Note on fractional difference Gronwall inequalities

This note is a reaction on a bunch of fractional inequalities that appeared in the last few years, and that are all based on what claims to be a fractional discrete Gronwall inequality. However, we show by a counterexample that this inequality is not correct. Stimulated by this, the main aim of this note is to propose new inequalities and illustrate the results on examples. Asymptotic properties of a solution of a linear equation are studied as well. Moreover, a brief discussion of other related results is given.


Introduction
Recently, Deekshitulu and Mohan published a set of papers on fractional difference inequalities.All the main results in [12,13,15] are proved using a fractional discrete version of Gronwall inequality given in [11].In this short note, we show that the proof of this inequality is not correct, the inequality does not hold, and hence the validity of the implied results is questionable.Stimulated by these results, the main aim of this paper is to prove new inequalities of Gronwall type for fractional difference inequalities with linear right-hand side and constant or variable coefficients.Asymptotic properties of a solution of a linear homogeneous fractional difference equation with constant coefficient are also studied.In our results, we use a convenient Green function.In Section 4, we illustrate our results on a linear example from [14] which is also corrected in this note.
Throughout the present paper, ∇ denotes the backward difference operator defined as ∇u(n) = u(n) − u(n − 1), and having the next properties.
So we will consider fractional differences corresponding to backward difference operator.Of course, other related achievements are already done than the above-mentioned: we refer the reader to similar interesting results in [1,8,16].We only note here that in [1], inequality of the form ∇ µ * u(k) ≤ a(k)u(k), k ∈ N 0 was investigated (cf.equation (4.1)); in [8], the authors used Riemann-Liouville type fractional difference; and in [16], the author focused on fractional difference corresponding to forward difference operator.Hence our results are not covered in these papers.
We denote by N a = {a, a + 1, . . .} the shifted set of positive integers, for simplicity N = N 1 , and −N a = {. . ., a − 1, a}.We assume the property of empty sum and empty product, i.e.,

Caputo like fractional difference
In this section, we recall some definitions of ∇-based fractional operators, and we show that fractional difference considered in [9] is of Caputo type.
Analogically to forward difference based Riemann-Liouville [5] and Caputo [3,4] like fractional difference, Deekshitulu and Mohan proposed in [9] the next definition of a ∇-based fractional difference.Definition 2.2.Let µ ∈ (0, 1) and f be defined on N 0 .Then we define the µ-th fractional difference of a function f as Here we used the lower index k to denote the variable affected by operator ∇.
In reality, Σ −µ is closely related to Riemann-Liouville like ∇-based fractional difference discussed in [6], while the following lemma states that ∇ µ is of Caputo type.Lemma 2.4.Let µ ∈ (0, 1), ν = 1 − µ and function f be defined on Proof.For k ∈ N we expand the left-hand side by (2.4), and apply Lemma 1.1 to get which, by (2.1) with a = 1, p = 0, is exactly what has to be proved.
In the sense of the above lemma, we add the lower index * as done in [3,4] to denote the Caputo nature of the difference, i.e., ∇ µ * := ∇ µ in the rest of the paper.Next, in [9, Remark  3.2] the properties of Σ µ were translated to ∇ µ * .The above discussion implies that this is also a mistake, and ∇ µ * does not have to possess such properties.Nevertheless, we do not go into details, as we do not need the properties in the present paper.

Linear fractional difference equation
In this section, we derive a solution of a nonhomogeneous linear fractional difference equation in terms of a Green function.Particular case of a constant coefficient at linear term is investigated in details.
Note that the above corollary holds true with ≥ instead of ≤.
Next, we derive a solution of a linear initial value problem.Let h(k) denote the solution of the problem ∇ and define a Green function {g j (k)} k∈N 0 , k, j ∈ N 0 as where and Here we note that g j (k) = 0 for 0 ≤ k ≤ j and g j (j + 1) = 1.So setting while (3.4) gives for j ∈ N 0 .Using ∇ µ * g j (j) = 1, we can directly verify that (3.8) solves (3.4).Thus there are two different ways, (3.6) and (3.8), how to define a solution u of (3.7).The following lemma uses (3.6), and concludes the above arguments.

Lemma 3.3. The initial value problem
Proof.The considered problem is decomposed to a homogeneous equation with a nontrivial initial condition, of the form (3.5), and a nonhomogeneous equation with a zero initial condition, of the form (3.7).Consequently, the superposition principle is applied.
The rest of this section is devoted to the case of constant function a(k).
Then u has the form Proof.First, we apply Lemma 3.1 to get a corresponding fractional sum equation Consequently, We claim that then what is the statement of the proposition.

