Mathematical formulations are crucial in understanding the dynamics of disease spread within a community. The objective of this research is to investigate the SEIR model of SARS-COVID-19 (C-19) with the inclusion of vaccinated effects for low immune individuals. A mathematical model is developed by incorporating vaccination individuals based on a proposed hypothesis. The fractal-fractional operator (FFO) is then used to convert this model into a fractional order. The newly developed SEVIR system is examined in both a qualitative and quantitative manner to determine its stable state. The boundedness and uniqueness of the model are examined to ensure reliable findings, which are essential properties of epidemic models. The global derivative is demonstrated to verify the positivity with linear growth and Lipschitz conditions for the rate of effects in each sub-compartment. The system is investigated for global stability using Lyapunov first derivative functions to assess the overall impact of vaccination. In fractal-fractional operators, fractal represents the dimensions of the spread of the disease, and fractional represents the fractional ordered derivative operator. We use combine operators to see real behavior of spread as well as control of COVID-19 with different dimensions and continuous monitoring. Simulations are conducted to observe the symptomatic and asymptomatic effects of the corona virus disease with vaccinated measures for low immune individuals, providing insights into the actual behavior of the disease control under vaccination effects. Such investigations are valuable for understanding the spread of the virus and developing effective control strategies based on justified outcomes.
Citation: Huda Alsaud, Muhammad Owais Kulachi, Aqeel Ahmad, Mustafa Inc, Muhammad Taimoor. Investigation of SEIR model with vaccinated effects using sustainable fractional approach for low immune individuals[J]. AIMS Mathematics, 2024, 9(4): 10208-10234. doi: 10.3934/math.2024499
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Mathematical formulations are crucial in understanding the dynamics of disease spread within a community. The objective of this research is to investigate the SEIR model of SARS-COVID-19 (C-19) with the inclusion of vaccinated effects for low immune individuals. A mathematical model is developed by incorporating vaccination individuals based on a proposed hypothesis. The fractal-fractional operator (FFO) is then used to convert this model into a fractional order. The newly developed SEVIR system is examined in both a qualitative and quantitative manner to determine its stable state. The boundedness and uniqueness of the model are examined to ensure reliable findings, which are essential properties of epidemic models. The global derivative is demonstrated to verify the positivity with linear growth and Lipschitz conditions for the rate of effects in each sub-compartment. The system is investigated for global stability using Lyapunov first derivative functions to assess the overall impact of vaccination. In fractal-fractional operators, fractal represents the dimensions of the spread of the disease, and fractional represents the fractional ordered derivative operator. We use combine operators to see real behavior of spread as well as control of COVID-19 with different dimensions and continuous monitoring. Simulations are conducted to observe the symptomatic and asymptomatic effects of the corona virus disease with vaccinated measures for low immune individuals, providing insights into the actual behavior of the disease control under vaccination effects. Such investigations are valuable for understanding the spread of the virus and developing effective control strategies based on justified outcomes.
In the last two decades, the fractional difference equations have recently received considerable attention in many fields of science and engineering, see [1,2,3,4] and the references therein. On the other hand, the q-difference equations have numerous applications in diverse fields in recent years and has gained intensive interest [5,6,7,8,9]. It is well know that the q-fractional difference equations can be used as a bridge between fractional difference equations and q-difference equations, many papers have been published on this research direction, see [10,11,12,13,14,15] for examples. We recommend the monograph [16] and the papers cited therein.
For 0<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 [17], Abdeljawad et.al generalized the q-fractional Gronwall-type inequality in [18], they obtained the following q-fractional Gronwall-type inequality.
Theorem 1.1 ([17]). Let α>0, u and ν be nonnegative functions and w(t) be nonnegative and nondecreasing function for t∈[a,∞)q such that w(t)≤M where M is a constant. If
u(t)≤ν(t)+w(t)q∇−αau(t), |
then
u(t)≤ν(t)+∞∑k=1(w(t)Γq(α))kq∇−kαaν(t). | (1.1) |
Based on the above result, Abdeljawad et al. investigated the following nonlinear delay q-fractional difference system:
{qCαax(t)=A0x(t)+A1x(τt)+f(t,x(t),x(τt)),t∈[a,∞)q,x(t)=ϕ(t),t∈Iτ, | (1.2) |
where qCαa means the Caputo fractional difference of order α∈(0,1), ˉIτ={τa,q−1τa,q−2τa,...,a}, τ=qd∈Tq with d∈N0={0,1,2,...}.
