Springer Science and Business Media; 1997:174‐185. · Zbl 1077.34505 |
| [16] | AgarwalR, HristovaS, O’ReganD. A survey of Lyapunov functions, stability and impulsive Caputo fractional differential equations. Frac Calc Appl Anal. 2016;19(2):290‐318. · Zbl 1343.34006 |
| [17] | WangJ, Fec̆kanM, ZhouY. A survey on impulsive fractional differential equations. Frac Calc Appl Anal. 2016;19(4):806‐831. · Zbl 1344.35169 |
| [18] | HernándezE, O’ReganD. On a new class of abstract impulsive differential equations. Proc Amer Math Soc. 2013;141(5):1641‐1649. · Zbl 1266.34101 |
| [19] | AhmadB, SivasundaramS. Existence results for nonlinear impulsive hybrid boundary value problems involving fractional differential equations. Nonlin Anal Hybrid Syst. 2009;3(3):251‐258. · Zbl 1193.34056 |
| [20] | FuX, BaoX. Some existence results for nonlinear fractional differential equations with impulsive and fractional integral boundary conditions. Adv Differ Equ. 2014;2014:129. · Zbl 1417.34050 |
| [21] | MardanovMJ, MahmudovNI, SharifovYA. Existence and uniqueness theorems for impulsive fractional differential equations with two-point and integral boundary conditions. Sci World J. 2014;2014:918730. |
| [22] | ZadaA, AlzabutJ, WaheedH, PopaI‐L. Ulam-Hyers stability of impulsive integrodifferential equations with Riemann-Liouville boundary conditions. Adv Differ Equ. 2020;2020:64. · Zbl 1487.34048 |
| [23] | WangJR, LiX. Periodic BVP for integer/fractional order nonlinear differential equations with non‐instantaneous impulses. J Appl Math Comput. 2014;46:321‐334. · Zbl 1296.34036 |
| [24] | WangJR, ZhouY, LinZ. On a new class of impulsive fractional differential equations. Appl Math Comput. 2014;242:649‐657. · Zbl 1334.34022 |
| [25] | LiP, XuC. Boundary Value Problems of Fractional Order Differential Equation with Integral Boundary Conditions and Not Instantaneous Impulses. J Func Spaces. 2015;2015:954925. · Zbl 1325.34013 |
| [26] | AgarwalR, HristovaS, O’ReganD. Non‐instantaneous impulses in Caputo fractional differential equations. Frac Calc Appl Anal. 2017;20(3):595‐622. · Zbl 1370.34008 |
| [27] | AgarwalR, HristovaS, O’ReganD. Caputo fractional differential equations with non‐instantaneous impulses and strict stability by Lyapunov functions. Filomat. 2017;31(16):5217‐5239. · Zbl 1499.34018 |
| [28] | AgarwalR, HristovaS, O’ReganD. Lipschitz stability for non‐instantaneous impulsive Caputo fractional differential equations with state dependent delays. Axioms. 2019;8:4. · Zbl 1432.34100 |
| [29] | AgarwalR, HristovaS, O’ReganD. Ulam type stability for non‐instantaneous impulsive Caputo fractional differential equations with finite state dependent delay. Georgian Math J. 2020. https://doi.org/10.1515/gmj‐2020‐2061 · Zbl 1476.34158 · doi:10.1515/gmj‐2020‐2061 |
| [30] | AgarwalR, O’ReganD, HristovaS. Stability of Caputo fractional differential equations with non‐instantaneous impulses. Commun Appl Anal. 2016;20:149‐174. · Zbl 1353.34005 |
| [31] | ZadaA, AliS, LiY. Ulam‐type stability for a class of implicit fractional differential equations with non‐instantaneous integral impulses and boundary condition. Adv Differ Equ. 2017;2017:317. · Zbl 1444.34083 |
| [32] | ZadaA, AliS. Stability analysis of multi‐point boundary value problem for sequential fractional differential equations with non‐instantaneous impulses. Int J Nonlinear Sci Numer Simul. 2018;19(7):763‐774. · Zbl 1461.34014 |
| [33] | ZadaA, AliS. Stability of Integral Caputo‐Type Boundary Value Problem with Noninstantaneous Impulses. Int J Appl Comp Math. 2019;5:55. · Zbl 1419.34045 |
| [34] | SamkoSG, KilbasAA, MarichevOI. Fractional Integrals and Derivatives. Theory and Applications. Yverdon: Gordon and Breach; 1993. · Zbl 0818.26003 |
| [35] | AgarwalR, HristovaS, O’ReganD. Non‐Instantaneous Impulses in Differential Equations Springer; 2017. · Zbl 1426.34001 |
| [36] | Fec̆kanM, ZhouY, WangJR. On the concept and existence of solution for impulsive fractional differential equations. Commun Nonlin Sci Numer Simul. 2012;17:3050‐3060. · Zbl 1252.35277 |
| [37] | YuX. Existence and β‐Ulam‐Hyers stability for a class of fractional differential equations with non‐instantaneous impulses. Adv Differ Equ. 2015;2015:104. · Zbl 1348.34030 |
| [38] | AngurajA, KanjanadeviS. Existence of mild solutions of abstract fractional differential equations with non‐instantaneous impulsive conditions. J Stat Sci Appl. 2016;4:53‐64. |
| [39] | GautamGR, DabasJ. Existence of mild solutions for impulsive fractional functional differential equations of order α ∈ (1, 2). In: Differential and Difference Equations with Applications, Proc. of ICDDEA. Amadora, Portugal; 2015:141‐148. · Zbl 1360.34162 |
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