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Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013, Article ID 543839, 7 pageshttp://dx.doi.org/10.1155/2013/543839

Research ArticleThe 𝑞-Fractional Analogue for Gronwall-Type Inequality

Thabet Abdeljawad1,2 and Jehad O. Alzabut2

1 Department of Mathematics and Computer Science, Çankaya University, Ögretmenler Caddesi 14, Balgat, 06530 Ankara, Turkey2Department of Mathematics and Physical Sciences, Prince Sultan University, P.O. Box 66833, Riyadh 11586, Saudi Arabia

Correspondence should be addressed to Jehad O. Alzabut; [email protected]

Received 8 May 2013; Accepted 7 July 2013

Academic Editor: Dashan Fan

Copyright © 2013 T. Abdeljawad and J. O. Alzabut. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

We utilize 𝑞-fractional Caputo initial value problems of order 0 < 𝛼 ≤ 1 to derive a 𝑞-analogue for Gronwall-type inequality. Someparticular cases are derived where 𝑞-Mittag-Leffler functions and 𝑞-exponential type functions are used. An example is given toillustrate the validity of the derived inequality.

1. Introduction

The fractional differential equations have conspicuously re-ceived considerable attention in the last two decades. Manyresearchers have investigated these equations due to theirsignificant applications in various fields of science and engi-neering such as in viscoelasticity, capacitor theory, electricalcircuits, electroanalytical chemistry, neurology, diffusion,control theory, and statistics; see, for instance, the mono-graphs [1–3].The study of 𝑞-difference equations, on the otherhand, has gained intensive interest in the last years. It hasbeen shown that these types of equations have numerousapplications in diverse fields and thus have evolved intomultidisciplinary subjects [4–11]. For more details on 𝑞-calculus, we refer the reader to the reference [12]. Thecorresponding fractional difference equations, however, havebeen comparably less considered. Indeed, the notions offractional calculus and 𝑞-calculus are tracked back to theworks of Jackson [13], respectively. However, the idea offractional difference equations is considered to be very recent;we suggest the new papers [14–28] whose authors havetaken the lead to promote the theory of fractional differenceequations.

The 𝑞-fractional difference equations which serve asa bridge between fractional difference equations and 𝑞-difference equations have become a main object of researchin the last years. Recently, many papers have appeared whichstudy 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 inequal-ities which are considered as effective tools for studyingsolutions properties have been also under consideration. Inparticular, we are interested in Gronwall’s inequality whichhas been amain target formany researchers.There are severalversions forGronwall’s inequality in the literature; we list herethose results which are concernedwith fractional order equa-tions [37–41]. To the best of authors’ observation, however,the 𝑞-fractional analogue for Gronwall-type inequality hasnot been investigated yet.

A primary purpose of this paper is to utilize the 𝑞-fractional Caputo initial value problems of order 0 < 𝛼 ≤ 1to derive a 𝑞-analogue for Gronwall-type inequality. Someparticular cases are derived where 𝑞-Mittag-Leffler functionsand 𝑞-exponential type functions are used. An example isgiven to illustrate the validity of the derived inequality.

2. Preliminary Assertions

Before stating and proving our main results, we introducesome definitions and notations that will be used throughoutthe paper. For 0 < 𝑞 < 1, we define the time scaleT𝑞 as follows:

T𝑞 = {𝑞𝑛: 𝑛 ∈ Z} ∪ {0} , (1)

2 Journal of Function Spaces and Applications

whereZ is the set of integers. In general, if 𝛼 is a nonnegativereal number then we define the time scale

T𝛼

𝑞= {𝑞𝑛+𝛼

: 𝑛 ∈ Z} ∪ {0} , (2)

and thus we may write T0𝑞= T𝑞. For a function 𝑓 : T𝑞 → R,

the nabla 𝑞-derivative of 𝑓 is given by

∇𝑞𝑓 (𝑡) =𝑓 (𝑡) − 𝑓 (𝑞𝑡)

(1 − 𝑞) 𝑡, 𝑡 ∈ T𝑞 − {0} . (3)

The nabla 𝑞-integral of 𝑓 is given by

∫

𝑡

0

𝑓 (𝑠) ∇𝑞𝑠 = (1 − 𝑞) 𝑡

∞

∑

𝑖=0

𝑞𝑖𝑓 (𝑡𝑞𝑖) ,

∫

𝑡

𝑎

𝑓 (𝑠) ∇𝑞𝑠 = ∫

𝑡

0

𝑓 (𝑠) ∇𝑞𝑠 − ∫

𝑎

0

𝑓 (𝑠) ∇𝑞𝑠, for 0 ≤ 𝑎 ∈ 𝑇𝑞.

