# Theq-Fractional Analogue for Gronwall-Type Inequality

Authors: Thabet Abdeljawad; Jehad O. Alzabut
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We utilizeq-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.


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## Body

## 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.Theq-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 theq-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. For0<q<1, we define the time scale 𝕋q as follows:
(1)𝕋q={qn:n∈ℤ}∪{0},
where ℤ is the set of integers. In general, if α is a nonnegative real number then we define the time scale
(2)𝕋qα={qn+α:n∈ℤ}∪{0},
and thus we may write 𝕋q0=𝕋q. For a function f:𝕋q→ℝ, the nabla q-derivative of f is given by
(3)∇qf(t)=f(t)-f(qt)(1-q)t,t∈𝕋q-{0}.
The nabla q-integral of f is given by
(4)∫0t‍f(s)∇qs=(1-q)t∑i=0∞‍qif(tqi),∫at‍f(s)∇qs=∫0t‍f(s)∇qs-∫0a‍f(s)∇qs,for0≤a∈Tq.
The q-factorial function for n∈ℕ is defined by
(5)(t-s)qn=∏i=0n-1‍(t-qis).
In case α is a nonpositive integer, the q-factorial function is defined by
(6)(t-s)qα=tα∏i=0∞‍1-(s/t)qi1-(s/t)qi+α.
In the following lemma, we present some properties of q-factorial functions.Lemma 1 (see [32]).
Forα,γ,β∈ℝ, one has the following.  (I)
(t-s)qβ+γ=(t-s)qβ(t-qβs)qγ.(II)
(at-as)qβ=aβ(t-s)qβ.(III)
The nablaq-derivative of the q-factorial function with respect to t is
(7)∇q(t-s)qα=1-qα1-q(t-s)qα-1.(IV)
The nablaq-derivative of the q-factorial function with respect to s is
(8)∇q(t-s)qα=-1-qα1-q(t-qs)qα-1.For a functionf:𝕋qα→ℝ, the left q-fractional integral q∇a-α of order α≠0,-1,-2,… and starting at 0<a∈𝕋q is defined by
(9)q∇a-αf(t)=1Γq(α)∫at‍(t-qs)qα-1f(s)∇qs,
where
(10)Γq(α+1)=1-qα1-qΓq(α),Γq(1)=1,α>0.
One should note that the left q-fractional integral q∇a-α maps functions defined on 𝕋q to functions defined on 𝕋q.Definition 2 (see [14]).
If0<α∉ℕ, then the Caputo left q-fractional derivative of order α of a function f is defined by
(11)qCaαf(t)≔q∇a-(n-α)∇qnf(t)=1Γ(n-α)∫at‍(t-qs)qn-α-1∇qnf(s)∇qs,
where n=[α]+1. In case α∈ℕ, we may write qCaαf(t)≔∇qnf(t).Lemma 3 (see [14]).
Assume thatα>0 and f is defined in a suitable domain. Then
(12)q∇a-αqCaαf(t)=f(t)-∑k=0n-1‍(t-a)qkΓq(k+1)∇qkf(a),
and if 0<α≤1, then
(13)q∇a-αqCaαf(t)=f(t)-f(a).For solving linearq-fractional equations, the following identity is essential:
(14)q∇a-α(x-a)qμ=Γq(μ+1)Γq(α+μ+1)(x-a)qμ+α,0<a<x<b,
where α∈ℝ+ and μ∈(-1,∞). See, for instance, the recent papers [14, 15] for more information.Theq-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]).
Forz,z0∈C and ℜ(α)>0, the q-Mittag-Leffler function is defined by
(15)qEα,β(λ,z-z0)=∑k=0∞‍λk(z-z0)qαkΓq(αk+β).
In case β=1, we may use qEα(λ,z-z0)≔qEα,1(λ,z-z0).The following example clarifies howq-Mittag-Leffler functions can be used to express the solutions of Caputo q-fractional linear initial value problems.Example 5 (see [14]).
Let0<α≤1 and consider the left Caputo q-fractional difference equation
(16)qCaαy(t)=λy(t)+f(t),y(a)=a0,t∈Tq.
Applying q∇a-α to (16) and using (13), we end up with
(17)y(t)=a0+λq∇a-αy(t)+q∇a-αf(t).
To obtain an explicit form for the solution, we apply the method of successive approximation. Set y0(t)=a0 and
(18)ym(t)=a0+λq∇a-αym-1(t)+q∇a-αf(t),m=1,2,3,….
For m=1, we have by the power formula (14)
(19)y1(t)=a0[1+λ(t-a)q(α)Γq(α+1)]+q∇a-αf(t).
For m=2, we also see that
(20)y2(t)=a0+λa0q∇a-α[1+(t-a)qαΓq(α+1)]+q∇a-αf(t)+λq∇a-2αf(t)=a0[1+λ(t-a)qαΓq(α+1)+λ2(t-a)q2αΓq(2α+1)]+q∇a-αf(t)+λq∇a-2αf(t).
If we proceed inductively and let m→∞, we obtain the solution
(21)y(t)=a0[1+∑k=1∞‍λk(t-a)qkαΓq(kα+1)]+∫at‍[∑k=1∞‍λk-1Γq(αk)(t-qs)qαk-1]f(s)∇qs=a0[1+∑k=1∞‍λk(t-a)qkαΓq(kα+1)]+∫at‍[∑k=0∞‍λkΓq(αk+α)(t-qs)qαk+(α-1)]f(s)∇qs=a0[1+∑k=1∞‍λk(t-a)qkαΓq(kα+1)]+∫at‍(t-qs)q(α-1)[∑k=0∞‍λkΓq(αk+α)(t-qαs)q(αk)]f(s)∇qs.
That is,
(22)y(t)=a0qEα(λ,t-a)+∫at‍(t-qs)qα-1qEα,α(λ,t-qαs)f(s)∇qs.Remark 6.
If instead we use the modifiedq-Mittag-Leffler function
(23)qeα,β(λ,z-z0)=∑k=0∞‍λk(z-z0)qαk+(β-1)Γq(αk+β)
then, the solution representation (17) becomes
(24)y(t)=a0qeα(λ,t-a)+∫at‍qeα,α(λ,t-qs)f(s)∇qs.Remark 7.
