# Finite-Time Stability of Solutions for Nonlinearq-Fractional Difference Coupled Delay Systems

Authors: Jingfeng Wang; Chuanzhi Bai
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

In this paper, we investigate and prove a new discreteq-fractional version of the coupled Gronwall inequality. By applying this result, the finite-time stability criteria of solutions for a class of nonlinear q-fractional difference coupled delay systems are obtained. As an application, an example is provided to demonstrate the effectiveness of our result.


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## Body

## 1. Introduction

Theq-difference equations have numerous applications in diverse fields in recent years and have gained intensive interest [1–4]. For more details on q-calculus, we recommend the readers to [5]. In the last two decades, the fractional difference equations have recently received considerable attention in many fields of science and engineering, see [6–9] and the references therein. We know that the q-fractional difference equations can be used as a bridge between fractional difference equations and q-difference equations, and there are many papers on this research direction which have been appeared in [10–16]. And, we recommend [17] and the papers cited therein.For0<q<1, we define the time scale Tq=qn:n∈ℤ∪0, where ℤ is the set of integers. For a=qn0 and n0∈ℤ, we denote Ta=a,∞q=q−ia:i=0,1,2,….In [18], Abdeljawad and Alzabut established a discrete q-fractional version of the Gronwall-type inequality as follows:Theorem 1 (see [18]).
Letα>0, u and μ be nonnegative real valued functions such that 0≤μt<1/tα1−qα, for all t∈Ta, and(1)ut≤ua+q∇a−αutμt,
Then,(2)ut≤ua∑k=0∞Eqμk1,where qEμk1=μkt−aqkα/Γqkα+1.Abdeljawad et al., in [19], extended the above inequality and obtained the following generalized q-fractional Gronwall-type inequality.Theorem 2 (see [19]).
Letα>0, u and ν be nonnegative functions, and wt be nonnegative and nondecreasing function, for t∈a,∞q, such that wt≤M, where M is a constant. If(3)ut≤νt+wtq∇a−αut,then(4)ut≤νt+∑k=1∞wtΓqαk∇qa−kανt.Based on the above result, Abdeljawad et al. investigated the following nonlinear delayq-fractional difference system:(5)Cqaαxt=A0xt+A1xτt+ft,xt,xτt,t∈a,∞q,xt=ϕt,t∈Iτ,where Cqaα means the Caputo fractional difference of order α, Iτ=τa,q−1τa,q−2τa,…,a, τ=qd∈Tq, with d∈ℕ0=0,1,2,….An interesting topic in control theory is finite-time control, and the objective of finite-time control is to design a control law, making the system state converge to the origin in finite time. In [20], Sun et al. developed the finite-time state feedback stabilisation scheme for nonlinear time-delay systems with high-order and low-order nonlinearities. Recently, Wang and Xiang [21, 22] presented a finite-time output feedback control scheme for a class of nonlinear systems and nonlinear time-delay in the p-normal form, respectively. In [23], Modiri and Mobayen studied the synchronization of fractional-order uncertain chaotic systems in the finite time. Mofid et al., in [24], considered the sliding mode disturbance observer control of a class of fractional-order chaotic systems by using adaptive synchronization. Moreover, the observer-based state feedback stabilizer design for a class of chaotic systems and the fixed-time attitude control for a flexible spacecraft in the presence of actuator faults, external disturbances, and coupling effect of flexible modes have been considered in [25, 26], respectively.On the contrary, finite-time stability is a method which is much valuable to analyze the transient behavior of nature of a system within a finite interval of time. In recent decades, the finite-time stability analysis of fractional differential systems have recently considerable attention, see, for instance, [27–31] and the references therein. However, till now, few researchers focus on finite-time stability of fractional delay difference systems.Motivated by the above works, we will to extend theq-fractional Gronwall-type inequality (Theorem 2) to coupled q-fractional Gronwall inequality. As an application, we establish a finite-time stability criterion of the following nonlinear coupled delay q-fractional difference system:(6)Cqaαxt=A0yt+A1yτt+ft,yt,yτt,t∈a,1q,Cqaαyt=B0xt+B1xτt+gt,xt,xτt,t∈a,1q,xt=ϕt,yt=ψt,t∈Iτ,where a,1q=a,1∩Ta, Iτ=τa,q−1τa,q−2τa,…,a, τ=qd∈Tq with d∈ℕ0=0,1,2,…, Cqaα and Cqaβ mean the Caputo fractional difference of order α∈0,1 and order β∈0,1, respectively, and the constant matrices A0, A1, B0, and B1 are of appropriate dimensions.In this paper, the coupledq-fractional Gronwall inequality is studied and given for the first time, which is a powerful tool and method to deal with finite-time stability and other stability of nonlinear coupled delay q-fractional difference systems. And, we studied the finite-time stability of a class of nonlinear coupled delay q-fractional difference system by using this inequality.The organization of this paper is given as follows. In Section2, we give some notations, definitions, and preliminaries. Section 3 is devoted to proving a coupled q-fractional Gronwall inequality. In Section 4, the finite-time stability theorem of nonlinear coupled delay q-fractional difference system is proved. In Section 5, an example is given to illustrate our theoretical result. Finally, the paper is concluded in Section 6.


