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Stability analysis of multi-point boundary conditions for fractional differential equation with non-instantaneous integral impulse


  • Received: 27 November 2022 Revised: 15 January 2023 Accepted: 20 January 2023 Published: 09 February 2023
  • This paper considers the stability of a fractional differential equation with multi-point boundary conditions and non-instantaneous integral impulse. Some sufficient conditions for the existence, uniqueness and at least one solution of the aforementioned equation are studied by using the Diaz-Margolis fixed point theorem. Secondly, the Ulam stability of the equation is also discussed. Lastly, we give one example to support our main results. It is worth pointing out that these two non-instantaneous integral impulse and multi-point boundary conditions factors are simultaneously considered in the fractional differential equations studied for the first time.

    Citation: Guodong Li, Ying Zhang, Yajuan Guan, Wenjie Li. Stability analysis of multi-point boundary conditions for fractional differential equation with non-instantaneous integral impulse[J]. Mathematical Biosciences and Engineering, 2023, 20(4): 7020-7041. doi: 10.3934/mbe.2023303

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  • This paper considers the stability of a fractional differential equation with multi-point boundary conditions and non-instantaneous integral impulse. Some sufficient conditions for the existence, uniqueness and at least one solution of the aforementioned equation are studied by using the Diaz-Margolis fixed point theorem. Secondly, the Ulam stability of the equation is also discussed. Lastly, we give one example to support our main results. It is worth pointing out that these two non-instantaneous integral impulse and multi-point boundary conditions factors are simultaneously considered in the fractional differential equations studied for the first time.



    We consider numerical invariants associated with polynomial identities of algebras over a field of characteristic zero. Given an algebra A, one can construct a sequence of non-negative integers {cn(A)},n=1,2,, called the codimensions of A, which is an important numerical characteristic of identical relations of A. In general, the sequence {cn(A)} grows faster than n!. However, there is a wide class of algebras with exponentially bounded codimension growth. This class includes all associative PI-algebras [2], all finite-dimensional algebras [2], Kac-Moody algebras [12], infinite-dimensional simple Lie algebras of Cartan type [9], and many others. If the sequence {cn(A)} is exponentially bounded then the following natural question arises: does the limit

    limnncn(A) (1.1)

    exist and what are its possible values? In case of existence, the limit (1.1) is called the PI-exponent of A, denoted as exp(A). At the end of 1980's, Amitsur conjectured that for any associative PI-algebra, the limit (1.1) exists and is a non-negative integer. Amitsur's conjecture was confirmed in [5,6]. Later, Amitsur's conjecture was also confirmed for finite-dimensional Lie and Jordan algebras [4,15]. Existence of exp(A) was also proved for all finite-dimensional simple algebras [8] and many others.

    Nevertheless, the answer to Amitsur's question in the general case is negative: a counterexample was presented in [14]. Namely, for any real α>1, an algebra Rα was constructed such that the lower limit of ncn(A) is equal to 1, whereas the upper limit is equal to α. It now looks natural to describe classes of algebras in which for any algebra A, its PI-exponent exp(A) exists. One of the candidates is the class of all finite-dimensional algebras. Another one is the class of so-called special Lie algebras. The next interesting class consists of unital algebras, it contains in particular, all algebras with an external unit. Given an algebra A, we denote by A the algebra obtained from A by adjoining the external unit. There is a number of papers where the existence of exp(A) has been proved, provided that exp(A) exists [11,16,17]. Moreover, in all these cases, exp(A)=exp(A)+1.

    The main goal of the present paper is to construct a series of unital algebras such that exp(A) does not exist, although the sequence {cn(A) is exponentially bounded (see Theorem 3.1 and Corollary 3.1 below). All details about polynomial identities and their numerical characteristics can be found in [1,3,7].

    Let A be an algebra over a field F and let F{X} be a free F-algebra with an infinite set X of free generators. The set Id(A)F{X} of all identities of A forms an ideal of F{X}. Denote by Pn=Pn(x1,,xn) the subspace of F{X} of all multilinear polynomials on x1,,xnX. Then PnId(A) is actually the set of all multilinear identities of A of degree n. An important numerical characteristic of Id(A) is the sequence of non-negative integers {cn(A)},n=1,2,, where

    cn(A)=dimPnPnId(A).

