Research article Special Issues

Analytical and numerical negative boundedness of fractional differences with Mittag–Leffler kernel

  • We show that a class of fractional differences with Mittag–Leffler kernel can be negative and yet monotonicity information can still be deduced. Our results are complemented by numerical approximations. This adds to a growing body of literature illustrating that the sign of a fractional difference has a very complicated and subtle relationship to the underlying behavior of the function on which the fractional difference acts, regardless of the particular kernel used.

    Citation: Pshtiwan Othman Mohammed, Rajendra Dahal, Christopher S. Goodrich, Y. S. Hamed, Dumitru Baleanu. Analytical and numerical negative boundedness of fractional differences with Mittag–Leffler kernel[J]. AIMS Mathematics, 2023, 8(3): 5540-5550. doi: 10.3934/math.2023279

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  • We show that a class of fractional differences with Mittag–Leffler kernel can be negative and yet monotonicity information can still be deduced. Our results are complemented by numerical approximations. This adds to a growing body of literature illustrating that the sign of a fractional difference has a very complicated and subtle relationship to the underlying behavior of the function on which the fractional difference acts, regardless of the particular kernel used.



    Define the sets Na and Nba, for any a,bR with ba a nonnegative integer, to be Na:={a,a+1,a+2,} and Nba:={a,a+1,a+2,,b}, respectively. It worth recalling that the nabla (or backward) difference of a function g : NaR can be expressed as follows

    (g)(t):=g(t)g(t1)fortNa+1.

    As is known, there is a strong correlation between the sign of the nabla operator and whether g is either monotone increasing or monotone decreasing. For example, if (g)(t)0, then g is increasing function on Na.

    In recent years a nonlocal version of the discrete calculus has been proposed. This nonlocal version is known as the "discrete fractional calculus", a research area popularized by the seminal papers of Atici and Eloe [9,10,11] in the late 2000s and then further extended by the subsequent work of Lizama [25]. One of the reasons for the interest in the discrete fractional calculus is its emerging applications in biological mathematics – see, for example, the recent work of Atici, et al. [7,8], in which the authors apply discrete fractional calculus to the modeling of tumors.

    To understand the nonlocal nature of the fractional calculus, consider a commonly utilized version of the discrete fractional difference – namely, for ν>0 the ν-th order Riemann-Liouville difference, which for a function g : NaR is denoted by RLaΔνg and defined pointwise by

    (RLaΔνg)(t):=t+νs=aΓ(ts)Γ(ν)Γ(ts+ν+1)g(s)tNa+Nν, (1.1)

    where a positive integer N satisfying N1<νN. The key fact about (1.1) is that it is nonlocal, very much unlike the integer-order difference mentioned in the first paragraph of this section. Consequently, the relationships between the sign of (RLaΔνg)(t) and the monotone behavior of g are quite muddled and complex. This question was initially investigated by Dahal and Goodrich [13] in 2014, and then subsequently investigated by many authors including Abdeljawad and Baleanu [1], Bravo, Lizama, and Rueda [12], Goodrich and Jonnalagadda [16], Goodrich and Lizama [18,19], Goodrich and Muellner [22], and Jia, Erbe, and Peterson [15,24].

    Very recently Goodrich, Lyons, and Velcsov [20] together with Jonnalagadda [17] and Scapellato [21] demonstrated that a function can increase (under certain conditions) even if its fractional difference is negative. This is something that plainly cannot happen in the integer-order case. And this phenomenon provides further evidence of the highly complicating nature of the nonlocal structure of fractional-order difference operators. It also has serious implications for the use of fractional calculus in modeling since one of the most important uses of calculus in modeling is to identify where functions are increasing or decreasing.

    So, all in all, there is a large body of evidence that nonlocal discrete operators behave in ways that are very complicated, particularly as concerns their ability to detect the qualitative behaviors of the functions on which they operate. At the same time, there are a variety of definitions for discrete fractional differences and sums. Therefore, it is relevant to determine whether these aberrant behaviors are exhibited by all such nonlocal difference operators – or only some of them.

    Consequently, in this brief note, we propose to continue investigating these questions in the specific context of the fractional difference with exponential-type kernels (see Definitions 2.1 and 2.2). In particular, we demonstrate that, as with other definitions of the fractional difference, the type of difference studied here can be negative even though the function on which it acts is increasing, and this observation confirms that fractional differences with Mittag-Leffler kernels exhibit the same sort of aberrant behavior as other nonlocal difference operators. This complements not only the already mentioned reference [17] but also builds upon earlier work by Abdeljawad, Al-Mdallal, and Hajji [6]. And it continues to demonstrate some of the surprising properties of this class of discrete nonlocal operators. To see the development of fractional calculus with Mittag-Leffler kernels we advise the readers to see the recently published articles [28,29].