(4)

The 𝑞-factorial function for 𝑛 ∈ N is defined by

(𝑡 − 𝑠)𝑛

𝑞=

𝑛−1

∏

𝑖=0

(𝑡 − 𝑞𝑖𝑠) . (5)

In case 𝛼 is a nonpositive integer, the 𝑞-factorial function isdefined by

(𝑡 − 𝑠)𝛼

𝑞= 𝑡𝛼

∞

∏

𝑖=0

1 − (𝑠/𝑡) 𝑞𝑖

1 − (𝑠/𝑡) 𝑞𝑖+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)𝛽+𝛾𝑞

= (𝑡 − 𝑠)𝛽

𝑞(𝑡 − 𝑞𝛽𝑠)𝛾

𝑞.

(II) (𝑎𝑡 − 𝑎𝑠)𝛽𝑞= 𝑎𝛽(𝑡 − 𝑠)

𝛽

𝑞.

(III) The nabla 𝑞-derivative of the 𝑞-factorial function withrespect to 𝑡 is

∇𝑞(𝑡 − 𝑠)𝛼

𝑞=1 − 𝑞𝛼

1 − 𝑞(𝑡 − 𝑠)

𝛼−1

𝑞. (7)

(IV) The nabla 𝑞-derivative of the 𝑞-factorial function withrespect to 𝑠 is

∇𝑞(𝑡 − 𝑠)𝛼

𝑞= −

1 − 𝑞𝛼

1 − 𝑞(𝑡 − 𝑞𝑠)

𝛼−1

𝑞. (8)

For a function 𝑓 : T𝛼𝑞→ R, the left 𝑞-fractional integral

𝑞∇−𝛼

𝑎of order 𝛼 ̸= 0, −1, −2, . . . and starting at 0 < 𝑎 ∈ T𝑞 is

defined by

𝑞∇−𝛼

𝑎𝑓 (𝑡) =

1

Γ𝑞 (𝛼)∫

𝑡

𝑎

(𝑡 − 𝑞𝑠)𝛼−1

𝑞𝑓 (𝑠) ∇𝑞𝑠, (9)

where

Γ𝑞 (𝛼 + 1) =1 − 𝑞𝛼

1 − 𝑞Γ𝑞 (𝛼) , Γ𝑞 (1) = 1, 𝛼 > 0. (10)

One should note that the left 𝑞-fractional integral 𝑞∇−𝛼

𝑎maps

functions defined on T𝑞 to functions defined on T𝑞.

Definition 2 (see [14]). If 0 < 𝛼 ∉ N, then the Caputo left𝑞-fractional derivative of order 𝛼 of a function 𝑓 is definedby

𝑞𝐶𝛼

𝑎𝑓 (𝑡) := 𝑞∇

−(𝑛−𝛼)

𝑎∇𝑛

𝑞𝑓 (𝑡)

=1

Γ (𝑛 − 𝛼)∫

𝑡

𝑎

(𝑡 − 𝑞𝑠)𝑛−𝛼−1

𝑞∇𝑛

𝑞𝑓 (𝑠) ∇𝑞𝑠,

(11)

where 𝑛 = [𝛼] + 1. In case 𝛼 ∈ N, we may write 𝑞𝐶𝛼

𝑎𝑓(𝑡) :=

∇𝑛

𝑞𝑓(𝑡).