    If the sequence {cn(A)} is exponentially bounded, then the lower and the upper PI-exponents of A, defined as follows

    exp_(A)=lim infnncn(A),¯exp(A)=lim supnncn(A),

    are well-defined. An existence of ordinary PI-exponent (1.1) is equivalent to the equality exp_(A)=¯exp(A).

    In [14], an algebra R=R(α) such that exp_(R)=1,¯exp(R)=α, was constructed for any real α>0. Slightly modifying the construction from [14], we want to get for any real α>2, an algebra Rα with exp_(Rα)=2 and α¯exp(R)α+1.

    Clearly, polynomial identities of A strongly depend on the identities of A. In particular, we make the following observation. Note that if f=f(x1,,xn) is a multilinear polynomial from F{X} then f(1+x1,,1+xn)F{X} is the sum

    f=fi1,,ik,{i1,,ik}{1,,n},0kn, (2.1)

    where fi1,,ik is a multilinear polynomial on xi1,,xik obtained from f by replacing all xj,ji1,,ik with 1.

    Remark 2.1. A multilinear polynomial f=f(x1,,xn) is an identity of A if and only if all of its components fi1,,ik on the left hand side of (2.1) are identities of A.

    The next statement easily follows from Remark 2.1.

    Remark 2.2. Suppose that an algebra A satisfies all multilinear identities of an algebra B of degree degf=kn for some fixed n. Then A satisfies all identities of B of degree kn.

    Using results of [13], we obtain the following inequalities.

    Lemma 2.1. ([13,Theorem 2]) Let A be an algebra with an exponentally bounded codimension growth. Then ¯exp(A)¯exp(A)+1.

    Lemma 2.2. ([13,Theorem 3]) Let A be an algebra with an exponentally bounded codimension growth satisfying the identity (2.2). Then exp_(A)exp_(A)+1.

    Given an integer T2, we define an infinite-dimensional algebra BT by its basis

    {a,b,zi1,,ziT|i=1,2,}

    and by the multiplication table

    zija={zij+1ifjT1,0ifj=T

    for all i1 and

    ziTb=zi+11,i1.

    All other products of basis elements are equal to zero. Clearly, algebra BT is right nilpotent of class 3, that is

    x1(x2x3)0 (2.2)

    is an identity of BT. Due to (2.2), any nonzero product of elements of BT must be left-normed. Therefore we omit brackets in the left-normed products and write (y1y2)y3=y1y2y3 and (y1yk)yk+1=y1yk+1 if k3.

    We will use the following properties of algebra BT.

    Lemma 2.3. ([14,Lema 2.1]) Let nT. Then cn(BT)2n3.

    Lemma 2.4. ([14,Lema 2.2]) Let n=kT+1. Then

    cn(BT)k!=(N1T)!.

    Lemma 2.5. ([14,Lema 2.3]) Any multilinear identity f=f(x1,,xn) of degree nT of algebra BT is an identity of BT+1.

    Let F[θ] be a polynomial ring over F on one indeterminate θ and let F[θ]0 be its subring of all polynomials without free term. Denote by QN the quotient algebra

    QN=F[θ]0(QN+1),

    where (QN+1) is an ideal of F[θ] generated by QN+1. Fix an infinite sequence of integers T1<N1<T2<N2 and consider the algebra

    R=B(T1,N1)B(T2,N2), (2.3)

    where B(T,N)=BTQN.

    Let R be an algebra of the type (2.3). Then the following lemma holds.

    Lemma 2.6. For any i1, the following equalities hold:

    (a) if TinNi then

    PnId(R)=PnId(B(Ti,Ni)B(Ti+1,Ni+1))=PnId(BTiBTi+1);

    (b) if Ni<nTi+1 then

    PnId(R)=PnId(B(Ti+1,Ni+1))=Pn(Id(BTi+1)).

    Proof. This follows immediately from the equality B(Ti,Ni)Ni+1=0 and from Lemma 2.5.

    The folowing remark is obvious.

    Remark 2.3. Ler R be an algebra of type (2.3). Then

    Id(R)=Id(B(T1,N1)B(T2,N2)).