## 2. Preliminaries

In this section, we provided some basic definitions and lemmas which are used in the sequel.Letf:Tq⟶ℝ, and we define the nabla q-derivative of f by(7)∇qft=ft−fqt1−qt,t∈Tq∖0.The nablaq-integral of f has the following form:(8)∫0tfs∇qs=1−qt∑i=0∞qiftqi,and for 0≤a∈Tq,(9)∫atfs∇qs=∫0tfs∇qs−∫0afs∇qs.The definition of theq-factorial function for a nonpositive integer α is given by(10)t−sqα=tα∏i=0∞1−s/tqi1−s/tqi+α.For a functionf:Tq⟶ℝ, the left q-fractional integral ∇qa−α of order α≠0,−1,−2,… and starting at 0<a∈Tq is defined by(11)∇qa−αft=1Γqα∫att−qsqα−1fs∇qs,where(12)Γqα+1=1−qα1−qΓqα,Γq1=1,α>0.Definition 1 (see [11]).
Let0<α∉ℕ. Then, the Caputo left q-fractional derivative of order α of a function f defined on Tq is defined by(13)Cqaαft≔∇qa−n−α∇qnft=1Γqn−α∫att−qsqn−α−1∇qnfs∇qs,where n=α+1.Lemma 1 (see [11]).
Letα>0 and f be defined in a suitable domain. Thus,(14)∇qa−αCqaαft=ft−∑k=0n−1t−aqkΓqk+1∇qkfa,and if 0<α≤1, we have(15)∇qa−αCqaαft=ft−fa.The following identity plays a crucial role in solving the linearq-fractional equations:(16)∇qa−αx−aqμ=Γqμ+1Γqα+μ+1x−aqμ+α0<a<x<b,where α∈ℝ+ and μ∈−1,∞. The q-analog of the Mittag–Leffler function with double index α,β is introduced as follows.Definition 2 (see [11]).
Forz,z0∈ℂ and ℜα>0, the q-Mittag–Leffler function is defined by(17)Eqα,βλ,z−z0=∑k=0∞λkz−z0qαkΓqαk+β.
In the caseβ=1, we utilize Eqαλ,z−z0=Eqα,1λ,z−z0.
Moreover, the modifiedq-Mittag–Leffler function is used in [19] as follows:(18)eqα,βλ,z−z0=∑k=0∞λkz−z0qαk+β−1Γqαk+β.


