The 𝑞-Fractional Analogue for Gronwall-Type Inequality

Abstract

We utilize 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.

1. 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, electroanalytical chemistry, neurology, diffusion, control theory, and statistics; see, for instance, the monographs [1–3]. 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 [4–11]. For more details on 𝑞-calculus, we refer the reader to the reference [12]. The corresponding fractional difference equations, however, have been comparably less considered. Indeed, the notions of fractional calculus and 𝑞-calculus are tracked back to the works of Jackson [13], respectively. However, the idea of fractional difference equations is considered to be very recent; we suggest the new papers [14–28] 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, many papers have appeared which study the qualitative properties of solutions for 𝑞-fractional differential equations [29–33], whereas few results exist for 𝑞-fractional difference equations [34–36]. The integral inequalities which are considered as effective tools for studying solutions properties have been also under consideration. In particular, we are interested in 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 are concerned with fractional order equations [37–41]. 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.

2. 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 𝕋_𝑞 as follows: 𝕋_𝑞=𝑞^𝑛∶𝑛∈ℤ∪{0}, (1) 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}, (2) and thus we may write 𝕋⁰_𝑞=𝕋_𝑞. For a function 𝑓∶𝕋_𝑞→ℝ, the nabla 𝑞-derivative of 𝑓 is given by ∇_𝑞𝑓(𝑡)=𝑓(𝑡)−𝑓𝑞𝑡1−𝑞𝑡,𝑡∈𝕋_𝑞−{0}. (3) The nabla 𝑞-integral of 𝑓 is given by ^𝑡₀𝑓(𝑠)∇_𝑞𝑠=1−𝑞𝑡^∞_𝑖=0𝑞^𝑖𝑓𝑡𝑞^𝑖,_𝑡𝑎𝑓(𝑠)∇_𝑞𝑠=^𝑡₀𝑓(𝑠)∇_𝑞𝑠−^𝑎₀𝑓(𝑠)∇_𝑞𝑠,for0≤𝑎∈𝑇_𝑞. (4) The 𝑞-factorial function for 𝑛∈ℕ is defined by (𝑡−𝑠)^𝑛_𝑞=^𝑛−1_𝑖=0𝑡−𝑞^𝑖𝑠. (5) In case 𝛼 is a nonpositive integer, the 𝑞-factorial function is defined by (𝑡−𝑠)^𝛼_𝑞=𝑡^𝛼∞_𝑖=01−(𝑠/𝑡)𝑞^𝑖1−(𝑠/𝑡)𝑞^𝑖+𝛼. (6) In the following lemma, we present some properties of 𝑞-factorial functions.

Lemma 1 (see [32]). For 𝛼,𝛾,𝛽∈ℝ, one has the following. (I)(𝑡−𝑠)_𝑞^𝛽+𝛾=(𝑡−𝑠)^𝛽_𝑞(𝑡−𝑞^𝛽𝑠)^𝛾_𝑞. (II)(𝑎𝑡−𝑎𝑠)^𝛽_𝑞=𝑎^𝛽(𝑡−𝑠)^𝛽_𝑞. (III)The nabla 𝑞-derivative of the 𝑞-factorial function with respect to 𝑡 is ∇_𝑞(𝑡−𝑠)^𝛼_𝑞=1−𝑞^𝛼()1−𝑞𝑡−𝑠_𝑞^𝛼−1. (7)(IV)The nabla 𝑞-derivative of the 𝑞-factorial function with respect to 𝑠 is ∇_𝑞(𝑡−𝑠)^𝛼_𝑞=−1−𝑞^𝛼1−𝑞𝑡−𝑞𝑠_𝑞^𝛼−1. (8)

For a function 𝑓∶𝕋^𝛼_𝑞→ℝ, the left 𝑞-fractional integral _𝑞∇_𝑎^−𝛼 of order 𝛼≠0,−1,−2,… and starting at 0<𝑎∈𝕋_𝑞 is defined by _𝑞∇_𝑎^−𝛼𝑓(𝑡)=1Γ_𝑞(𝛼)^𝑡_𝑎𝑡−𝑞𝑠_𝑞^𝛼−1𝑓(𝑠)∇_𝑞𝑠, (9) where Γ_𝑞(𝛼+1)=1−𝑞^𝛼Γ1−𝑞^𝑞(𝛼),Γ_𝑞(1)=1,𝛼>0. (10) One should note that the left 𝑞-fractional integral _𝑞∇_𝑎^−𝛼 maps functions defined on 𝕋_𝑞 to functions defined on 𝕋_𝑞.