Lemma 3 (see [14]). Assume that 𝛼 > 0 and 𝑓 is defined in asuitable domain. Then

𝑞∇−𝛼

𝑎 𝑞𝐶𝛼

𝑎𝑓 (𝑡) = 𝑓 (𝑡) −

𝑛−1

∑

𝑘=0

(𝑡 − 𝑎)𝑘

𝑞

Γ𝑞 (𝑘 + 1)∇𝑘

𝑞𝑓 (𝑎) , (12)

and if 0 < 𝛼 ≤ 1, then

𝑞∇−𝛼

𝑎 𝑞𝐶𝛼

𝑎𝑓 (𝑡) = 𝑓 (𝑡) − 𝑓 (𝑎) . (13)

For solving linear 𝑞-fractional equations, the followingidentity is essential:

𝑞∇−𝛼

𝑎(𝑥 − 𝑎)

𝜇

𝑞=

Γ𝑞 (𝜇 + 1)

Γ𝑞 (𝛼 + 𝜇 + 1)(𝑥 − 𝑎)

𝜇+𝛼

𝑞,

0 < 𝑎 < 𝑥 < 𝑏,

(14)

where 𝛼 ∈ R+ and 𝜇 ∈ (−1,∞). See, for instance, the recentpapers [14, 15] for more information.

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

Definition 4 (see [14]). For 𝑧, 𝑧0 ∈ C and R(𝛼) > 0, the 𝑞-Mittag-Leffler function is defined by

𝑞𝐸𝛼,𝛽 (𝜆, 𝑧 − 𝑧0) =

∞

∑

𝑘=0

𝜆𝑘(𝑧 − 𝑧0)

𝛼𝑘

𝑞

Γ𝑞 (𝛼𝑘 + 𝛽). (15)

In case 𝛽 = 1, we may use 𝑞𝐸𝛼(𝜆, 𝑧 − 𝑧0) := 𝑞𝐸𝛼,1(𝜆, 𝑧 − 𝑧0).

The following example clarifies how 𝑞-Mittag-Lefflerfunctions 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 leftCaputo 𝑞-fractional difference equation

𝑞𝐶𝛼

𝑎𝑦 (𝑡) = 𝜆𝑦 (𝑡) + 𝑓 (𝑡) , 𝑦 (𝑎) = 𝑎0, 𝑡 ∈ 𝑇𝑞. (16)

Journal of Function Spaces and Applications 3

Applying 𝑞∇−𝛼

𝑎to (16) and using (13), we end up with

𝑦 (𝑡) = 𝑎0 + 𝜆𝑞∇−𝛼

𝑎𝑦 (𝑡) + 𝑞∇

−𝛼

𝑎𝑓 (𝑡) . (17)

To obtain an explicit form for the solution, we apply themethod of successive approximation. Set 𝑦0(𝑡) = 𝑎0 and

𝑦𝑚 (𝑡) = 𝑎0 + 𝜆𝑞∇−𝛼

𝑎𝑦𝑚−1 (𝑡) + 𝑞∇

−𝛼

𝑎𝑓 (𝑡) ,

𝑚 = 1, 2, 3, . . . .

(18)

For𝑚 = 1, we have by the power formula (14)

𝑦1 (𝑡) = 𝑎0[

[

1 +

𝜆(𝑡 − 𝑎)(𝛼)

𝑞

Γ𝑞 (𝛼 + 1)

]

]

+ 𝑞∇−𝛼

𝑎𝑓 (𝑡) . (19)

For𝑚 = 2, we also see that

𝑦2 (𝑡) = 𝑎0 + 𝜆𝑎0 𝑞∇−𝛼

𝑎[1 +

(𝑡 − 𝑎)𝛼

𝑞

Γ𝑞 (𝛼 + 1)]

+ 𝑞∇−𝛼

𝑎𝑓 (𝑡) + 𝜆𝑞∇

−2𝛼

𝑎𝑓 (𝑡)

= 𝑎0 [1 +

𝜆(𝑡 − 𝑎)𝛼

𝑞

Γ𝑞 (𝛼 + 1)+

𝜆2(𝑡 − 𝑎)

2𝛼

𝑞

Γ𝑞 (2𝛼 + 1)]

+ 𝑞∇−𝛼

𝑎𝑓 (𝑡) + 𝜆𝑞∇

−2𝛼

𝑎𝑓 (𝑡) .