If we setα=1, λ=1, a=0, and f(t)=0, we reach 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]q with [r]q=(1-qr)/(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
(25)Eq(t)=1ϕ0(0;q,t)=∏n=0∞‍(1-qnt)-1=∑n=0∞‍tn(q)n,
where (q)n=(1-q)(1-q2)⋯(1-qn) 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 value problem:
(26)qCaαy(t)=f(t,y(t)),a∈𝕋q,y(a)=y0.
Applying q∇a-α to both sides of (26), we obtain
(27)y(t)=y0+q∇a-αf(t,y(t)).
Set
(28)f(t,y(t))=x(t)y(t),
where
(29)0≤x(t)≤1tα(1-q)α.
In the following, we present a comparison result for the fractional summation operator.Theorem 8.
Letw and v satisfy
(30)w(t)≥w(a)+q∇a-αx(t)w(t),(31)v(t)≤v(a)+q∇a-αx(t)v(t),
respectively. If w(a)≥v(a), then w(t)≥v(t) for t∈Λa={a=qn0,qn0-1,…}.Proof.
Setu(t)=v(t)-w(t). We claim that u(t)≤0 for t∈Λa. Let us assume that u(s)≤0 is valid for s=qn0,qn0-1,…,qn-1, where n<n0. Then, for t=qn we have
(32)u(t)=v(t)-w(t)≤[v(a)-w(a)]+q∇a-αx(t)[v(t)-w(t)],
or
(33)v(t)-w(t)≤[v(a)-w(a)]+1Γq(α)∫at‍(t-qs)qα-1x(s)(v(s)-w(s))∇qs.
It follows that
(34)v(t)-w(t)≤[v(a)-w(a)]+1Γq(α)∫aqt‍(t-qs)qα-1x(s)(v(s)-w(s))∇qs+1Γq(α)∫qtt‍(t-qs)qα-1x(s)(v(s)-w(s))∇qs.
Since v(t)-w(t)≤0 and ∫ρ(t)t‍f(s)∇s=(t-ρ(t))f(t), (34) can be written in the form
(35)v(t)-w(t)≤1Γq(α)(t-qt)(t-qt)qα-1x(t)(v(t)-w(t))=(1-q)αtαx(t)(v(t)-w(t)),
where Γq(α)=(1-q)qα-1/(1-q)α-1 is used. It follows that
(36)(1-x(t)(1-q)αtα)(v(t)-w(t))≤0.
By (29), we conclude that v(t)-w(t)≤0.Define the following operator(37)qΩxϕ=q∇a-αx(t)ϕ(t).
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 constantλ, one has
(38)|qΩλ1|≤qΩ|λ|1.Lemma 10.
For any constantλ, one has
(39)qΩλn1=λn(t-a)qnαΓq(nα+1),wheren∈ℕ.Lemma 11.
Letλ>0 be such that |y(t)|≤λ for t∈Λa. Then
(40)|qΩyn1|≤qΩλn1,n∈ℕ.The next result together with Theorem8 will give us the desired q-fractional Gronwall-type inequality.Theorem 12.
Let|x(t)|≤1/(1-q)αtα for t∈Λa∩[a,b]. Then, the q-fractional integral equation
(41)y(t)=y(a)+q∇a-αx(t)y(t),
for t∈Λa∩[a,b] where b∈ℝ, has a solution
(42)y(t)=y(a)∑k=0∞‍qΩxk1.Proof.
The proof is achieved by utilizing the successive approximation method. Set(43)y0(t)=y(a),yn(t)=y(a)+q∇a-αx(t)yn-1(t),forn≥1.
We observe that
(44)y1(t)=y(a)+q∇a-αx(t)y0(t)=y(a)+qΩxy(a),y2(t)=y(a)+qΩx(y(a)+qΩxy(a))=y(a)+qΩxy(a)+qΩx2y(a).
Inductively, we end up with
(45)yn(t)=y(a)∑k=0n‍qΩxk1,n≥0.
Taking the limit as n→∞, we have
(46)y(t)=y(a)∑k=0∞‍qΩxk1.
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
(47)∑k=0∞‍qΩxk1≤∑k=0∞‍qΩ1/tα(1-q)αk1≤∑k=0∞‍(q∇0-α)k(1tα(1-q)α)≤1(1-q)α∑k=0∞‍(q∇0-α)k(t-α).
However, for k=1 we observe that
(48)q∇0-αt-α=Γ(1-α)Γ(0+1)t0=Γq(1-α).
For k=2, it follows that
(49)q∇0-α(Γq(1-α))=Γq(1-α)q∇0-αt0=Γq(1-α)Γq(α+1)tα.
For k=3, we have
(50)q∇0-α(Γq(1-α)Γq(α+1)tα)=Γq(1-α)Γq(α+1)Γq(α+1)Γq(α+α+1)t2α,
or
(51)q∇0-α(Γq(1-α)Γq(α+1)tα)=Γq(1-α)Γq(2α+1)t2α.
For k=4, we get
(52)q∇0-α(Γq(1-α)Γq(2α+1)t2α)=Γq(1-α)Γq(3α+1)t3α.
Therefore, (47) becomes
(53)∑k=0∞‍qΩxk1≤1(1-q)α[1+Γq(1-α)+Γq(1-α)∑k=1∞‍tkαΓq(kα+1)].
Let ak=t(k-1)α/Γq((k-1)α+1). Then,
(54)akak-1=tkαΓq(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).
We observe that
(55)Γq((k-1)α+1)Γq(kα+1)=(1-q)q(k-1)α(1-q)(1-k)α(1-q)qkα(1-q)-kα=(1-q)q(k-1)α(1-q)α(1-q)qkα.
Setting
(56)(1-q)q(k-1)α(1-q)α(1-q)qkα≔(1-q)α∏i=0∞((1-qi+1)/(1-qiqkα-α+1))∏i=0∞((1-qi+1)/(1-qiqkα+1)),
we deduce that
(57)limk→∞(1-q)α∏i=0∞((1-qi+1)/(1-qiqkα-α+1))∏i=0∞((1-qi+1)/(1-qiqkα+1))=(1-q)α∏i=0∞(1-qi+1)∏i=0∞(1-qi+1)=(1-q)α<1.
Hence, convergence is guaranteed. In case a>0, we can proceed in a similar way taking into account that (t-a)qkα/(t-a)qkα-α=(t-qkαqαa)qα→tα as k→∞.Theorem 13 (q-fractional Gronwall's lemma).
Letv and μ be nonnegative real valued functions such that 0≤μ(t)<1/tα(1-q)α for all t∈Λa (in particular if 0≤μ(t)<1/(1-q)α) and
(58)v(t)≤v(a)+q∇a-αv(t)μ(t).
Then
(59)v(t)≤v(a)∑k=0∞‍Ωμk1.The proof of the previous statement is a straightforward implementation of Theorems8 and 12 by setting w(t)=v(a)∑k=0∞‍(Ωμk1)(t).In caseα=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.
Let0≤δ(t)<1/(1-q) for all t∈Λa. If
(60)v(t)≤v(a)+∫at‍δ(s)v(s)∇qs,
then
(61)v(t)≤v(a)eq(t,a),
where eq(t,a)=qΩ1(1,t-a) is the nabla q-exponential function on the time scale 𝕋q.