Definition 2 (see [14]). If 0<𝛼∉ℕ, then the Caputo left 𝑞-fractional derivative of order 𝛼 of a function 𝑓 is defined by _𝑞𝐶^𝛼_𝑎𝑓(𝑡)∶=_𝑞∇_𝑎^{−(𝑛−𝛼)}∇^𝑛_𝑞𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡_𝑎𝑡−𝑞𝑠_𝑞^{𝑛−𝛼−1}∇^𝑛_𝑞𝑓(𝑠)∇_𝑞𝑠, (11) where 𝑛=[𝛼]+1. In case 𝛼∈ℕ, we may write _𝑞𝐶^𝛼_𝑎𝑓(𝑡)∶=∇^𝑛_𝑞𝑓(𝑡).

Lemma 3 (see [14]). Assume that 𝛼>0 and 𝑓 is defined in a suitable domain. Then _𝑞∇^−𝛼_𝑎𝑞𝐶^𝛼_𝑎𝑓(𝑡)=𝑓(𝑡)−^𝑛−1_𝑘=0(𝑡−𝑎)^𝑘_𝑞Γ_𝑞()∇𝑘+1^𝑘𝑞𝑓(𝑎), (12) and if 0<𝛼≤1, then _𝑞∇^−𝛼_𝑎𝑞𝐶^𝛼_𝑎𝑓(𝑡)=𝑓(𝑡)−𝑓(𝑎). (13)

For solving linear 𝑞-fractional equations, the following identity is essential: _𝑞∇_𝑎^−𝛼(𝑥−𝑎)^𝜇_𝑞=Γ_𝑞𝜇+1Γ_𝑞()𝛼+𝜇+1𝑥−𝑎_𝑞^𝜇+𝛼,0<𝑎<𝑥<𝑏, (14) where 𝛼∈ℝ⁺ and 𝜇∈(−1,∞). See, for instance, the recent papers [14, 15] for more information.

The 𝑞-analogue of Mittag-Leffler function with double index (𝛼,𝛽) is first introduced in [14]. Indeed, it was defined as follows.

Definition 4 (see [14]). For 𝑧,𝑧₀∈𝐂 and ℜ(𝛼)>0, the 𝑞-Mittag-Leffler function is defined by _𝑞𝐸_𝛼,𝛽𝜆,𝑧−𝑧₀=^∞_𝑘=0𝜆^𝑘𝑧−𝑧_0𝑞^𝛼𝑘Γ_𝑞.𝛼𝑘+𝛽 (15) In case 𝛽=1, we may use _𝑞𝐸_𝛼(𝜆,𝑧−𝑧₀)∶=_𝑞𝐸_𝛼,1(𝜆,𝑧−𝑧₀).

The following example clarifies how 𝑞-Mittag-Leffler functions can be used to express the solutions of Caputo 𝑞-fractional linear initial value problems.