Remark 1.1. The domain of t in (1.2) is inaccurate, please see the reference [19].
In [20], Sheng and Jiang gave the following extended form of the fractional Gronwall inequality :
Theorem 1.2 ([20]). Suppose α>0, β>0, a(t) is a nonnegative function locally integrable on [0,T), ˜g(t), and ˉg(t) are nonnegative, nondecreasing, continuous functions defined on [0,T); ˜g(t)≤˜M, ˉg(t)≤ˉM, where ˜M and ˉM are constants. Suppose x(t) is a nonnegative and locally integrable on [0,T) with
x(t)≤a(t)+˜g(t)∫t0(t−s)α−1x(s)ds+ˉg(t)∫t0(t−s)β−1x(s)ds,t∈[0,T). |
Then
x(t)≤a(t)+∫t0∞∑n=1[g(t)]nn∑k=0Ckn[Γ(α)]n−k[Γ(β)]kΓ[(n−k)α+kβ](t−s)(n−k)α+kβ−1a(s)ds, | (1.3) |
where t∈[0,T), g(t)=˜g(t)+ˉg(t) and Ckn=n(n−1)⋯(n−k+1)k!.
Corollary 1.3 [20] Under the hypothesis of Theorem 1.2, let a(t) be a nondecreasing function on [0,T). Then
x(t)≤a(t)Eγ[g(t)(Γ(α)tα+Γ(β)tβ)], | (1.4) |
where γ=min{α,β}, Eγ is the Mittag-Leffler function defined by Eγ(z)=∞∑k=0zkΓ(kγ+1).
Finite-time stability is a more practical method which is much valuable to analyze the transient behavior of nature of a system within a finite interval of time. It has been widely studied of integer differential systems. In recent decades, the finite-time stability analysis of fractional differential systems has received considerable attention, for instance [21,22,23,24,25] and the references therein. In [26], Du and Jia studied the finite-time stability of a class of nonlinear fractional delay difference systems by using a new discrete Gronwall inequality and Jensen inequality. Recently, Du and Jia in [27] obtained a criterion on finite time stability of fractional delay difference system with constant coefficients by virtue of a discrete delayed Mittag-Leffler matrix function approach. In [28], Ma and Sun investigated the finite-time stability of a class of fractional q-difference equations with time-delay by utilizing the proposed delayed q-Mittag-Leffler type matrix and generalized q-Gronwall inequality, respectively. Based on the generalized fractional (q,h)-Gronwall inequality, Du and Jia in [19] derived the finite-time stability criterion of nonlinear fractional delay (q,h)-difference systems.
Motivated by the above works, we will extend the q-fractional Gronwall-type inequality (Theorem 1.1) to the spreading form of the q-fractional Gronwall inequality. As applications, we consider the existence and uniqueness of the solution of the following nonlinear delay q-fractional difference damped system :
{qCαax(t)−A0qCβax(t)=B0x(t)+B1x(τt)+f(t,x(t),x(τt)),t∈[a,b)q,x(t)=ϕ(t),∇qx(t)=ψ(t),t∈Iτ, | (1.5) |
where [a,b)q=[a,b)∩Ta, b∈Ta, Iτ={qτa,τa,q−1τa,q−2τa,...,a}, τ=qd∈Tq with d∈N0={0,1,2,...}, qCαa and qCβa mean the Caputo fractional difference of order α∈(1,2) and order β∈(0,1), respectively, and the constant matrices A0, B0 and B1 are of appropriate dimensions. Moreover, a novel criterion of finite-time stability criterion of (1.5) is established. We generalized the main results of [17] in this paper.
The organization of this paper is given as follows: In Section 2, we give some notations, definitions and preliminaries. Section 3 is devoted to proving a spreading form of the q-fractional Gronwall inequality. In Section 4, the existence and uniqueness of the solution of system (1.5) are given and proved, and the finite-time stability theorem of nonlinear delay q-fractional difference damped system is obtained. In Section 5, an example is given to illustrate our theoretical result. Finally, the paper is concluded in Section 6.
In this section, we provided some basic definitions and lemmas which are used in the sequel.