    In this section, we begin with recalling the necessary fundamental discrete operators for our main results briefly. The interest reader may visit the monograph [23] by Goodrich and Peterson and recently published articles [1,2,3,26] for additional information and mathematical background regarding the discrete fractional calculus.

    The first and well-known definition in this article is the discrete Mittag-Leffler functions. We then provide the definition of the discrete fractional difference defined using the Mittag-Leffler kernel on the set Na. The discrete Mittag-Leffler function of 2parameters is given by (see [27]):

    E¯ν,β(λ,t):=k=0λkt¯kν+β1Γ(kν+β),

    for λR such that 1>|λ|, and ν,β,tC such that Re(ν)>0. It is essential to see that t¯ν is the rising function and given as follows

    t¯ν:=Γ(t+ν)Γ(t),

    for νR and tR apart from the elements {,2,1,0}. As a special case of the above definition, the discrete Mittag-Leffler function of 1parameter is given as follows

    E¯ν(λ,t):=k=0λkt¯kνΓ(kν+1)(for|λ|<1).

    Remark 2.1. Considering Remark 1 in [4], we can obtain the following for λ1=ν12ν and 1<ν<32:

    E¯ν1(λ1,0)=1,

    E¯ν1(λ1,1)=2ν,

    E¯ν1(λ1,2)=ν(2ν)2,

    E¯ν1(λ1,3)=2ν2[(ν1)3(2ν3)3(ν1)2+2],

    0<E¯ν1(λ1,t)<1 for each 1<ν<32 and t=1,2,3,. At the same time, we have that E¯ν1(λ1,t) is monotonically decreasing for each 1<ν<32 and t=0,1,2,.

    Definition 2.1. (see [1,Definition 2.24]) Let 0<ν<12 and λ0=ν1ν. Then, the discrete fractional difference operators with Mittag-Leffler kernels of order ν denoted by ABCaνg and ABRaνg, respectively, defined by

    (ABCaνg)(t):=H(ν)1νttr=a+1g(r)E¯ν(λ0,tr+1),

    and

    (ABRaνg)(t):=H(ν)1νtr=a+1(rg)(r)E¯ν(λ0,tr+1),

    for each tNa+1. Here the function νH(ν) is a normalization constant satisfying 0<H(ν).

    Definition 2.2. (see [5]) For g:NaR with <ν+12 and N0, Then, the discrete fractional difference operators with Mittag-Leffler kernels of order ν, respectively, are defined by

    (ABCaνg)(t)=(ABCaνg)(t):=H(ν)+1νttr=a+1(rg)(r)E¯ν(λ,tr+1),

    and

    (ABRaνg)(t)=(ABRaνg)(t):=H(ν)+1νtr=a+1(+1rg)(r)E¯ν(λ,tr+1),

    for tNa+1. Here λ=ν+1ν.

    The following is the essential lemma which brings us to the main results.

    Lemma 2.1. Let the function g be defined on Na and 1<ν<32. Then we have that

    (ABRaνg)(t)=H(ν1){(g)(t)+12ν[E¯ν1(λ1,ta)E¯ν1(λ1,ta1)](g)(a+1)+12νt1r=a+2[E¯ν1(λ1,tr+1)E¯ν1(λ1,tr)](rg)(r)},

    for each tNa+3.

    Proof. From Definitions 2.1 and 2.2, the following can be deduced for 1<ν<32:

    (ABRaνg)(t)=H(ν1)2ν{tr=a+1E¯ν1(λ1,tr+1)(rg)(r)t1r=a+1E¯ν1(λ1,tr)(rg)(r)}=H(ν1)2ν{(2ν)(g)(t)+t1r=a+1[E¯ν1(λ1,tr+1)E¯ν1(λ1,tr)](rg)(r)}=H(ν1){(g)(t)+12ν[E¯ν1(λ1,ta)E¯ν1(λ1,ta1)](g)(a+1)+12νt1r=a+2[E¯ν1(λ1,tr+1)E¯ν1(λ1,tr)](rg)(r)},

    for each tNa+3, and hence the proof is complete.

    The first main result we present, Theorem 2.1, demonstrates that (ABRaνg)(a+3) can be negative even though (g)(a+3)>0 – cf., [14,Theorem 3.1].

    Theorem 2.1. Let the function g be defined on Na, and let 1<ν<1.5 and ε>0. Assume that

    (ABRaνg)(a+3)>ε(g)(a+1)H(ν1). (2.1)

    If (g)(a+1)0, (g)(a+2)0, and 12(ν1)2(2ν25ν+2)<ε, then (g)(a+3)0.