## 4. Applications

In this section, we show, by the help of theq-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.Letf(t,y) satisfy a Lipschitz condition with constant 0≤L<1 for all t and y.Example 15.
Consider the followingq-fractional initial value problems:
(62)q∇aαφ(t)=f(t,φ(t)),0<α≤1,a∈𝕋q,t∈Λa,φ(a)=γ,q∇aαψ(t)=f(t,ψ(t)),0<α≤1,a∈𝕋q,t∈Λa,ψ(a)=β.
It follows that
(63)φ(t)-ψ(t)=(γ-β)+q∇a-α[f(t,φ(t))-f(t,ψ(t))].
Taking the absolute value, we obtain
(64)|φ(t)-ψ(t)|≤|γ-β|+|q∇a-αf(t,φ(t))-f(t,ψ(t))|,
or
(65)|φ(t)-ψ(t)|≤|γ-β|+Lq∇a-α|φ(t)-ψ(t)|.
By using Theorem 13, we get
(66)|φ(t)-ψ(t)|≤|γ-β|∑i=0∞‍qΩLi1=|γ-β|qΩα(L,t-a).
Consider the following q-fractional initial value problem:
(67)q∇aαϕ(t)=f(t,ϕ(t)),0<α≤1,a∈𝕋q,t∈Λa,ϕ(a)=γn,
where γn→γ. If the solution of (67) is denoted by ϕn, then for all t∈Λa we have
(68)|φ(t)-ϕn(t)|≤|γ-γn|∑i=0∞‍qΩLi1=|γ-γn|qΩα(L,t-a).
Hence |φ(t)-ϕn(t)|→0 as γn→γ. This clearly verifies the dependence of solutions on the initial conditions.



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