Let f:Tq→R (q∈(0,1)), the nabla q-derivative of f is defined by Thabet et al. as follows:
∇qf(t)=f(t)−f(qt)(1−q)t,t∈Tq∖{0}, |
and q-derivatives of higher order by
∇nqf(t)=∇q(∇n−1qf)(t),n∈N. |
The nabla q-integral of f has the following form
∫t0f(s)∇qs=(1−q)t∞∑i=0qif(tqi) | (2.1) |
and for 0≤a∈Tq
∫taf(s)∇qs=∫t0f(s)∇qs−∫a0f(s)∇qs. | (2.2) |
The definition of the q-factorial function for a nonpositive integer α is given by
(t−s)αq=tα∞∏i=01−stqi1−stqi+α. | (2.3) |
For a function f:Tq→R, the left q-fractional integral q∇−αa of order α≠0,−1,−2,... and starting at 0<a∈Tq is defined by
q∇−αaf(t)=1Γq(α)∫ta(t−qs)α−1qf(s)∇qs, | (2.4) |
where
Γq(α+1)=1−qα1−qΓq(α),Γq(1)=1, α>0. | (2.5) |
The left q-fractional derivative q∇βa of order β>0 and starting at 0<a∈Tq is defined by
q∇βaf(t)=(q∇maq∇−(m−β)af)(t), | (2.6) |
where m is the smallest integer greater or equal than β.
Definition 2.1 ([11]). Let 0<α∉N and f:Ta→R. Then the Caputo left q-fractional derivative of order α of a function f is defined by
qCαaf(t):=q∇−(n−α)a∇nqf(t)=1Γq(n−α)∫ta(t−qs)n−α−1q∇nqf(s)∇qs,t∈Ta, | (2.7) |
where n=[α]+1.
Let us now list some properties which are needed to obtain our results.
Lemma 2.1 ([29]). Let α,β>0 and f be a function defined on (0,b). Then the following formulas hold:
(q∇−βaq∇−αaf)(t)=q∇−(α+β)af(t),0<a<t<b, |
(q∇αaq∇−αaf)(t)=f(t),0<a<t<b. |
Lemma 2.2 ([11]). Let α>0 and f be defined in a suitable domain. Thus
q∇−αaqCαaf(t)=f(t)−n−1∑k=0(t−a)kqΓq(k+1)∇kqf(a) | (2.8) |
and if 0<α≤1 we have
q∇−αaqCαaf(t)=f(t)−f(a). | (2.9) |
The following identity plays a crucial role in solving the linear q-fractional equations:
q∇−αa(x−a)μq=Γq(μ+1)Γq(α+μ+1)(x−a)μ+αq,0<a<x<b, | (2.10) |
where α∈R+ and μ∈(−1,∞).
Apply q∇αa on both sides of (2.10), by virtue of Lemma 2.1, one can obtain
q∇αa(x−a)μ+αq=Γq(α+μ+1)Γq(μ+1)(x−a)μq,0<a<x<b, | (2.11) |
where α∈R+ and μ∈(−1,∞).
By Theorem 7 in [11], for any 0<β<1, one has
(qCβaf)(t)=(q∇βaf)(t)−(t−a)−βqΓq(1−β)f(a). | (2.12) |
For any 1<α≤2, by (2.8), one has
q∇−αaqCαaf(t)=f(t)−f(a)−(t−a)1q∇qf(a). | (2.13) |
Apply q∇αa on both sides of (2.13), by Lemma 2.1 and (2.11), we get
(qCαaf)(t)=(q∇αaf)(t)−f(a)q∇αa(t−a)0q−f(a)q∇αa(t−a)1q=(q∇αaf)(t)−(t−a)−αqΓq(1−α)f(a)−(t−a)1−αqΓq(2−α)∇qf(a). | (2.14) |
In this section, we give and prove the following spreading form of generalized q-fractional Gronwall inequality, which extend a q-fractional Gronwall inequality in Theorem 1.1.
Theorem 3.1. Let α>0 and β>0. Assume that u(t) and g(t) are nonnegative functions for t∈[a,T)q. Let wi(t) (i=1,2) be nonnegative and nondecreasing functions for t∈[a,T)q with wi(t)≤Mi, where Mi are positive constants (i=1,2) and
[Γq(α)Tα(1−q)α+Γq(β)Tβ(1−q)β]max{M1Γq(α), M2Γq(β)}<1. | (3.1) |
If
u(t)≤g(t)+w1(t)q∇−αau(t)+w2(t)q∇−βau(t),t∈[a,T)q, | (3.2) |
then
u(t)≤g(t)+∞∑n=1w(t)nn∑k=0CknΓq(α)n−kΓq(β)kq∇−((n−k)α+kβ)ag(t),t∈[a,T)q, | (3.3) |
where w(t)=max{w1(t)Γq(α), w2(t)Γq(β)}.