(20)

If we proceed inductively and let 𝑚 → ∞, we obtain thesolution𝑦 (𝑡)

= 𝑎0[

[

1 +

∞

∑

𝑘=1

𝜆𝑘(𝑡 − 𝑎)

𝑘𝛼

𝑞

Γ𝑞 (𝑘𝛼 + 1)

]

]

+ ∫

𝑡

𝑎

[

∞

∑

𝑘=1

𝜆𝑘−1

Γ𝑞 (𝛼𝑘)(𝑡 − 𝑞𝑠)

𝛼𝑘−1

𝑞]𝑓 (𝑠) ∇𝑞𝑠

= 𝑎0[

[

1 +

∞

∑

𝑘=1


𝑘𝛼

𝑞

Γ𝑞 (𝑘𝛼 + 1)

]

]

+ ∫

𝑡

𝑎

[

∞

∑

𝑘=0

𝜆𝑘

Γ𝑞 (𝛼𝑘 + 𝛼)(𝑡 − 𝑞𝑠)

𝛼𝑘+(𝛼−1)

𝑞]𝑓 (𝑠) ∇𝑞𝑠

= 𝑎0[

[

1 +

∞

∑

𝑘=1


𝑘𝛼

𝑞

Γ𝑞 (𝑘𝛼 + 1)

]

]

+ ∫

𝑡

𝑎

(𝑡 − 𝑞𝑠)(𝛼−1)

𝑞[

∞

∑

𝑘=0

𝜆𝑘

Γ𝑞 (𝛼𝑘 + 𝛼)(𝑡 − 𝑞

𝛼𝑠)(𝛼𝑘)

𝑞]𝑓 (𝑠) ∇𝑞𝑠.

(21)

That is,

𝑦 (𝑡) = 𝑎0𝑞𝐸𝛼 (𝜆, 𝑡 − 𝑎)

+ ∫

𝑡

𝑎

(𝑡 − 𝑞𝑠)𝛼−1

𝑞 𝑞𝐸𝛼,𝛼 (𝜆, 𝑡 − 𝑞

𝛼𝑠) 𝑓 (𝑠) ∇𝑞𝑠.

(22)

Remark 6. If instead we use the modified 𝑞-Mittag-Lefflerfunction

𝑞𝑒𝛼,𝛽 (𝜆, 𝑧 − 𝑧0) =

∞

∑

𝑘=0

𝜆𝑘(𝑧 − 𝑧0)

𝛼𝑘+(𝛽−1)

𝑞

Γ𝑞 (𝛼𝑘 + 𝛽)(23)

then, the solution representation (17) becomes

𝑦 (𝑡) = 𝑎0 𝑞𝑒𝛼 (𝜆, 𝑡 − 𝑎) + ∫

𝑡

𝑎𝑞𝑒𝛼,𝛼 (𝜆, 𝑡 − 𝑞𝑠) 𝑓 (𝑠) ∇𝑞𝑠. (24)

Remark 7. If we set 𝛼 = 1, 𝜆 = 1, 𝑎 = 0, and 𝑓(𝑡) = 0, wereach to the 𝑞-exponential formula 𝑒𝑞(𝑡) = ∑

∞

𝑘=0(𝑡𝑘/Γ𝑞(𝑘+1))

on the time scale T𝑞, where Γ𝑞(𝑘+1) = [𝑘]𝑞! = [1]𝑞[2]𝑞 ⋅ ⋅ ⋅ [𝑘]𝑞with [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 hypergeometricseries, given by

𝐸𝑞 (𝑡) = 1𝜙0 (0; 𝑞, 𝑡) =

∞

∏

𝑛=0

(1 − 𝑞𝑛𝑡)−1=

∞

∑

𝑛=0

𝑡𝑛

(𝑞)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 that0 < 𝛼 ≤ 1. Consider the following 𝑞-fractional initial valueproblem:

𝑞𝐶𝛼

𝑎𝑦 (𝑡) = 𝑓 (𝑡, 𝑦 (𝑡)) , 𝑎 ∈ T𝑞,

𝑦 (𝑎) = 𝑦0.

(26)

Applying 𝑞∇−𝛼

𝑎to both sides of (26), we obtain

𝑦 (𝑡) = 𝑦0 + 𝑞∇−𝛼

𝑎𝑓 (𝑡, 𝑦 (𝑡)) . (27)

Set

𝑓 (𝑡, 𝑦 (𝑡)) = 𝑥 (𝑡) 𝑦 (𝑡) , (28)

where

0 ≤ 𝑥 (𝑡) ≤1

𝑡𝛼(1 − 𝑞)𝛼 . (29)

In the following, we present a comparison result for thefractional summation operator.

