The q-fractional analogue for Gronwall-type inequality

In this article, we utilize \emph{q}--fractional Caputo initial value problems of order $0<\alpha\leq 1$ to derive a \emph{q}--analogue for Gronwall--type inequality. Some particular cases are derived where \emph{q}--Mittag--Leffler functions and \emph{q}--exponential type functions are used. An example is given to illustrate the validity of the derived inequality.


Introduction
The fractional differential equations have conspicuously received considerable attention in the last two decades. Many researchers have investigated these equations due to their significant applications in various fields of science and engineering such as in viscoelasticity, capacitor theory, electrical circuits, electro-analytical chemistry, neurology, diffusion, control theory and statistics; see for instance the monographs [1,2,3]. The study of q-difference equations, on the other hand, has gained intensive interest in the last years. It has been shown that these types of equations have numerous applications in diverse fields and thus have evolved into multidisciplinary subjects [4,5,6,7,8,9,10]. For more details on q-calculus, we refer the reader to the references [11,12]. The corresponding fractional difference equations, however, have been comparably less considered. Indeed, the notions of fractional calculus and qcalculus are tracked back to the works of Euler and Jackson [13], respectively. However, the idea of fractional difference equations is considered to be very recent; we suggest the new papers [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28] whose authors have taken the lead to promote the theory of fractional difference equations.
The q-fractional difference equations which serve as a bridge between fractional difference equations and q-difference equations have become a main object of research in the last years. Recently, there have appeared many papers which study the qualitative properties of solutions for q-fractional differential equations [29,30,31,32,33] whereas few results exist for q-fractional difference equations [34,35,36]. The integral inequalities which are considered as an effective tools for studying solutions properties have been also under consideration. In particular, we are interested with Gronwall's inequality which has been a main target for many researchers. There are several versions for Gronwall's inequality in the literature; we list here those results which concern with fractional order equations [37,38,39,40,41].
To the best of authors' observation, however, the q-fractional analogue for Gronwall-type inequality has not been investigated yet.
A primary purpose of this paper is to utilize the q-fractional Caputo initial value problems of order 0 < α ≤ 1 to derive a q-analogue for Gronwall-type inequality. Some particular cases are derived where q-Mittag-Leffler functions and q-exponential type functions are used. An example is given to illustrate the validity of the derived inequality.

Preliminary assertions
Before stating and proving our main results, we introduce some definitions and notations that will be used throughout the paper. For 0 < q < 1, we define the time scale Tq as follows where Z is the set of integers. In general, if α is a nonnegative real number then we define the time scale T α q = {q n+α : n ∈ Z} ∪ {0} and thus we may write T 0 q = Tq. For a function f : Tq → R, the nabla q−derivative of f is given by The nabla q-integral of f is given by and The q-factorial function for n ∈ N is defined by In case α is a non positive integer, the q-factorial function is defined by In the following lemma, we present some properties of q-factorial functions.
IV. The nabla q-derivative of the q-factorial function with respect to s is For a function f : T α q → R, the left q-fractional integral q ∇ −α a of order α = 0, −1, −2, . . . and starting at 0 < a ∈ Tq is defined by where One should note that the left q-fractional integral q ∇ −α a maps functions defined on Tq to functions defined on Tq.
Then the Caputo left q-fractional derivative of order α of a function f is defined by Lemma 2.2. [14] Assume that α > 0 and f is defined in a suitable domain. Then and if 0 < α ≤ 1 then For solving linear q-fractional equations, the following identity is essential where α ∈ R + and µ ∈ (−1, ∞). See for instance the recent paper [?] for more information.
The following example clarifies how q-Mittag-Leffler functions can be used to express the solutions of Caputo q-fractional linear initial value problems.
Applying q ∇ −α a to equation (13) and using (10), we end up with To obtain an explicit form for the solution, we apply the method of successive approximation. Set y0(t) = a0 and For m = 1, we have by the power formula (11) For m = 2, we also see that If we proceed inductively and let m → ∞, we obtain the solution That is, [k]q with [r]q = 1−q r 1−q . It is known that eq(t) = Eq((1 − q)t), where Eq(t) is a special case of the basic hypergeometric series, given by

The Main Results
Throughout the remaining part of the paper, we assume that 0 < α ≤ 1. Consider the following q−fractional initial value problem q C α a y(t) = f (t, y(t)), a ∈ Tq, y(a) = y0.
Applying q ∇ −α a to both sides of (15), we obtain Set where In the following, we present a comparison result for the fractional summation operator.
Theorem 3.1. Let w and v satisfy and We claim that u(t) ≤ 0 for t ∈ Λa. Let us assume that u(s) ≤ 0 is valid for s = q n 0 , q n 0 −1 , . . . , q n−1 , where n < n0. Then, for t = q n we have Since v(t) − w(t) ≤ 0 and t ρ(t) f (s)∇s = (t − ρ(t))f (t), (21) can be written in the form where Γq(α) = Define the following operator The following lemmas are essential in the proof of the main theorem. We only state these statements as their proofs are straightforward.  Lemma 3.4. Let λ > 0 be such that |y(t)| ≤ λ for t ∈ Λa. Then q Ω n y 1 ≤ q Ω n λ 1, n ∈ N.
Taking the limit as n → ∞, we have It remains to prove the convergence of the series in (25). The subsequent analysis are carried out for a = 0. In virtue of (18), we obtain However, for k = 1 we observe that For k = 2, it follows that For k = 3, we have For k = 4, we get Therefore, (26) becomes We observe that Hence, convergence is guaranteed. In case a > 0, we can proceed in a similar way taking into account that Theorem 3.6 (q-Fractional Gronwall's Lemma). Let v and µ be nonnegative real valued functions such that 0 ≤ µ(t) < The proof of the above statement is a straightforward implementation of Theorem 3.1 and Theorem 3.5 by setting w(t) = v(a) ∞ k=0 (Ω k µ 1)(t).
In case α = 1, we deduce the following immediate consequence of Theorem 3.6 which can be considered as the well known q-Gronwall's Lemma; consult for instance the paper [42]. Then where eq(t, a) = q Ω1(1, t − a) is the nabla q−exponential function on the time scale Tq.

Applications
In this section we show, by the help of the q−fractional Gronwall inequality proved in the previous section, that small changes in the initial conditions of Caputo q−fractional initial value problems lead to small changes in the solution.
Let f (t, y) satisfy a Lipschitz condition with constant 0 ≤ L < 1 for all t and y.