    Proof. Due to Lemma 2.1 and the condition (2.1) we get

    (g)(t)(g)(a+1){12ν[E¯ν1(λ1,ta)E¯ν1(λ1,ta1)]+ε}12νt1r=a+2[E¯ν1(λ1,tr+1)E¯ν1(λ1,tr)](rg)(r), (2.2)

    for each tNa+3. Setting t=a+3 in (2.2), yields

    (g)(a+3)(g)(a+1){12ν[E¯ν1(λ1,3)E¯ν1(λ1,2)]+ε}12νa+2r=a+2[E¯ν1(λ1,a+4r)E¯ν1(λ1,a+3r)](rg)(r).

    Since (g)(a+2)0 by assumption, it follows that

    12νa+2r=a+2[E¯ν1(λ1,a+4r)E¯ν1(λ1,a+3r)](rg)(r)=12ν>0[(2ν)(ν1)2]<0(g)(a+2)00. (2.3)

    Also, we know that (g)(a+1)0. So, we can use the inequalities (2.2) and (2.3) to deduce that (g)(a+3)0, using especially that

    12ν[E¯ν1(λ1,3)E¯ν1(λ1,2)]+ε=12(ν1)2(2ν25ν+2)+ε<0

    by the assumption given in the statement of the theorem, and this ends the proof.

    Remark 2.2. Figure 1 shows the graph of ν12(ν1)2(2ν25ν+2) for ν(1,32). Observe that in order for Theorem 2.1 to be applied it must hold that ε(0,12(ν1)2(2ν25ν+2)) for a fixed ν(1,32). This admissible region for ε is shown by the light grey region in the figure.

    Figure 1.  Graph of ν12(ν1)2(2ν25ν+2) for ν[1,32]. The admissible range of ε for ν fixed is shaded in light grey.

    Now, define the set Fk,ε(1,32) as follows:

    Fk,ε:={ν(1,32):12ν[E¯ν1(λ1,ka)E¯ν1(λ1,ka1)]<ε}(1,1.5),kNa+3.

    Lemma 2.2 proves that the nested collection {Fk,ε}k=a+1 is decreasing (whenever ε>0). This is a phenomenon that has been observed in similar contexts (e.g., [14,Lemma 3.3], [20,Lemma 3.2], [21,Lemma 3.2]).

    Lemma 2.2. Let 1<ν<32. Then for each ε>0 and kNa+3 we have that Fk+1,εFk,ε.

    Proof. Let ε>0 and νFk+1,ε be arbitrary for some fixed kNa+3. Then we have

    12ν[E¯ν1(λ1,k+1a)E¯ν1(λ1,ka)]=λ12νE¯ν1,ν1(λ1,k+1a)<ε.

    Since E¯ν1,ν1(λ1,k+1a) is decreasing for each kNa+3 (see [4]), 1<ν<32, and λ1<0, we have

    λ12νE¯ν1,ν1(λ1,ka)<λ12νE¯ν1,ν1(λ1,k+1a)<ε.

    This implies that νFk,ε, and, therefore, Fk+1,εFk,ε. Thus, we have accomplished the result.

    Theorem 2.1 and Lemma 2.2 now lead to the following corollary, which is the principal analytical result of this note – cf., [14,Corollary 3.4]. Corollary 2.1 asserts that the same pathological behavior observed with other discrete fractional differences carries over to the Mittag-Leffler kernel setting.

    Corollary 2.1. Let the assumptions of Theorem 2.1 be fulfilled together with

    (ABRaνg)(t)>εH(ν1)(g)(a+1), (2.4)

    for each ν(1,32), tNsa+3 and some fixed sNa+3. Now, if we assume that (g)(a+1)0, (g)(a+2)0, and νFs,ε, then we have (g)(t)0, for all tNsa+1.

    Proof. Due to the assumption that νFs,ε and Lemma 2.2, we have

    νFs,ε=Fs,εk=a+3s1Fk,ε.

    This leads to

    12ν[E¯ν1(λ1,k+1a)E¯ν1(λ1,ka)]<ε, (2.5)

    for each kNsa+3.

    We now can proceed by induction to complete the proof as follows. At first, for t=a+3 we can obtain (g)(a+3)0 immediately as in Theorem 2.1 with the help of inequalities (2.4) and (2.5) just as in the proof of Theorem 2.1. Accordingly, we can continue and inductively iterate inequality (2.2) to get (g)(t)0, for all tNsa+2 as requested. Note that in this last step we are using the fact that E¯ν1(λ1,tr+1)E¯ν1(λ1,tr)0, for each (r,t)Nt1a+2×Nsa+3, which is true since the partial function tE¯ν1(λ1,t) is decreasing – see Remark 2.1. Thus, we have completed the proof.