## 3. A Generalized Coupledq-Fractional Gronwall Inequality

In this section, we give and prove the following a generalized coupledq-fractional Gronwall inequality, which extend a generalized q-fractional Gronwall inequality in Theorem 2.Theorem 3.
Assume thatut, vt, and git (i=1,2) are nonnegative functions for t∈Ta. Let wit (i=1,2) be nonnegative and nondecreasing functions for t∈Ta with wit≤Mi, where Mi are constants (i=1,2). If(19)ut≤g1t+w1tq∇a−αvt,t∈a,1q,vt≤g2t+w2tq∇a−βut,t∈a,1q,and(20)M1M21−qα+β<1,hold, then(21)ut≤g1t+w1tΓqα∫att−qsqα−1g2s∇qs+∑k=1∞w1tkw2tk∇qa−kα+βg1t+w1tΓqα∫qtt−qsqα−1g2s∇qs,and(22)vt≤g2t+w2tΓqβ∫att−qsqβ−1g1s∇qs+∑k=1∞w1tkw2tk∇qa−kα+βg2t+w2tΓqβ∫att−qsqβ−1g1s∇qs.Proof.
Let(23)Avt=w1tΓqα∫att−qsqα−1vs∇qs,t∈Ta,and(24)But=w2tΓqβ∫att−qsqβ−1us∇qs,t∈Ta.
According to (19), one has(25)ut≤g1t+Avt,vt≤g2t+But.
By (25) and the monotonicity of the operators A and B, we obtain(26)ut≤g1t+Ag2t+But=g1t+Ag2t+ABut≤g1t+Ag2t+ABg1t+Ag2t+ABut=g1t+ABg1t+Ag2t+ABAg2t+AB2ut≤∑k=0n−1ABkg1t+∑k=0n−1ABkAg2t+Anut,t∈Ta.
Similarly, we obtain(27)vt≤∑k=0n−1BAkg2t+∑k=0n−1BAkBg1t+BAnvt,t∈Ta,where AB0g1t=g1t and BA0g2t=g2t.
In the following, we will prove that(28)ABnut≤w1tnw2tn∇qa−nα+βut,(29)BAnvt≤w1tnw2tn∇qa−nα+βvt,where t∈Ta and(30)limn⟶∞ABnut=0,limn⟶∞BAnvt=0.
We know that (28) and (29) are true for n=1. In fact, one has(31)ABut=ABut=w1tΓqα∫att−qsqα−1Bus∇qs=w1tΓqαΓqβ∫att−qsqα−1w2s∫ast−qrqβ−1ur∇qr∇qs≤w1tw2tΓqαΓqβ∫att−qsqα−1∫ast−qrqβ−1ur∇qr∇qs=w1tw2tΓqαΓqβ∫at∫rtt−qsqα−1s−qrqβ−1ur∇qr∇qs=w1tw2tΓqβ∫at1Γqα∫rtt−qsqα−1s−qrqβ−1∇qsur∇qr=w1tw2tΓqβ∫at∇qqr−αt−qrqβ−1ur∇qr,where ∇qqr−αut=1/Γqα∫qrtt−qsqα−1us∇qs has been used. By (16), we have(32)∇qa−αx−aqμ=Γqμ+1Γqα+μ+1x−aqμ+α0<a<x<b,μ>−1.
Combining (31) with (32), one has(33)ABut≤w1tw2tΓqβ∫atΓqβΓqα+βt−qrqα+β−1ur∇qr=w1tw2tΓqα+β∫att−qrqα+β−1ur∇qr=w1tw2tq∇a−α+βut.
Similarly, one has(34)BAvt≤w1tw2tq∇a−α+βvt.
Thus, (28) and (29) are valid for n=1. Assume that (28) and (29) are true, for n=k, which are(35)ABkut≤w1tkw2tk∇qa−kα+βut,(36)BAkvt≤w1tkw2tk∇qa−kα+βvt.
Forn=k+1 and t∈ℕa+1+kν+μ, by using (33) and (35) and the nondecreasing functions w1t and w2t, we obtain(37)ABk+1ut=ABABkut≤w1tw2tΓqα+β∫att−qsqα+β−1∫asw1skw2skΓqkα+βt−qrqkα+β−1ur∇qr∇qs≤w1tk+1w2tk+1Γqkα+β∫at∫rt1Γqα+βt−qsqα+β−1s−qrqkα+β−1∇qsur∇qr=w1tk+1w2tk+1Γqkα+β∫at∇qqr−α+βt−qrqkα+β−1ur∇qr,where ∇qqr−α+βut=1/Γqα∫qrtt−qsqα+β−1us∇qs has been used. By using (16) and (37), we obtain(38)ABk+1ut≤w1tk+1w2tk+1Γqkα+β∫att−qrqk+1α+β−1Γqkα+βΓqk+1α+βur∇qr=w1tk+1w2tk+1Γqk+1α+β∫att−qrqk+1α+β−1ur∇qr=w1tk+1w2tk+1∇qa−k+1α+βut.