Theorem 8. Let 𝑤 and V satisfy

𝑤 (𝑡) ≥ 𝑤 (𝑎) + 𝑞∇−𝛼

𝑎𝑥 (𝑡) 𝑤 (𝑡) , (30)

V (𝑡) ≤ V (𝑎) + 𝑞∇−𝛼

𝑎𝑥 (𝑡) V (𝑡) , (31)

respectively. If 𝑤(𝑎) ≥ V(𝑎), then 𝑤(𝑡) ≥ V(𝑡) for 𝑡 ∈ Λ 𝑎 = {𝑎 =𝑞𝑛0 , 𝑞𝑛0−1, . . .}.


Proof. Set 𝑢(𝑡) = V(𝑡) − 𝑤(𝑡). We claim that 𝑢(𝑡) ≤ 0for 𝑡 ∈ Λ 𝑎. Let us assume that 𝑢(𝑠) ≤ 0 is valid for 𝑠 =𝑞𝑛0 , 𝑞𝑛0−1, . . . , 𝑞

𝑛−1, where 𝑛 < 𝑛0. Then, for 𝑡 = 𝑞𝑛 we have

𝑢 (𝑡) = V (𝑡) − 𝑤 (𝑡)

≤ [V (𝑎) − 𝑤 (𝑎)] + 𝑞∇−𝛼

𝑎𝑥 (𝑡) [V (𝑡) − 𝑤 (𝑡)] ,

(32)

or

V (𝑡) − 𝑤 (𝑡) ≤ [V (𝑎) − 𝑤 (𝑎)]

+1

Γ𝑞 (𝛼)∫

𝑡

𝑎

(𝑡 − 𝑞𝑠)𝛼−1

𝑞𝑥 (𝑠) (V (𝑠) − 𝑤 (𝑠)) ∇𝑞𝑠.

(33)

It follows that

V (𝑡) − 𝑤 (𝑡)

≤ [V (𝑎) − 𝑤 (𝑎)]

+1

Γ𝑞 (𝛼)∫

𝑞𝑡

𝑎

(𝑡 − 𝑞𝑠)𝛼−1

𝑞𝑥 (𝑠) (V (𝑠) − 𝑤 (𝑠)) ∇𝑞𝑠

+1

Γ𝑞 (𝛼)∫

𝑡

𝑞𝑡

(𝑡 − 𝑞𝑠)𝛼−1

𝑞𝑥 (𝑠) (V (𝑠) − 𝑤 (𝑠)) ∇𝑞𝑠.

(34)

Since V(𝑡) − 𝑤(𝑡) ≤ 0 and ∫𝑡𝜌(𝑡)

𝑓(𝑠)∇𝑠 = (𝑡 − 𝜌(𝑡))𝑓(𝑡), (34)can be written in the form

V (𝑡) − 𝑤 (𝑡)

≤1

Γ𝑞 (𝛼)(𝑡 − 𝑞𝑡) (𝑡 − 𝑞𝑡)

𝛼−1

𝑞𝑥 (𝑡) (V (𝑡) − 𝑤 (𝑡))

= (1 − 𝑞)𝛼𝑡𝛼𝑥 (𝑡) (V (𝑡) − 𝑤 (𝑡)) ,

(35)

where Γ𝑞(𝛼) = (1 − 𝑞)𝛼−1

𝑞/(1 − 𝑞)

𝛼−1 is used. It follows that

(1 − 𝑥 (𝑡) (1 − 𝑞)𝛼𝑡𝛼) (V (𝑡) − 𝑤 (𝑡)) ≤ 0. (36)

By (29), we conclude that V(𝑡) − 𝑤(𝑡) ≤ 0.

Define the following operator

𝑞Ω𝑥𝜙 = 𝑞∇−𝛼

𝑎𝑥 (𝑡) 𝜙 (𝑡) . (37)

The following lemmas are essential in the proof of the maintheorem. We only state these statements as their proofs arestraightforward.