    Theorem 3.1. For any real α>1, there exists an algebra Rα with exp_(Rα)=1,¯exp(Rα)=α such that exp_(Rα)=2 and α¯exp(Rα)α+1.

    Proof. Note that

    cn(A)ncn1(A) (3.1)

    for any algebra A satisfying (2.2). We will construct Rα of type (2.3) by a special choice of the sequence T1,N1,T2,N2, depending on α. First, choose T1 such that

    2m3<αm (3.2)

    for all mT1. By Lemma 2.4, algebra BT1 has an overexponential codimenson growth. Hence there exists N1>T1 such that

    cn(BT1)<αnfor allnN11andcN1(BT1)αN1.

    Consider an arbitrary n>N1. By Remark 2.1, we have

    cn(R)nk=0(nk)ck(R)=Σ1+Σ2,

    where

    Σ1=N1k=0(nk)ck(R),Σ2=nk=N1+1(nk)ck(R).

    By Lemma 2.6, we have Σ1+Σ2Σ1+Σ2, where

    Σ1=N1k=0(nk)ck(BT1),Σ2=nk=0(nk)ck(BT2).

    Then for any T2>N1, an upper bound for Σ2 is

    Σ2nk=0(nk)2k32n3nk=0(nk)=2n32n, (3.3)

    which follows from (3.2), provided that nT2.

    Let us find an upper bound for Σ1 assuming that n is sufficiently large. Clearly,

    Σ1N1αN1N1k=0(nk) (3.4)

    which follows from the choice of N1, relation (3.1), and the equality B(T1,N1)n=0 for all nN1+1. Since N1αN1 is a constant for fixed N1, we only need to estimate the sum of binomial coefficients.

    From the Stirling formula

    m!=2πm(me)me112m+θm,0<θm<1,

    it follows that

    (nk)nk(nk)nnkk(nk)nk. (3.5)

    Now we define the function Φ:[0;1]R by setting

    Φ(x)=1xx(1x)1x.

    It is not difficult to show that Φ increases on [0;1/2], Φ(0)=1, and Φ(x)2 on [0;1]. In terms of the function Φ we rewrite (3.5) as

    (nk)Φ(kn)Φ(kn)n<2Φ(kn)n2Φ(N1n)n (3.6)

    provided that n>2N1. Now (3.4) and (3.6) together with (3.3) imply

    Σ12N1αN1(N1+1)Φ(N1n)n,Σ22n32n.

    Since

    limnΦ(N1n)n=1

    and Φ(x) increases on (0;1/2], there exists n>2N1 such that

    2N1(N1+1)αN1Φ(N1n)n+2n32n<(2+12)n. (3.7)

    Now we take T2 to be equal to the minimal n>2N1 satisfying (3.7). Note that for such T2 we have

    cn(R)<(2+12)n

    for n=T2.

    As soon as T2 is choosen, we can take N2 as the minimal n such that cn(BT2)αn. Then again, cm(R)<mαm if m<N2. Repeating this procedure, we can construct an infinite chain T1<N1<T2<N2 such that

    cn(R)<αn+2n3 (3.8)

    for all nN1,N2,,

    αncn(R)<αn+n(αn1+2n3) (3.9)

    for all n=N1,N2, and

    2Nj(Nj+1)αNjΦ(NjTj+1)Tj+1+2T3j+12Tj+1<(2+12j)Tj+1 (3.10)

    for all j=1,2,.

    Let us denote by Rα the just constructed algebra R of type (2.3). Then (3.10) means that

    cn(Rα)<(2+12j)n (3.11)

    if n=Tj+1,j=1,2,. It follows from inequality (3.11) that

    exp_(Rα)2. (3.12)

    On the other hand, since Rα is not nilpotent, it follows that

    exp_(Rα)1. (3.13)

    Since the PI-exponent of non-nilpotent algebra cannot be strictly less than 1, relations (3.12), (3.13) and Lemma 2.2 imply

    exp_(Rα)=1,exp_(Rα)=2.