Similarly, we can obtain(39)BAk+1vt≤w1tk+1w2tk+1∇qa−k+1α+βvt.
Thus, (28) and (29) are proved.
Using Stirling’s formula of theq-gamma function [32] yields that(40)Γqx=2q1/2Γq2121−q1/2−xeθqx/1−q−qx,0<θ<1,that is,(41)Γqx∼D1−q1/2−x,x⟶∞,where D=2q1/2Γq21/2. Moreover, if t>a>0 and γ>0 (γ is not a positive integer), then 1−a/tqj<1−a/tqγ+j, for each j=0,1,…, and(42)t−aqγ=tγ∏j=0∞1−a/tqj1−a/tqγ+j<tγ.
Applying the first mean value theorem for definite integrals [33], (41) and (42), and w1t<M1 and w2t<M2, there exists a ξ∈a,1q such that(43)limn⟶∞ABnut≤limn⟶∞uξM1nM2nΓqnα+β∫att−qrqnα+β−1∇qr=limn⟶∞uξM1nM2nΓqnα+β+1t−aqnα+β≤limn⟶∞uξM1nM2nΓqnα+β+1tnα+β=limn⟶∞uξM1nM2nD1−q1/2−nα+β+1tnα+β≤limn⟶∞uξM1nM2nD1−q1/2−nα+β+1=limn⟶∞uξ1−qDM1M21−qα+βn.
From (20), for each t∈a,1q, one has(44)M1M21−qα+βn⟶0,asn⟶∞.
Thus,ABnut⟶0 as n⟶∞, Similarly, we can obtain that BAnut⟶0 as n⟶∞, for each t∈Ta. Therefore, (30) is proved.
Letn⟶∞ in (26); with the help of the semigroup property ∇qa−α∇qa−μ=∇qa−α+μ and the definition of A and B, one obtains(45)ut≤g1t+Ag2t+∑k=1∞ABkg1t+∑k=1∞ABkAg2t=g1t+w1tΓqα∫att−qsqα−1g2s∇qs+∑k=1∞w1tkw2tk∇qa−kα+βg1t+w1tΓqα∫att−qsqα−1g2s∇qs.
Similarly, letn⟶∞ in (27), and we obtain (22). This completes the proof.Corollary 1.
Under the hypothesis of Theorem3, let g1t and g2t be two nondecreasing functions on t∈Ta. Then,(46)ut≤g1tqEα+βw1tw2t,t−a+w1tg2tqeα+β,α+1w1tw2t,t−aand(47)vt≤g2tqEα+βw1tw2t,t−a+w2tg1tqeα+β,β+1w1tw2t,t−a.Proof.
By (16) and the assumption that g1t, g2t, and w1t are three nondecreasing functions for t∈Ta, we have(48)∇qa−αg2t≤g2t∇qa−α1=g2tΓqα+1t−aqα,and(49)∇qa−kα+βg1t+w1t∇qa−αg2t≤g1t+w1t∇qa−αg2t∇qa−kα+β1=g1t+w1t∇qa−αg2tt−aqkα+βΓqkα+β+1.
Thus, from (21), (48), and (49), one can conclude that(50)ut≤g1t+w1tg2tq∇a−α11+∑k=1∞w1tkw2tk∇qa−kα+β1=g1t1+∑k=1∞w1tkw2tk∇qa−kα+β1+w1tg2tΓqα+1t−aqα+w1tg2t∑k=1∞w1tkw2tk∇qa−kα+β∇qa−α1=g1t∑k=0∞w1tw2tkt−aqkα+βΓqkα+β+1+w1tg2tΓqα+1t−aqα+w1tg2t∑k=1∞w1tkw2tk∇qa−kα+β−α1=g1tqEα+βw1tw2t,t−a+w1tg2tΓqα+1t−aqα+w1tg2t∑k=1∞w1tw2tkt−aqkα+β+αΓqkα+β+α+1=g1tqEα+βw1tw2t,t−a+w1tg2t∑k=0∞w1tw2tkt−aqkα+β+αΓqkα+β+α+1=g1tqEα+βw1tw2t,t−a+w1tg2tqeα+β,α+1w1tw2t,t−a.
Similarly,we can obtain (47) holds.