Lemma 9. For any constant 𝜆, one has𝑞Ω𝜆1

≤ 𝑞Ω|𝜆|1. (38)

Lemma 10. For any constant 𝜆, one has

𝑞Ω𝑛

𝜆1 =

𝜆𝑛(𝑡 − 𝑎)

𝑛𝛼

𝑞

Γ𝑞 (𝑛𝛼 + 1), where 𝑛 ∈ N. (39)

Lemma 11. Let 𝜆 > 0 be such that |𝑦(𝑡)| ≤ 𝜆 for 𝑡 ∈ Λ 𝑎. Then𝑞Ω𝑛

𝑦1≤ 𝑞Ω𝑛

𝜆1, 𝑛 ∈ N. (40)

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

Theorem 12. Let |𝑥(𝑡)| ≤ 1/(1 − 𝑞)𝛼𝑡𝛼 for 𝑡 ∈ Λ 𝑎 ∩ [𝑎, 𝑏].Then, the 𝑞-fractional integral equation

𝑦 (𝑡) = 𝑦 (𝑎) + 𝑞∇−𝛼

𝑎𝑥 (𝑡) 𝑦 (𝑡) , (41)

for 𝑡 ∈ Λ 𝑎 ∩ [𝑎, 𝑏] where 𝑏 ∈ R, has a solution

𝑦 (𝑡) = 𝑦 (𝑎)

∞

∑

𝑘=0

𝑞Ω𝑘

𝑥1. (42)

Proof. The proof is achieved by utilizing the successiveapproximation method. Set

𝑦0 (𝑡) = 𝑦 (𝑎) ,

𝑦𝑛 (𝑡) = 𝑦 (𝑎) + 𝑞∇−𝛼

𝑎𝑥 (𝑡) 𝑦𝑛−1 (𝑡) , for 𝑛 ≥ 1.

(43)

We observe that

𝑦1 (𝑡) = 𝑦 (𝑎) + 𝑞∇−𝛼

𝑎𝑥 (𝑡) 𝑦0 (𝑡)

= 𝑦 (𝑎) + 𝑞Ω𝑥𝑦 (𝑎) ,

𝑦2 (𝑡) = 𝑦 (𝑎) + 𝑞Ω𝑥 (𝑦 (𝑎) + 𝑞Ω𝑥𝑦 (𝑎))

= 𝑦 (𝑎) + 𝑞Ω𝑥𝑦 (𝑎) + 𝑞Ω2

𝑥𝑦 (𝑎) .

(44)

Inductively, we end up with

𝑦𝑛 (𝑡) = 𝑦 (𝑎)

𝑛

∑

𝑘=0

𝑞Ω𝑘

𝑥1, 𝑛 ≥ 0. (45)

Taking the limit as 𝑛 → ∞, we have

𝑦 (𝑡) = 𝑦 (𝑎)

∞

∑

𝑘=0

𝑞Ω𝑘

𝑥1. (46)

It remains to prove the convergence of the series in (46). Thesubsequent analyses are carried out for 𝑎 = 0.

In virtue of (29), we obtain∞

∑

𝑘=0

𝑞Ω𝑘

𝑥1 ≤

∞

∑

𝑘=0

𝑞Ω𝑘

1/𝑡𝛼(1−𝑞)𝛼1

≤

∞

∑

𝑘=0

(𝑞∇−𝛼

0)𝑘

(1

𝑡𝛼(1 − 𝑞)𝛼)

≤1

(1 − 𝑞)𝛼

∞

∑

𝑘=0

(𝑞∇−𝛼

0)𝑘

(𝑡−𝛼) .

(47)

However, for 𝑘 = 1 we observe that

𝑞∇−𝛼

0𝑡−𝛼

=Γ (1 − 𝛼)

Γ (0 + 1)𝑡0= Γ𝑞 (1 − 𝛼) . (48)


For 𝑘 = 2, it follows that

𝑞∇−𝛼

0(Γ𝑞 (1 − 𝛼))

= Γ𝑞(1 − 𝛼)𝑞∇−𝛼

0𝑡0=Γ𝑞 (1 − 𝛼)

Γ𝑞 (𝛼 + 1)𝑡𝛼.