    We next provide an example in order to demonstrate the application of the preceding ideas.

    Example 2.1. Considering Lemma 2.1 with t:=a+3:

    (ABRaνg)(a+3)=H(ν1){(g)(a+3)+12ν[E¯ν1(λ1,3)E¯ν1(λ1,2)](g)(a+1)+12νa+2r=a+2[E¯ν1(λ1,a+4r)E¯ν1(λ1,a+3r)](rg)(r)}.

    For a=0, it follows that

    (ABR0νg)(3)=H(ν1){(g)(3)+12(ν1)2(2ν25ν+2)(g)(1)+12ν2r=2[E¯ν1(λ1,4r)E¯ν1(λ1,3r)](rg)(r)}=H(ν1){(g)(3)+12(ν1)2(2ν25ν+2)(g)(1)(ν1)2(g)(2)}=H(ν1){g(3)g(2)+12(ν1)2(2ν25ν+2)[g(1)g(0)](ν1)2[g(2)g(1)]}.

    If we take ν=1.99,g(0)=0.01,g(1)=1.01,g(2)=1.001,g(3)=1.005, and ϵ=0.002, we have

    (ABR01.99g)(3)=0.0018H(0.99)>0.002H(0.99)=ϵH(0.99)(g)(1).

    Note that (ABR01.99g)(3)<0. Yet, as Theorem 2.1 correctly predicts, it, nonetheless, holds that (g)(1)>0. Thus, the collection of functions to which Theorem 2.1 applies is non-void.

    We conclude this note by providing a brief numerical analysis of the set Fk,ε, which plays a key role in Corollary 2.1; throughout we take a=0 purely for convenience. Let us first consider Figure 2 above. This is a heat map, which identifies the cardinality of the set {kN : νFk,ε}. It is worth mentioning that the warmer colors (i.e., oranges and reds) are associated to higher cardinality values as indicated by the vertical sidebar in Figure 2. We see that there is a concentration of larger cardinalities in a roughly triangular region as indicated in the figure. The largest cardinalities seem to be concentrated for 1.07ν1.10 and ε>0 close to zero; this implies that the analytical results presented earlier (i.e., Corollary 2.1) should be valid for the greatest number of time steps t when ν and ε are in this region of the (ν,ε)-parameter space.

    Figure 2.  Heat map for the cardinality of {k : νFk,ε}.

    On the other hand, Figures 36 plot the interval of k values such that Fk,ε for different choices of both ν and ε. Consistent with the heat map in Figure 2, we see that there is a relative maximum when ν is away from the boundary values ν=1 and ν=1.5, though the precise value depends on the value of ε. In particular, as ε0+, the maximum seems to approach about 1.07, just as indicated by Figure 2. In addition, we see that the length of the intervals drops off sharply both as ν1+ and as ν1.5 – again, precisely as depicted in Figure 2.

    Figure 3.  Fk,0.01.
    Figure 4.  Fk,0.005.
    Figure 5.  Fk,0.0005.
    Figure 6.  Fk,0.0001.

    Finally, the data contained in Figures 26 is not entirely dissimilar from the observations in [14,Figures 14], which analyzed the Riemann-Liouville fractional nabla difference. In each of the Mittag-Leffler kernel and the Riemann-Liouville settings the ν-values for which the respective monotonicity-type theorems – i.e., Corollary 2.1 and [14,Corollary 3.4] – seem to be most applicable are apparently concentrated for ν close to 1. A possible, albeit non-rigorous, explanation for this common observation is that when ν1 the fractional difference is "more like" the first-order difference, which is closely connected to monotonicity. But we do not have a precise analytical explanation for this numerical observation, and we hope that this sort of curiosity provides motivation to analyze further these types of fractional difference operators in the future. Nonetheless, the results of this note show that this is a common feature across multiple types of fractional difference operators

    .

    In this brief note we have demonstrated that the fractional difference with Mittag-Leffler kernel behaves in an aberrant manner, similar to that of other classes of nonlocal difference operators. In particular, we have shown that such a difference acting on a function can be negative even if the function on which it acts is increasing. This sort of unusual behavior is not possible when considering a local difference operator, but it seems to be an almost defining feature of nonlocal discrete operators as the results of this note demonstrate.

    This research was supported by Taif University Researchers Supporting Project (No. TURSP-2020/155), Taif University, Taif, Saudi Arabia.

    The authors declare that they have no conflicts interests.



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