Example 5 (see [14]). Let 0<𝛼≤1 and consider the left Caputo 𝑞-fractional difference equation _𝑞𝐶^𝛼_𝑎𝑦(𝑡)=𝜆𝑦(𝑡)+𝑓(𝑡),𝑦(𝑎)=𝑎₀,𝑡∈𝑇_𝑞. (16) Applying _𝑞∇_𝑎^−𝛼 to (16) and using (13), we end up with 𝑦(𝑡)=𝑎₀+𝜆_𝑞∇_𝑎^−𝛼𝑦(𝑡)+_𝑞∇_𝑎^−𝛼𝑓(𝑡). (17) To obtain an explicit form for the solution, we apply the method of successive approximation. Set 𝑦₀(𝑡)=𝑎₀ and 𝑦_𝑚(𝑡)=𝑎₀+𝜆_𝑞∇_𝑎^−𝛼𝑦_𝑚−1(𝑡)+_𝑞∇_𝑎^−𝛼𝑓(𝑡),𝑚=1,2,3,…. (18) For 𝑚=1, we have by the power formula (14) 𝑦₁(𝑡)=𝑎₀𝜆()1+𝑡−𝑎_𝑞^(𝛼)Γ_𝑞()+𝛼+1^𝑞∇_𝑎^−𝛼𝑓(𝑡). (19) For 𝑚=2, we also see that 𝑦₂(𝑡)=𝑎₀+𝜆𝑎_0𝑞∇_𝑎^−𝛼()1+𝑡−𝑎^𝛼_𝑞Γ_𝑞()+𝛼+1_𝑞∇_𝑎^−𝛼𝑓(𝑡)+𝜆_𝑞∇_𝑎^−2𝛼𝑓(𝑡)=𝑎₀𝜆()1+𝑡−𝑎^𝛼_𝑞Γ_𝑞()+𝜆𝛼+1²(𝑡−𝑎)_𝑞^2𝛼Γ_𝑞()+2𝛼+1_𝑞∇_𝑎^−𝛼𝑓(𝑡)+𝜆_𝑞∇_𝑎^−2𝛼𝑓(𝑡). (20) If we proceed inductively and let 𝑚→∞, we obtain the solution 𝑦(𝑡)=𝑎₀1+^∞_𝑘=1𝜆^𝑘(𝑡−𝑎)_𝑞^𝑘𝛼Γ_𝑞()+𝑘𝛼+1_{𝑡𝑎∞𝑘=1}𝜆^𝑘−1Γ_𝑞(𝛼𝑘)𝑡−𝑞𝑠_𝑞^𝛼𝑘−1𝑓(𝑠)∇_𝑞𝑠=𝑎₀1+^∞_𝑘=1𝜆^𝑘(𝑡−𝑎)_𝑞^𝑘𝛼Γ_𝑞(𝑘𝛼+1)+^𝑡_𝑎^∞_𝑘=0𝜆^𝑘Γ_𝑞(𝛼𝑘+𝛼)𝑡−𝑞𝑠_𝑞^{𝛼𝑘+(𝛼−1)}𝑓(𝑠)∇_𝑞𝑠=𝑎₀1+^∞_𝑘=1𝜆^𝑘(𝑡−𝑎)_𝑞^𝑘𝛼Γ_𝑞()+𝑘𝛼+1_𝑡𝑎𝑡−𝑞𝑠_𝑞^{(𝛼−1)∞}_𝑘=0𝜆^𝑘Γ_𝑞(𝛼𝑘+𝛼)𝑡−𝑞^𝛼𝑠_𝑞^(𝛼𝑘)𝑓(𝑠)∇_𝑞𝑠. (21) That is, 𝑦(𝑡)=𝑎_0𝑞𝐸_𝛼(𝜆,𝑡−𝑎)+^𝑡_𝑎𝑡−𝑞𝑠_𝑞𝑞^𝛼−1𝐸_𝛼,𝛼𝜆,𝑡−𝑞^𝛼𝑠𝑓(𝑠)∇_𝑞𝑠. (22)

Remark 6. If instead we use the modified 𝑞-Mittag-Leffler function _𝑞𝑒_𝛼,𝛽𝜆,𝑧−𝑧₀=^∞_𝑘=0𝜆^𝑘𝑧−𝑧_0𝑞^{𝛼𝑘+(𝛽−1)}Γ_𝑞𝛼𝑘+𝛽 (23) then, the solution representation (17) becomes 𝑦(𝑡)=𝑎_0𝑞𝑒_𝛼(𝜆,𝑡−𝑎)+^𝑡_𝑎𝑞𝑒_𝛼,𝛼𝜆,𝑡−𝑞𝑠𝑓(𝑠)∇_𝑞𝑠. (24)

Remark 7. If we set 𝛼=1, 𝜆=1, 𝑎=0, and 𝑓(𝑡)=0, we reach to the 𝑞-exponential formula 𝑒_𝑞(𝑡)=∑^∞_𝑘=0(𝑡^𝑘/Γ_𝑞(𝑘+1)) on the time scale 𝕋_𝑞, where Γ_𝑞(𝑘+1)=[𝑘]_𝑞!=[1]_𝑞[2]_𝑞⋯[𝑘]_𝑞 with [𝑟]_𝑞=(1−𝑞^𝑟)/(1−𝑞). It is known that 𝑒_𝑞(𝑡)=𝐸_𝑞((1−𝑞)𝑡), where 𝐸_𝑞(𝑡) is a special case of the basic hypergeometric series, given by 𝐸_𝑞(𝑡)=₁𝜙₀0;𝑞,𝑡=^∞_𝑛=01−𝑞^𝑛𝑡⁻¹=^∞_𝑛=0𝑡^𝑛𝑞_𝑛, (25) where (𝑞)_𝑛=(1−𝑞)(1−𝑞²)⋯(1−𝑞^𝑛) is the 𝑞-Pochhammer symbol.