    Finally, relations (3.8), (3.9) imply the equality ¯exp(Rα)=α. Applying Lemma 2.1, we see that ¯exp(Rα)α+1. The inequality α=¯exp(Rα)¯exp(Rα) is obvious, since Rα is a subalgebra of Rα, Thus we have completed the proof of Theorem 3.1.

    As a consequence of Theorem 3.1 we get an infinite family of unital algebras of exponential codimension growth without ordinary PI-exponent.

    Corollary 1. Let β>2 be an arbitrary real number. Then the ordinary PI-exponent of unital algebra Rβ from Theorem 3.1 does not exist. Moreover, exp_(Rβ)=2, whereas β¯exp(Rβ)β+1.

    We would like to thank the referee for comments and suggestions.



    [1] I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1998.
    [2] J. Klafter, S. Lim, R. Metzler, Fractional Dynamics in Physics, World Scientific, Sinapore, 2011.
    [3] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional Calculus Models and Numerical Methods, World Scientific, Singapore, 3 (2012). https://doi.org/10.1142/10044
    [4] F. Mainardi, P. Pironi, The fractional Langevin equation: Brownian motion revisited, Extr. Math., 10 (1996), 140–154. https://doi.org/10.48550/arXiv.0806.1010 doi: 10.48550/arXiv.0806.1010
    [5] K. M. Saad, D. Baleanu, A. Atangana, New fractional derivatives applied to the Korteweg–de Vries and Korteweg–de Vries–Burger's equations, Comput. Appl. Math., 37 (2018), 5203–5216. https://doi.org/10.1007/s40314-018-0627-1 doi: 10.1007/s40314-018-0627-1
    [6] V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Berlin, 2011. https://doi.org/10.1007/978-3-642-14003-7
    [7] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Nort-Holand Mathematics Studies, Amsterdam, 204 (2006), 1–540.
    [8] K. Zhao, Multiple positive solutions of integral BVPs for high–order nonlinear fractional differential equations with impulses and distributed delays, Dyn. Syst., 30 (2015), 208–223. https://doi.org/10.1080/14689367.2014.995595 doi: 10.1080/14689367.2014.995595
    [9] K. Zhao, Impulsive integral boundary value problems of the higher–order fractional differential equation with eigenvalue arguments, Adv. Differ. Equations, 2015 (2015), 1–16. https://doi.org/10.1186/s13662-015-0725-y doi: 10.1186/s13662-015-0725-y
    [10] Y. Tian, Z. Bai, Impulsive boundary value problem for differential equations with fractional order, Differ. Equations Dyn. Syst., 21 (2013), 253–260. https://doi.org/10.1007/s12591-012-0150-6 doi: 10.1007/s12591-012-0150-6
    [11] J. Wang, F. Michal, Y. Zhou, Presentation of solutions of impulsive fractional Langevin equations and existence results, Eur. Phys. J. Spec. Top., 222 (2013), 1857–1874. https://doi.org/10.1140/epjst/e2013-01969-9 doi: 10.1140/epjst/e2013-01969-9
    [12] J. Wang, Z. Yong, L. Zeng, On a new class of impulsive fractional differential equations, Appl. Math. Comput., 242 (2014), 649–657. https://doi.org/10.1016/j.amc.2014.06.002 doi: 10.1016/j.amc.2014.06.002
    [13] S. Ulam, A Collection of Mathematical Problems, New York: Interscience Publishers, 1960.
    [14] D. H. Hyers, On the stability of the linear functional equation, PNAS, 27 (1941), 222–224. https://doi.org/10.1073/pnas.27.4.222 doi: 10.1073/pnas.27.4.222
    [15] T. Rassias, On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300. https://doi.org/10.1090/S0002-9939-1978-0507327-1 doi: 10.1090/S0002-9939-1978-0507327-1
    [16] H. Khan, J. F. Gómez-Aguilar, A. Khan, T. S. Khan, Stability analysis for fractional order advection–reaction diffusion system, Physica A, 521 (2019), 737–751. https://doi.org/10.1016/j.physa.2019.01.102 doi: 10.1016/j.physa.2019.01.102