Proof. Define the operator
Au(t)=w(t)∫ta[(t−qs)α−1q+(t−qs)β−1q]u(s)∇qs,t∈[a,T)q. | (3.4) |
According to (3.2), one has
u(t)≤g(t)+Au(t). | (3.5) |
By (3.5) and the monotonicity of the operator A, we obtain
u(t)≤n−1∑k=0Akg(t)+Anu(t),t∈[a,T)q. | (3.6) |
In the following, we will prove that
Anu(t)≤w(t)nn∑k=0CknΓq(α)n−kΓq(β)kq∇−((n−k)α+kβ)au(t),t∈[a,T)q, | (3.7) |
and
limn→∞Anu(t)=0. | (3.8) |
Obviously, the inequality (3.7) holds for n=1. Assume that (3.7) is true for n=m, that is
Amu(t)≤w(t)mm∑k=0CkmΓq(α)m−kΓq(β)kq∇−((m−k)α+kβ)au(t)=w(t)mm∑k=0CkmΓq(α)m−kΓq(β)kΓq((m−k)α+kβ)∫ta(t−qs)(m−k)α+kβ−1qu(s)∇qs,t∈[a,T)q. | (3.9) |
When n=m+1, by using (3.4), (3.9), (2.10) and the nondecreasing of function w(t), we get
Am+1u(t)=A(Amu(t))
≤w(t)∫ta[(t−qs)α−1q+(t−qs)β−1q]
×(w(s)mm∑k=0CkmΓq(α)m−kΓq(β)kΓq((m−k)α+kβ)∫sa(s−qr)(m−k)α+kβ−1qu(r)∇qr)∇qs
≤w(t)m+1∫tam∑k=0CkmΓq(α)m−kΓq(β)kΓq((m−k)α+kβ)[(t−qs)α−1q+(t−qs)β−1q]
×[∫sa(s−qr)(m−k)α+kβ−1qu(r)∇qr]∇qs
=w(t)m+1m∑k=0CkmΓq(α)m−kΓq(β)kΓq((m−k)α+kβ)[∫ta(t−qs)α−1q∫sa(s−qr)(m−k)α+kβ−1qu(r)∇qr∇qs
+∫ta(t−qs)β−1q∫sa(s−qr)(m−k)α+kβ−1qu(r)∇qr∇qs]
=w(t)m+1m∑k=0CkmΓq(α)m−kΓq(β)kΓq((m−k)α+kβ)[∫ta∫tqr(t−qs)α−1q(s−qr)(m−k)α+kβ−1qu(r)∇qr∇qs
+∫ta∫tqr(t−qs)β−1q(s−qr)(m−k)α+kβ−1qu(r)∇qr∇qs]
=w(t)m+1m∑k=0CkmΓq(α)m−kΓq(β)kΓq((m−k)α+kβ)
×(Γq(α)∫ta[1Γq(α)∫tqr(t−qs)α−1q(s−qr)(m−k)α+kβ−1q∇qs]u(r)∇qr
+Γq(β)∫ta[1Γq(β)∫tqr(t−qs)β−1q(s−qr)(m−k)α+kβ−1q∇qs]u(r)∇qr)
=w(t)m+1m∑k=0CkmΓq(α)m−kΓq(β)kΓq((m−k)α+kβ)
×(Γq(α)∫taq∇−αqr(t−qr)(m−k)α+kβ−1qu(r)∇qr
+Γq(β)∫taq∇−βqr(t−qr)(m−k)α+kβ−1qu(r)∇qr)
=w(t)m+1m∑k=0CkmΓq(α)m−kΓq(β)kΓq((m−k)α+kβ)
×(Γq(α)Γq((m−k)α+kβ)Γq((m−k+1)α+kβ)∫ta(t−qr)(m−k+1)α+kβ−1qu(r)∇qr
+Γq(β)Γq((m−k)α+kβ)Γq((m−k)α+(k+1)β)∫ta(t−qr)(m−k)α+(k+1)β−1qu(r)∇qr)
=w(t)m+1m∑k=0CkmΓq(α)m−kΓq(β)k
×(Γq(α)q∇−((m−k+1)α+kβ)au(t)+Γq(β)q∇−((m−k)α+(k+1)β)au(t))
=w(t)m+1m∑k=0CkmΓq(α)m+1−kΓq(β)kq∇−((m−k+1)α+kβ)au(t)
+w(t)m+1m+1∑k=1Ck−1mΓq(α)m+1−kΓq(β)kq∇−((m+1−k)α+kβ)au(t)
=w(t)m+1[C0mΓq(α)m+1q∇−((m+1)α)au(t)
+m∑k=1(Ckm+Ck−1m)Γq(α)m+1−kΓq(β)kq∇−((m−k+1)α+kβ)au(t)
+CmmΓq(β)m+1q∇−((m+1)β)au(t)]
=w(t)m+1m+1∑k=0Ckm+1Γq(α)m+1−kΓq(β)kq∇−((m+1−k)α+kβ)au(t).