3. The Main Results

Throughout the remaining part of the paper, we assume that 0<𝛼≤1. Consider the following 𝑞-fractional initial value problem: _𝑞𝐶^𝛼_𝑎𝑦(𝑡)=𝑓𝑡,𝑦(𝑡),𝑎∈𝕋_𝑞,𝑦(𝑎)=𝑦₀. (26) Applying _𝑞∇_𝑎^−𝛼 to both sides of (26), we obtain 𝑦(𝑡)=𝑦₀+_𝑞∇_𝑎^−𝛼𝑓𝑡,𝑦(𝑡). (27) Set 𝑓𝑡,𝑦(𝑡)=𝑥(𝑡)𝑦(𝑡), (28) where (𝑡)≤10≤𝑥𝑡^𝛼1−𝑞^𝛼. (29) In the following, we present a comparison result for the fractional summation operator.

Theorem 8. Let 𝑤 and 𝑣 satisfy 𝑤(𝑡)≥𝑤(𝑎)+_𝑞∇_𝑎^−𝛼𝑥(𝑡)𝑤(𝑡),𝑣(𝑡)≤𝑣(𝑎)+_𝑞∇_𝑎^−𝛼𝑥(𝑡)𝑣(𝑡), (30)(31) respectively. If 𝑤(𝑎)≥𝑣(𝑎), then 𝑤(𝑡)≥𝑣(𝑡) for 𝑡∈Λ_𝑎={𝑎=𝑞^𝑛0,𝑞^𝑛0−1,…}.

Proof. Set 𝑢(𝑡)=𝑣(𝑡)−𝑤(𝑡). We claim that 𝑢(𝑡)≤0 for 𝑡∈Λ_𝑎. Let us assume that 𝑢(𝑠)≤0 is valid for 𝑠=𝑞^𝑛0,𝑞^𝑛0−1,…,𝑞^𝑛−1, where 𝑛<𝑛₀. Then, for 𝑡=𝑞^𝑛 we have 𝑢(𝑡)(𝑡)(𝑡)≤[𝑣(𝑎)(𝑎+=𝑣−𝑤−𝑤)]_𝑞∇_𝑎^−𝛼𝑥(𝑡)[𝑣(𝑡)−𝑤(𝑡)], (32) or 𝑣(𝑡)(𝑡)≤[𝑣(𝑎)(𝑎+1−𝑤−𝑤)]Γ_𝑞(𝛼)^𝑡_𝑎𝑡−𝑞𝑠_𝑞^𝛼−1𝑥(𝑠)(𝑣(𝑠)−𝑤(𝑠))∇_𝑞𝑠. (33) It follows that 𝑣(𝑡)(𝑡)≤[𝑣(𝑎)(𝑎+1−𝑤−𝑤)]Γ_𝑞(𝛼)_𝑎^𝑞𝑡(𝑡−𝑞𝑠)_𝑞^𝛼−1𝑥(𝑠)(𝑣(𝑠)−𝑤(𝑠))∇_𝑞𝑠+1Γ_𝑞(𝛼)^𝑡_𝑞𝑡𝑡−𝑞𝑠_𝑞^𝛼−1𝑥(𝑠)(𝑣(𝑠)−𝑤(𝑠))∇_𝑞𝑠. (34) Since 𝑣(𝑡)−𝑤(𝑡)≤0 and ∫^𝑡_𝜌(𝑡)𝑓(𝑠)∇𝑠=(𝑡−𝜌(𝑡))𝑓(𝑡), (34) can be written in the form 𝑣(𝑡)(𝑡)≤1−𝑤Γ_𝑞(𝛼)𝑡−𝑞𝑡𝑡−𝑞𝑡_𝑞^𝛼−1𝑥(𝑡𝑣(𝑡)(𝑡=)(−𝑤))1−𝑞^𝛼𝑡^𝛼𝑥(𝑡)(𝑣(𝑡)−𝑤(𝑡)), (35) where Γ_𝑞(𝛼)=(1−𝑞)_𝑞^𝛼−1/(1−𝑞)^𝛼−1 is used. It follows that 1−𝑥(𝑡)1−𝑞^𝛼𝑡^𝛼(𝑣(𝑡)−𝑤(𝑡))≤0. (36) By (29), we conclude that 𝑣(𝑡)−𝑤(𝑡)≤0.