## 4. Main Result

Throughout this paper, we make the following assumptions:(H1)f,g∈DTa×ℝn,ℝn are two Lipschitz-type functions. That is, for any x,y,u,v:Tτa⟶ℝn, there exist two positive constants L1,L2>0 such that(51)ft,yt,yτt−gt,vt,vτt≤L1yt−vt+yτt−vτt,gt,xt,xτt−gt,ut,uτt≤L2xt−ut+xτt−uτt,Fort∈a,1q,(52)H2ft,0,0=0,0,…,0︸nT,gt,0,0=0,0,…,0︸nT,H3A0+A1+2L1B0+B1+2L21−qα+β<1.Let us denoteϕ,ψ=maxmaxt∈Iτϕt,maxt∈Iτψt.Definition 3.
System (6) is finite-time stable with respect toδ,ε,T, with δ<ϵ, if and only if ϕ,ψ<δ implies xt,yt=maxxt,yt<ε, ∀t∈Tτa.Theorem 4.
Assume that conditions (H1)–(H3) hold. Then, system (6) is finite-time stable if the following conditions are satisfied:(53)1+A1+L1Γqα+1t−aqαEqα+βc,t−a+A0+A1+2L11+B1+L2Γqβ+1t−aqβeqα+β,α+1c,t−a<εδ,and(54)1+B1+L2Γqβ+1t−aqβEqα+βc,t−a+B0+B1+2L21+A1+L1Γqα+1t−aqαeqα+β,β+1c,t−a<εδ,where c=A0+A1+2L1B0+B1+2L2.Proof.
From Theorem4 in [19], it is easy to see that x,y:Tτa×Tτa⟶ℝn×ℝn is a solution of system (6) if and only if(55)xt=ϕα+1Γqα∫att−qsqα−1A0ys+A1yτs+fs,ys,yτs∇qs,yt=ψα+1Γqβ∫att−qsqβ−1B0xs+B1xτs+gs,xs,xτs∇qs,xt=ϕt,yt=ψt,t∈Iτ.
Fort∈Ta, we have, by (55), that(56)xt≤ϕa+1Γqα∫att−qsqα−1A0ys+A1yτs+fs,ysyτs∇qs≤ϕ+A0Γqα∫att−qsqα−1ys∇qs+A1Γqα∫att−qsqα−1yτs∇qs+1Γqα∫att−qsqα−1fs,ys,yτs∇qs≤ϕ+A0+L1Γqα∫att−qsqα−1ys∇qs+A1+L1Γqα∫att−qsqα−1yτs∇qs≤ϕ+A0+L1Γqα∫att−qsqα−1ys∇qs+A1+L1Γqα∫att−qsqα−1supθ∈Jτyθs+ψ∇qs=ϕ+ψA1+L1Γqα+1t−aqα+A0+L1Γqα∫att−qsqα−1ys∇qs+A1+L1Γqα∫att−qsqα−1supθ∈Jτyθs∇qs,where Jτ=τ,τq−1,…,1. Similarly, we can obtain(57)yt≤ψ+ϕB1+L2Γqβ+1t−aqβ+B0+L2Γqβ∫att−qsqβ−1xs∇qs+B1+L2Γqβ∫att−qsqβ−1supθ∈Jτxθs∇qs.
Letg1t=ϕ+ψA1+L1/Γqα+1t−aqα and g2t=ψ+ϕB1+L2/Γqβ+1t−aqβ; then, g1 and g2 are two nondecreasing functions.
Setx¯t=supθ∈Jτxθt and y¯t=supθ∈Jτyθt; then, by (56), we obtain(58)x¯t≤g1t+A0+L1Γqα∫att−qsqα−1y¯s∇qs+A1+L1Γqα∫att−qsqα−1y¯s∇qs=g1t+A0+A1+2L1Γqα∫att−qsqα−1y¯s∇qs=g1t+A0+A1+2L1q∇a−αy¯t.
Similarly, we obtain(59)y¯t≤g2t+B0+B1+2L2q∇a−βx¯t.
Hence, by (53) and (54) and applying the result of Corollary 1, we have(60)xt≤x¯t≤g1tqEα+βc,t−a+A0+A1+2L1g2tqeα+β,α+1c,t−a=1+A1+L1Γqα+1t−aqαϕ,ψqEα+βc,t−a+A0+A1+2L11+B1+L2Γqβ+1t−aqβϕ,ψqeα+β,α+1c,t−a≤δ1+A1+L1Γqα+1t−aqαqEα+βc,t−a+δA0+A1+2L11+B1+L2Γqβ+1t−aqβqeα+β,α+1c,t−a<ε,and(61)yt≤y¯t≤g2tqEα+βc,t−a+B1+B0+2L2g1tqeα+β,β+1c,t−a=ψ+‖ϕ‖B1+L2Γqβ+1t−aqβqEα+βc,t−a+B1+B0+2L2ϕ+ψA1+L1Γqα+1t−aqαqeα+β,β+1c,t−a.≤δ1+B1+L2Γqβ+1t−aqβqEα+βc,t−a+δB1+B0+2L21+A1+L1Γqα+1t−aqαqeα+β,β+1c,t−a<ε.Remark 1.
In [34], Lyu and Vong proposed and analyzed a scheme on nonuniform mesh for solving the q-Volterra equation which provides the same solution to the q-fractional initial value problem. We will apply the proposed scheme as in (3.12) in [32] to solve Example 1.