(49)

For 𝑘 = 3, we have

𝑞∇−𝛼

0(Γ𝑞 (1 − 𝛼)

Γ𝑞 (𝛼 + 1)𝑡𝛼) =

Γ𝑞 (1 − 𝛼)

Γ𝑞 (𝛼 + 1)

Γ𝑞 (𝛼 + 1)

Γ𝑞 (𝛼 + 𝛼 + 1)𝑡2𝛼, (50)

or

𝑞∇−𝛼

0(Γ𝑞 (1 − 𝛼)

Γ𝑞 (𝛼 + 1)𝑡𝛼) =

Γ𝑞 (1 − 𝛼)

Γ𝑞 (2𝛼 + 1)𝑡2𝛼. (51)

For 𝑘 = 4, we get

𝑞∇−𝛼

0(Γ𝑞 (1 − 𝛼)

Γ𝑞 (2𝛼 + 1)𝑡2𝛼) =

Γ𝑞 (1 − 𝛼)

Γ𝑞 (3𝛼 + 1)𝑡3𝛼. (52)

Therefore, (47) becomes

∞

∑

𝑘=0

𝑞Ω𝑘

𝑥1

≤1

(1 − 𝑞)𝛼 [1 + Γ𝑞 (1 − 𝛼) + Γ𝑞 (1 − 𝛼)

∞

∑

𝑘=1

𝑡𝑘𝛼

Γ𝑞 (𝑘𝛼 + 1)] .

(53)

Let 𝑎𝑘 = 𝑡(𝑘−1)𝛼

/Γ𝑞((𝑘 − 1)𝛼 + 1). Then,

𝑎𝑘

𝑎𝑘−1

=𝑡𝑘𝛼

Γ𝑞 (𝑘𝛼 + 1)

Γ𝑞 ((𝑘 − 1) 𝛼 + 1)

𝑡(𝑘−1)𝛼

= 𝑡𝛼Γ𝑞 ((𝑘 − 1) 𝛼 + 1)

Γ𝑞 (𝑘𝛼 + 1)≤Γ𝑞 ((𝑘 − 1) 𝛼 + 1)

Γ𝑞 (𝑘𝛼 + 1).

(54)

We observe that

Γ𝑞 ((𝑘 − 1) 𝛼 + 1)

Γ𝑞 (𝑘𝛼 + 1)

=

(1 − 𝑞)(𝑘−1)𝛼

𝑞(1 − 𝑞)

(1−𝑘)𝛼

(1 − 𝑞)𝑘𝛼

𝑞(1 − 𝑞)

−𝑘𝛼=

(1 − 𝑞)(𝑘−1)𝛼

𝑞(1 − 𝑞)

𝛼

(1 − 𝑞)𝑘𝛼

𝑞

.

(55)

Setting

(1 − 𝑞)(𝑘−1)𝛼

𝑞(1 − 𝑞)

𝛼

(1 − 𝑞)𝑘𝛼

𝑞

:= (1 − 𝑞)𝛼∏∞

𝑖=0((1 − 𝑞

𝑖+1) / (1 − 𝑞

𝑖𝑞𝑘𝛼−𝛼+1

))

∏∞

𝑖=0((1 − 𝑞𝑖+1) / (1 − 𝑞𝑖𝑞𝑘𝛼+1))

,

(56)

we deduce that

lim𝑘→∞

(1 − 𝑞)𝛼∏∞

𝑖=0((1 − 𝑞

𝑖+1) / (1 − 𝑞

𝑖𝑞𝑘𝛼−𝛼+1

))

∏∞

𝑖=0((1 − 𝑞𝑖+1) / (1 − 𝑞𝑖𝑞𝑘𝛼+1))

= (1 − 𝑞)𝛼∏∞

𝑖=0(1 − 𝑞

𝑖+1)

∏∞

𝑖=0(1 − 𝑞𝑖+1)

= (1 − 𝑞)𝛼< 1.

(57)

Hence, convergence is guaranteed. In case 𝑎 > 0, we canproceed in a similar way taking into account that (𝑡−𝑎)𝑘𝛼

𝑞/(𝑡−

𝑎)𝑘𝛼−𝛼

𝑞= (𝑡 − 𝑞

𝑘𝛼𝑞𝛼𝑎)𝛼

𝑞→ 𝑡𝛼 as 𝑘 → ∞.

Theorem 13 (𝑞-fractional Gronwall’s lemma). Let V and 𝜇 benonnegative real valued functions such that 0 ≤ 𝜇(𝑡) < 1/𝑡𝛼(1−𝑞)𝛼 for all 𝑡 ∈ Λ 𝑎 (in particular if 0 ≤ 𝜇(𝑡) < 1/(1 − 𝑞)

𝛼) and

V (𝑡) ≤ V (𝑎) + 𝑞∇−𝛼

𝑎V (𝑡) 𝜇 (𝑡) . (58)

Then

V (𝑡) ≤ V (𝑎)∞

∑

𝑘=0

Ω𝑘

𝜇1. (59)

The proof of the previous statement is a straightforwardimplementation of Theorems 8 and 12 by setting 𝑤(𝑡) =V(𝑎)∑

∞

𝑘=0(Ω𝑘

𝜇1)(𝑡).