Define the following operator _𝑞Ω_𝑥𝜙=_𝑞∇_𝑎^−𝛼𝑥(𝑡)𝜙(𝑡). (37) The following lemmas are essential in the proof of the main theorem. We only state these statements as their proofs are straightforward.

Lemma 9. For any constant 𝜆, one has _𝑞Ω_𝜆1≤_𝑞Ω_|𝜆|1. (38)

Lemma 10. For any constant 𝜆, one has _𝑞Ω^𝑛_𝜆𝜆1=^𝑛(𝑡−𝑎)_𝑞^𝑛𝛼Γ_𝑞(𝑛𝛼+1),𝑤ℎ𝑒𝑟𝑒𝑛∈ℕ. (39)

Lemma 11. Let 𝜆>0 be such that |𝑦(𝑡)|≤𝜆 for 𝑡∈Λ_𝑎. Then _𝑞Ω^𝑛_𝑦1≤_𝑞Ω^𝑛_𝜆1,𝑛∈ℕ. (40)

The next result together with Theorem 8 will give us the desired 𝑞-fractional Gronwall-type inequality.

Theorem 12. Let |𝑥(𝑡)|≤1/(1−𝑞)^𝛼𝑡^𝛼 for 𝑡∈Λ_𝑎∩[𝑎,𝑏]. Then, the 𝑞-fractional integral equation 𝑦(𝑡)=𝑦(𝑎)+_𝑞∇_𝑎^−𝛼𝑥(𝑡)𝑦(𝑡), (41) for 𝑡∈Λ_𝑎∩[𝑎,𝑏] where 𝑏∈ℝ, has a solution 𝑦(𝑡)=𝑦(𝑎)^∞_𝑞𝑘=0Ω^𝑘_𝑥1. (42)