    [17] A. Khan, J. F. Gómez-Aguilar, T. S. Khan, H. Khan, Stability analysis and numerical solutions of fractional order HIV/AIDS model, Chaos, Solitons Fractals, 122 (2019), 119–128. https://doi.org/10.1016/j.chaos.2019.03.022 doi: 10.1016/j.chaos.2019.03.022
    [18] R. Rizwan, A. Zada, X. Wang, Stability analysis of nonlinear implicit fractional Langevin equation with noninstantaneous impulses, Adv. Differ. Equations, 2019 (2019), 1–31. https://doi.org/10.1186/s13662-019-1955-1 doi: 10.1186/s13662-019-1955-1
    [19] I. Rus, Ulam stability of ordinary differential equations in a Banach space, Carpathian J. Math., 26 (2010), 103–107. Available form: http://www.jstor.org/stable/43999438.
    [20] J. Wang, A. Zada, W. Ali, Ulam's-type stability of first–order impulsive differential equations with variable delay in quasi-Banach spaces, Int. J. Nonlinear Sci. Numer. Simul., 19 (2018), 553–560. https://doi.org/10.1515/ijnsns-2017-0245 doi: 10.1515/ijnsns-2017-0245
    [21] J. Wang, K. Shah, A. Ali, Existence and Hyers–Ulam stability of fractional nonlinear impulsive switched coupled evolution equations, Math. Methods Appl. Sci., 41 (2018), 2392–2402. https://doi.org/10.1002/mma.4748 doi: 10.1002/mma.4748
    [22] K. Zhao, P. Gong, Positive solutions of m-point multi–term fractional integral BVP involving time–delay for fractional differential equations, Boundary Value Probl., 2015 (2015), 1–19. https://doi.org/10.1186/s13661-014-0280-6 doi: 10.1186/s13661-014-0280-6
    [23] A. Zada, S. Ali, Y. Li, Ulam–type stability for a class of implicit fractional differential equations with non-instantaneous integral impulses and boundary condition, Adv. Differ. Equations, 2017 (2017), 1–26. https://doi.org/10.1186/s13662-017-1376-y doi: 10.1186/s13662-017-1376-y
    [24] A. Zada, S. Ali, Stability analysis of multi–point boundary value problem for sequential fractional differential equations with non-instantaneous impulses, Int. J. Nonlinear Sci. Numer. Simul., 19 (2018), 763–774. https://doi.org/10.1515/ijnsns-2018-0040 doi: 10.1515/ijnsns-2018-0040
    [25] J. D. Stein, On generalized complete metric spaces, Bull. Amer. Math. Soc., 75 (1969), 113–116. https://doi.org/10.1090/S0002-9904-1969-12210-X doi: 10.1090/S0002-9904-1969-12210-X
    [26] J. Diaz, B. Margolis, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bull. Am. Math. Soc., 74 (1968), 305–309. https://doi.org/10.1090/S0002-9904-1968-11933-0 doi: 10.1090/S0002-9904-1968-11933-0
    [27] W. Li, J. Ji, L. Huang, Global dynamics analysis of a water hyacinth fish ecological system under impulsive control, J. Franklin Inst., 359 (2022), 10628–10652. https://doi.org/10.1016/j.jfranklin.2022.09.030 doi: 10.1016/j.jfranklin.2022.09.030
    [28] W. Li, J. Ji, L. Huang, L. Zhang, Global dynamics and control of malicious signal transmission in wireless sensor networks, Nonlinear Anal. Hybrid Syst., 48 (2023), 101324. https://doi.org/10.1016/j.nahs.2022.101324 doi: 10.1016/j.nahs.2022.101324
    [29] Z. Cai, L. Huang, Generalized Lyapunov approach for functional differential inclusions, Automatica, 113 (2020), 108740. https://doi.org/10.1016/j.automatica.2019.108740 doi: 10.1016/j.automatica.2019.108740
    [30] W. Li, J. Ji, L. Hunag, Y. Zhang, Complex dynamics and impulsive control of a chemostat model under the ratio threshold policy, Chaos, Solitons Fractals, 167 (2023), 113077. https://doi.org/10.1016/j.chaos.2022.113077 doi: 10.1016/j.chaos.2022.113077