In case 𝛼 = 1, we deduce the following immediateconsequence of Theorem 13 which can be considered as thewell-known 𝑞-Gronwall’s Lemma; consult, for instance, thepaper [42].

Corollary 14. Let 0 ≤ 𝛿(𝑡) < 1/(1 − 𝑞) for all 𝑡 ∈ Λ 𝑎. If

V (𝑡) ≤ V (𝑎) + ∫𝑡

𝑎

𝛿 (𝑠) V (𝑠) ∇𝑞𝑠, (60)

then

V (𝑡) ≤ V (𝑎) 𝑒𝑞 (𝑡, 𝑎) , (61)

where 𝑒𝑞(𝑡, 𝑎) = 𝑞Ω1(1, 𝑡 − 𝑎) is the nabla 𝑞-exponentialfunction on the time scale T𝑞.

4. Applications

In this section, we show, by the help of the 𝑞-fractionalGronwall inequality proved in the previous section, that smallchanges in the initial conditions of Caputo 𝑞-fractional initialvalue 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 valueproblems:

𝑞∇𝛼

𝑎𝜑 (𝑡) = 𝑓 (𝑡, 𝜑 (𝑡)) , 0 < 𝛼 ≤ 1, 𝑎 ∈ T𝑞, 𝑡 ∈ Λ 𝑎,

𝜑 (𝑎) = 𝛾,

𝑞∇𝛼

𝑎𝜓 (𝑡) = 𝑓 (𝑡, 𝜓 (𝑡)) , 0 < 𝛼 ≤ 1, 𝑎 ∈ T𝑞, 𝑡 ∈ Λ 𝑎,

𝜓 (𝑎) = 𝛽.

(62)


It follows that

𝜑 (𝑡) − 𝜓 (𝑡) = (𝛾 − 𝛽) + 𝑞∇−𝛼

𝑎[𝑓 (𝑡, 𝜑 (𝑡)) − 𝑓 (𝑡, 𝜓 (𝑡))] .

(63)

Taking the absolute value, we obtain

𝜑 (𝑡) − 𝜓 (𝑡) ≤

𝛾 − 𝛽 +

𝑞∇−𝛼

𝑎𝑓 (𝑡, 𝜑 (𝑡)) − 𝑓 (𝑡, 𝜓 (𝑡))

,

(64)

or𝜑 (𝑡) − 𝜓 (𝑡)

≤𝛾 − 𝛽

+ 𝐿 𝑞∇−𝛼

𝑎

𝜑 (𝑡) − 𝜓 (𝑡) . (65)

By usingTheorem 13, we get

𝜑 (𝑡) − 𝜓 (𝑡) ≤

𝛾 − 𝛽

∞

∑

𝑖=0

𝑞Ω𝑖

𝐿1 =

𝛾 − 𝛽 𝑞Ω𝛼 (𝐿, 𝑡 − 𝑎) .

(66)

Consider the following 𝑞-fractional initial value problem:

𝑞∇𝛼

𝑎𝜙 (𝑡) = 𝑓 (𝑡, 𝜙 (𝑡)) , 0 < 𝛼 ≤ 1, 𝑎 ∈ T𝑞, 𝑡 ∈ Λ 𝑎,

𝜙 (𝑎) = 𝛾𝑛,

(67)

where 𝛾𝑛 → 𝛾. If the solution of (67) is denoted by 𝜙𝑛, thenfor all 𝑡 ∈ Λ 𝑎 we have

𝜑 (𝑡) − 𝜙𝑛 (𝑡) ≤

𝛾 − 𝛾𝑛

∞

∑

𝑖=0

𝑞Ω𝑖

𝐿1 =

𝛾 − 𝛾𝑛 𝑞Ω𝛼 (𝐿, 𝑡 − 𝑎) .

(68)

Hence |𝜑(𝑡) − 𝜙𝑛(𝑡)| → 0 as 𝛾𝑛 → 𝛾. This clearly verifies thedependence of solutions on the initial conditions.

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