Proof. The proof is achieved by utilizing the successive approximation method. Set 𝑦₀(𝑡)(𝑎),𝑦=𝑦_𝑛(𝑡)=𝑦(𝑎)+_𝑞∇_𝑎^−𝛼𝑥(𝑡)𝑦_𝑛−1(𝑡),for𝑛≥1. (43) We observe that 𝑦₁(𝑡)=𝑦(𝑎)+_𝑞∇_𝑎^−𝛼𝑥(𝑡)𝑦₀(𝑡)=𝑦(𝑎)+_𝑞Ω_𝑥𝑦(𝑎),𝑦₂(𝑡)=𝑦(𝑎)+_𝑞Ω_𝑥𝑦(𝑎)+_𝑞Ω_𝑥𝑦(𝑎)=𝑦(𝑎)+_𝑞Ω_𝑥𝑦(𝑎)+_𝑞Ω²_𝑥𝑦(𝑎). (44) Inductively, we end up with 𝑦_𝑛(𝑡)=𝑦(𝑎)^𝑛_𝑞𝑘=0Ω^𝑘_𝑥1,𝑛≥0. (45) Taking the limit as 𝑛→∞, we have 𝑦(𝑡)=𝑦(𝑎)^∞_𝑞𝑘=0Ω^𝑘_𝑥1. (46) It remains to prove the convergence of the series in (46). The subsequent analyses are carried out for 𝑎=0.
In virtue of (29), we obtain ^∞_𝑞𝑘=0Ω^𝑘_𝑥1≤^∞_𝑞𝑘=0Ω^𝑘_{1/𝑡𝛼1−𝑞}^𝛼1≤^∞_𝑘=0𝑞∇₀^−𝛼𝑘1𝑡^𝛼1−𝑞^𝛼≤11−𝑞^𝛼∞_𝑘=0^𝑞∇₀^−𝛼𝑘𝑡^−𝛼. (47) However, for 𝑘=1 we observe that _𝑞∇₀^−𝛼𝑡^−𝛼=Γ(1−𝛼)Γ()𝑡0+1⁰=Γ_𝑞(1−𝛼). (48) For 𝑘=2, it follows that _𝑞∇₀^−𝛼Γ_𝑞(1−𝛼)=Γ_𝑞(1−𝛼)_𝑞∇₀^−𝛼𝑡⁰=Γ_𝑞(1−𝛼)Γ_𝑞()𝑡𝛼+1^𝛼. (49) For 𝑘=3, we have _𝑞∇₀^−𝛼Γ_𝑞(1−𝛼)Γ_𝑞()𝑡𝛼+1^𝛼=Γ_𝑞(1−𝛼)Γ_𝑞()Γ𝛼+1^𝑞(𝛼+1)Γ_𝑞()𝑡𝛼+𝛼+1^2𝛼, (50) or _𝑞∇₀^−𝛼Γ_𝑞(1−𝛼)Γ_𝑞()𝑡𝛼+1^𝛼=Γ_𝑞(1−𝛼)Γ_𝑞()𝑡2𝛼+1^2𝛼. (51) For 𝑘=4, we get _𝑞∇₀^−𝛼Γ_𝑞(1−𝛼)Γ_𝑞()𝑡2𝛼+1^2𝛼=Γ_𝑞(1−𝛼)Γ_𝑞()𝑡3𝛼+1^3𝛼. (52) Therefore, (47) becomes ^∞_𝑞𝑘=0Ω^𝑘_𝑥1≤11−𝑞^𝛼1+Γ_𝑞(1−𝛼)+Γ_𝑞(1−𝛼)^∞_𝑘=1𝑡^𝑘𝛼Γ_𝑞().𝑘𝛼+1 (53) Let 𝑎_𝑘=𝑡^{(𝑘−1)𝛼}/Γ_𝑞((𝑘−1)𝛼+1). Then, 𝑎_𝑘𝑎_𝑘−1=𝑡^𝑘𝛼Γ_𝑞()Γ𝑘𝛼+1^𝑞((𝑘−1)𝛼+1)𝑡^{(𝑘−1)𝛼}=𝑡^𝛼Γ_𝑞((𝑘−1)𝛼+1)Γ_𝑞()≤Γ𝑘𝛼+1^𝑞((𝑘−1)𝛼+1)Γ_𝑞().𝑘𝛼+1 (54) We observe that Γ_𝑞((𝑘−1)𝛼+1)Γ_𝑞()=𝑘𝛼+11−𝑞_𝑞^{(𝑘−1)𝛼}1−𝑞^{(1−𝑘)𝛼}1−𝑞_𝑞^𝑘𝛼1−𝑞^−𝑘𝛼=1−𝑞_𝑞^{(𝑘−1)𝛼}1−𝑞^𝛼1−𝑞_𝑞^𝑘𝛼. (55) Setting 1−𝑞_𝑞^{(𝑘−1)𝛼}1−𝑞^𝛼1−𝑞_𝑞^𝑘𝛼∶=1−𝑞^𝛼∏^∞_𝑖=01−𝑞^𝑖+1/1−𝑞^𝑖𝑞^{𝑘𝛼−𝛼+1}∏^∞_𝑖=01−𝑞^𝑖+1/1−𝑞^𝑖𝑞^𝑘𝛼+1, (56) we deduce that lim_𝑘→∞1−𝑞^𝛼∏^∞_𝑖=01−𝑞^𝑖+1/1−𝑞^𝑖𝑞^{𝑘𝛼−𝛼+1}∏^∞_𝑖=01−𝑞^𝑖+1/1−𝑞^𝑖𝑞^𝑘𝛼+1=1−𝑞^𝛼∏^∞_𝑖=01−𝑞^𝑖+1∏^∞_𝑖=01−𝑞^𝑖+1=1−𝑞^𝛼<1. (57) Hence, convergence is guaranteed. In case 𝑎>0, we can proceed in a similar way taking into account that (𝑡−𝑎)_𝑞^𝑘𝛼/(𝑡−𝑎)_𝑞^{𝑘𝛼−𝛼}=(𝑡−𝑞^𝑘𝛼𝑞^𝛼𝑎)^𝛼_𝑞→𝑡^𝛼 as 𝑘→∞.