    [31] Q. Zhu, H. Wang, Output feedback stabilization of stochastic feedforward systems with unknown control coefficients and unknown output function, Automatica, 87 (2018), 166–175. https://doi.org/10.1016/j.automatica.2017.10.004 doi: 10.1016/j.automatica.2017.10.004
    [32] B. Wang, Q. Zhu, Stability analysis of discrete time semi-markov jump linear systems, IEEE Trans. Autom. Control, 65 (2020), 5415–5421. https://doi.org/10.1109/TAC.2020.2977939 doi: 10.1109/TAC.2020.2977939
    [33] H. Wang, Q. Zhu, Global stabilization of a class of stochastic nonlinear time-delay systems with SISS inverse dynamics, IEEE Trans. Autom. Control, 65 (2020), 4448–4455. https://doi.org/10.1109/TAC.2020.3005149 doi: 10.1109/TAC.2020.3005149
    [34] K. Ding, Q. Zhu, Extended dissipative anti-disturbance control for delayed switched singular semi-Markovian jump systems with multi-disturbance via disturbance observer, Automatica, 18 (2021), 109556. https://doi.org/10.1016/j.automatica.2021.109556 doi: 10.1016/j.automatica.2021.109556
    [35] R. Rao, Z. Lin, X. Ai, J. Wu, Synchronization of epidemic systems with Neumann boundary value under delayed impulse, Mathematics, 10 (2022), 2064. https://doi.org/10.3390/math10122064 doi: 10.3390/math10122064
    [36] G. Wang, B. Ahmad, L. Zhang, Impulsive anti–periodic boundary value problem for nonlinear differential equations of fractional order, Nonlinear Anal. Theory Methods Appl., 74 (2011), 792–804. https://doi.org/10.1016/j.na.2010.09.030 doi: 10.1016/j.na.2010.09.030
    [37] D. Luo, M. Tian, Q. Zhu, Some results on finite-time stability of stochastic fractional–order delay differential equations, Chaos, Solitons Fractals, 158 (2022), 111996. https://doi.org/10.1016/j.chaos.2022.111996 doi: 10.1016/j.chaos.2022.111996
    [38] X. Wang, D. Luo, Q. Zhu, Ulam–Hyers stability of caputo type fuzzy fractional differential equations with time-delays, Chaos, Solitons Fractals, 156 (2022), 111822. https://doi.org/10.1016/j.chaos.2022.111822 doi: 10.1016/j.chaos.2022.111822
    [39] D. Luo, Q. Zhu, Z. Luo, A novel result on averaging principle of stochastic Hilfer–type fractional system involving non-Lipschitz coefficients, Appl. Math. Lett., 122 (2021), 107549. https://doi.org/10.1016/j.aml.2021.107549 doi: 10.1016/j.aml.2021.107549
    [40] I. Rus, Ulam stability of ordinary differential equations, Studia Universitatis Babes Bolyai Mathematica, 54 (2009), 125–133. Available form: https://www.cs.ubbcluj.ro/studia-m/2009-4/rus-final.pdf.
    [41] S. O. Shah, A. Zada, Existence, uniqueness and stability of solution to mixed integral dynamic systems with instantaneous and noninstantaneous impulses on time scales, Appl. Math. Comput., 359 (2019), 202–213. https://doi.org/10.1016/j.amc.2019.04.044 doi: 10.1016/j.amc.2019.04.044
    [42] F. Haq, K. Shah, G. ur Rahman, M. Shahzad, Hyers–Ulam stability to a class of fractional differential equations with boundary conditions, Int. J. Appl. Comput. Math., 3 (2017), 1135–1147. https://doi.org/10.1007/s40819-017-0406-5 doi: 10.1007/s40819-017-0406-5
    [43] W. Li, Y. Zhang, L. Huang, Dynamics analysis of a predator–prey model with nonmonotonic functional response and impulsive control, Math. Comput. Simul., 204 (2023), 529–555. https://doi.org/10.1016/j.matcom.2022.09.002 doi: 10.1016/j.matcom.2022.09.002
    [44] W. Li, J. Ji, L. Huang, Z. Guo, Global dynamics of a controlled discontinuous diffusive SIR epidemic system, Appl. Math. Lett., 121 (2021), 107420. https://doi.org/10.1016/j.aml.2021.107420 doi: 10.1016/j.aml.2021.107420
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