## 5. Example

Example 1.
Consider the nonlinear delayq-fractional differential difference coupled system:(62)Cqa0.7xt=0.7yt+sin18yτt,t∈a,1q,Cqa0.5yt=0.8xt+arctan19xτt,t∈a,1q,where α=0.7, β=0.5, q=0.6, a=q9=0.69, τ=q9=0.69, ft,yt,yτt=sin1/8yτt, gt,xt,xτt=arctan1/9xτt, ϕt=0.07, and ψt=0.09, and t∈Iτ=0.618,0.617,…,0.69.
Letδ=0.1 and ϵ=0.5. Obviously, ϕ,ψ=max0.07,0.09<0.1=δ. We can see that f,g satisfy conditions (H2) and (H1) with L1=1/8 and L2=1/9. Moreover, we have A0=0.7, A1=0, B0=0.8, and B1=0, and (H3) holds. In the following, the aim is to validate the FTS conditions (53) and (54), w.r.t. δ,ϵ,T. By using Matlab (the pseudocode to compute different values of Γqα, see [35]), when t=0.6∈a,1q,(63)1+A1+L1Γqα+1t−aqαqEα+βc,t−a+A0+A1+2L11+B1+L2Γqβ+1t−aqβqeα+β,α+1c,t−a≈3.4304<5=εδ,and(64)1+B1+L2Γqβ+1t−aqβqEα+βc,t−a+B0+B1+2L21+A1+L1Γqα+1t−aqαqeα+β,β+1c,t−a≈4.8461<5=εδ.
Thus, we obtain that the estimated time of FTS isT=0.6.
Within given parameters, we can observe the finite-time behavior. In Figures1 and 2, we can see that, within the finite-time T=0.6s, ϕ,ψ=0.09<δ=0.1, the norm x,y of solution xt,yt does not exceed ε=0.5 which supposes the theorem numerically. The phase plot of (62) is shown in Figure 3.Figure 1
Behavior of the solution of system (62) in the t,x plane within T=0.6s.Figure 2
Behavior of the solution of system (62) in the t,y plane within T=0.6s.Figure 3
Phase plot of (62).


## 6. Conclusion

The problem of finite-time stability of coupledq-fractional difference delay systems is emphasized in this work. For this, we obtained a generalized coupled q-fractional Gronwall inequality, and by applying this inequality, a novel and easy to verify sufficient conditions have been provided in this paper to determine the finite-time stability of the solutions for the considered system. Finally, an example is given to illustrate the effectiveness and feasibility of our criterion.In the future, we will consider moreq-fractional difference systems, such as q-fractional difference singular systems or q-fractional difference uncertain systems, and we will study the problems of finite-time stability, and the Ulam–Hyers stability for these systems.



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