Theorem 13 (𝑞-fractional Gronwall's lemma). Let 𝑣 and 𝜇 be nonnegative real valued functions such that 0≤𝜇(𝑡)<1/𝑡^𝛼(1−𝑞)^𝛼 for all 𝑡∈Λ_𝑎 (in particular if 0≤𝜇(𝑡)<1/(1−𝑞)^𝛼) and 𝑣(𝑡)≤𝑣(𝑎)+_𝑞∇_𝑎^−𝛼𝑣(𝑡)𝜇(𝑡). (58) Then 𝑣(𝑡)≤𝑣(𝑎)^∞_𝑘=0Ω^𝑘_𝜇1. (59)

The proof of the previous statement is a straightforward implementation of Theorems 8 and 12 by setting 𝑤(𝑡)=𝑣(𝑎)∑^∞_𝑘=0(Ω^𝑘_𝜇1)(𝑡).

In case 𝛼=1, we deduce the following immediate consequence of Theorem 13 which can be considered as the well-known 𝑞-Gronwall's Lemma; consult, for instance, the paper [42].

Corollary 14. Let 0≤𝛿(𝑡)<1/(1−𝑞) for all 𝑡∈Λ_𝑎. If 𝑣(𝑡)≤𝑣(𝑎)+^𝑡_𝑎𝛿(𝑠)𝑣(𝑠)∇_𝑞𝑠, (60) then 𝑣(𝑡)≤𝑣(𝑎)𝑒_𝑞(𝑡,𝑎), (61) where 𝑒_𝑞(𝑡,𝑎)=_𝑞Ω₁(1,𝑡−𝑎) is the nabla 𝑞-exponential function on the time scale 𝕋_𝑞.

4. 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.

Example 15. Consider the following 𝑞-fractional initial value problems: _𝑞∇^𝛼_𝑎𝜑(𝑡)=𝑓𝑡,𝜑(𝑡),0<𝛼≤1,𝑎∈𝕋_𝑞,𝑡∈Λ_𝑎,𝜑(𝑎)=𝛾,_𝑞∇^𝛼_𝑎𝜓(𝑡)=𝑓𝑡,𝜓(𝑡),0<𝛼≤1,𝑎∈𝕋_𝑞,𝑡∈Λ_𝑎,𝜓(𝑎)=𝛽. (62) It follows that 𝜑(𝑡)−𝜓(𝑡)=𝛾−𝛽+_𝑞∇_𝑎^−𝛼𝑓𝑡,𝜑(𝑡)−𝑓𝑡,𝜓(𝑡). (63) Taking the absolute value, we obtain 𝜑(𝑡)−𝜓(𝑡)≤𝛾−𝛽+_𝑞∇_𝑎^−𝛼𝑓𝑡,𝜑(𝑡)−𝑓𝑡,𝜓(𝑡), (64) or 𝜑(𝑡)−𝜓(𝑡)≤𝛾−𝛽+𝐿_𝑞∇_𝑎^−𝛼𝜑(𝑡)−𝜓(𝑡). (65) By using Theorem 13, we get 𝜑(𝑡)−𝜓(𝑡)≤𝛾−𝛽^∞_𝑞𝑖=0Ω^𝑖_𝐿1=𝛾−𝛽_𝑞Ω_𝛼(𝐿,𝑡−𝑎). (66) Consider the following 𝑞-fractional initial value problem: _𝑞∇^𝛼_𝑎𝜙(𝑡)=𝑓𝑡,𝜙(𝑡),0<𝛼≤1,𝑎∈𝕋_𝑞,𝑡∈Λ_𝑎,𝜙(𝑎)=𝛾_𝑛, (67) where 𝛾_𝑛→𝛾. If the solution of (67) is denoted by 𝜙_𝑛, then for all 𝑡∈Λ_𝑎 we have 𝜑(𝑡)−𝜙_𝑛(𝑡)≤𝛾−𝛾_𝑛^∞_𝑞𝑖=0Ω^𝑖_𝐿1=𝛾−𝛾_𝑛𝑞Ω_𝛼(𝐿,𝑡−𝑎). (68) Hence |𝜑(𝑡)−𝜙_𝑛(𝑡)|→0 as 𝛾_𝑛→𝛾. This clearly verifies the dependence of solutions on the initial conditions.