Fractional difference inequalities 7
We prove the claim by induction with respect to k.If k = 0, then due to the empty sum property u(0) = u 0 in both (3.13) and (3.14).Now, let us assume that (3.14) holds for 0, 1, . . ., k, and we show that it is true also for k + 1.Using (3.13) and the inductive hypothesis, we have Similarly in S 2 : . Note that the second sum is empty for q = k + 1. Hence Note that j − 1 takes values 0, 1, . . ., k − q.So we can denote i q+1 = j − 1 and merge the last two sums to obtain and after summing S 1 and S 2 , (3.14) is obtained for k + 1.

M. Fečkan and M. Pospíšil
Now we present an alternative proof without using induction principle.
Alternative proof of Proposition 3.4.If a = 0, then by (3.12), the statement is proved.From now on, we assume that a ̸ = 0.By using the formula [19,Problem 7,p. 15] it is clear that 1 lim sup So the power series has the radius of convergence 1.Furthermore, (3.16) gives that and using Next, we know that the sequence {|B µ (n)|} ∞ n=0 is decreasing (see the proof of Lemma 4.6) with lim n→∞ B µ (n) = 0 (see (3.16)).So the Leibnitz criterion implies the convergence of the series and the Abel theorem [23, p. 9] thus the power series has the radius of convergence R U greater than or equal to 1 1+|a| .Consequently, we start with 0 ̸ = |x| < 1 1+|a| .Then using (3.13), we derive By expanding the right-hand side of (3.21) and comparing the powers of x, we immediately get (3.14).
If u 0 = 0 then U(x) = 0.So we suppose that u 0 ̸ = 0. Next, using (3.16) we see that as n → ∞.So by results of [23, pp. 224-225],we have (3.20),thus the radius of convergence R U of U(x) is less than or equal to 1, and we get On the other hand, we know where we applied the estimation of the ratio of gamma functions from [17], and used the notation for the polylogarithm function [20,Section 7.12].Consequently, for ] .
We note that F is increasing on (0, 1).The right-hand side of (3.23) is positive if and only if |x| < F −1 (1).Note that F −1 (1) > 1 1+|a| .To see this, we estimate This improves (3.22) to Summarizing the above arguments, we obtain the next result.
Corollary 3.6.If the solution u(n) of (3.10) with u 0 ̸ = 0 satisfies u(n) → 0 as n → ∞, then the rate of convergence is slower than any exponential one, i.e., there are no constants c 1 > 0 and ϖ ∈ (0, 1) On the other hand, if a > 0, On the other hand, from (3.11), we get lim So we call the bracket in (3.11), the generalized binomial.Note that linear Riemann-Liouville fractional difference equations are solved in [7] leading to discrete Mittag-Leffler functions.
Next, we derive a formula for a Green function satisfying (3.4) with constant a.

Fractional difference inequalities
In this section, we explain where the problem lies of the fractional Gronwall inequality established in [11].Then we propose our alternative fractional difference inequalities of the Fractional difference inequalities 13 Gronwall type that can be used instead of the original one.For a better clarity, we use quotation marks when recalling the false result.At the end of the section, an example of linear fractional difference equation with a given initial condition from [14], is given.On this example, we illustrate different estimates of its solution.
"Lemma" 4.2.Let µ ∈ (0, 1), and u, a, b be real nonnegative functions defined on N 0 .If Originally, inequality (4.1) is transformed to a corresponding sum equation using the properties of operator ∇ µ * .However, from Section 2 we know that this does not have to be valid.Next, A µ (k, j) is considered only as a function of j and Lemma 4.1 is applied with p(j) = A µ (k − 1, j)a(j) and q(j) = A µ (k − 1, j)b(j).This can cause another problem, since in general, the discrete Gronwall inequality does not hold with p, q depending on k.To prove that, it really makes a problem and "Lemma" 4.2 is not correct, we provide the following counterexample.

Conclusion
In this note, we showed by Example 4.3 that the Caputo like ∇-based fractional Gronwall lemma ("Lemma" 4.2) is false.By this, many proofs of the recent results of Deekshitulu and Mohan are discarded, as they are based on the "lemma".This should motivate authors to carefully and critically approach any results published in more or less esteemed journals.One can try to find simple counterexamples to show that the inequalities from [12,13,15] are false as well, or use the proposed Lemma 4.6 to obtain valid results.
We note that in [10], the authors made a similar fault as in [11] -they neglected the dependence of a summed function on independent variable in discrete Langenhop inequality without any explanation.So one can doubt the correctness of this step.So we hope that our approach and results could help these authors improve their results.
On the other hand, their results were stimulations for us to propose new ∇-based fractional difference inequalities of the Gronwall type for constant or variable coefficients, which have been not yet studied.We also dealt with related linear fractional difference equations.