Thus, (3.7) is proved.
Using Stirling's formula of the q-gamma function [30], yields that
Γq(x)=[2]1/2qΓq2(1/2)(1−q)12−xeθqx(1−q)−qx,0<θ<1, |
that is
Γq(x)∼D(1−q)12−x,x→∞, | (3.10) |
where D=[2]1/2qΓq2(1/2). Moreover, if t>a>0 and γ>0 (γ is not a positive integer), then 1−atqj<1−atqγ+j for each j=0,1,..., and
(t−a)γq=tγ∞∏j=01−atqj1−atqγ+j<tγ. | (3.11) |
By w1(t)<M1 and w2(t)<M2, one has that w(t)<max{M1Γq(α), M2Γq(β)}:=M. Applying the first mean value theorem for definite integrals [31], (3.10) and (3.11), there exists a ξ∈[a,t]q such that
limn→∞Anu(t)≤limn→∞u(ξ)n∑k=0MnCknΓq(α)n−kΓq(β)kΓq((n−k)α+kβ)∫ta(t−qr)(n−k)α+kβ−1q∇qs=limn→∞u(ξ)n∑k=0MnCknΓq(α)n−kΓq(β)kΓq((n−k)α+kβ+1)(t−a)(n−k)α+kβq≤limn→∞u(ξ)n∑k=0MnCknΓq(α)n−kΓq(β)kΓq((n−k)α+kβ+1)t(n−k)α+kβ=limn→∞u(ξ)n∑k=0MnCknΓq(α)n−kΓq(β)kD(1−q)12−((n−k)α+kβ+1)t(n−k)α+kβ=limn→∞u(ξ)√1−qDn∑k=0MnCkn[Γq(α)tα(1−q)α]n−k[Γq(β)tβ(1−q)β]k=limn→∞u(ξ)√1−qD[M(Γq(α)(1−q)αtα+Γq(β)(1−q)βtβ)]n. |
From (3.1), for each t∈[a,T)q, we have
[M(Γq(α)(1−q)αtα+Γq(β)(1−q)βtβ)]n→0,as n→∞. |
Thus, Anu(t)→0 as n→∞. Let n→∞ in (3.6), by (3.8) we get
u(t)≤g(t)+∞∑k=1Akg(t). | (3.12) |
From (3.7) and (3.12), we obtain (3.3). This completes the proof.
Corollary 3.2. Under the hypothesis of Theorem 3.1, let g(t) be a nondecreasing function on t∈[a,T)q. Then
u(t)≤g(t)∞∑n=0w(t)nn∑k=0CknΓq(α)n−kΓq(β)kΓq((n−k)α+kβ+1)(t−a)(n−k)α+kβq | (3.13) |
Proof. By (3.3), (2.10) and the assumption that g(t) is nondecreasing function for t∈[a,T)q, we have
u(t)≤g(t)[1+∞∑n=1w(t)nn∑k=0CknΓq(α)n−kΓq(β)kq∇−((n−k)α+kβ)a1]=g(t)[1+∞∑n=1w(t)nn∑k=0CknΓq(α)n−kΓq(β)k1Γq((n−k)α+kβ+1)(t−a)(n−k)α+kβq]=g(t)∞∑n=0w(t)nn∑k=0CknΓq(α)n−kΓq(β)kΓq((n−k)α+kβ+1)(t−a)(n−k)α+kβq. |
Throughout this paper, we make the following assumptions:
(H1) f∈D(Tq×Rn×Rn,Rn) is a Lipschitz-type function. That is, for any x,y:Tτa→Rn, there exists a positive constant L>0 such that
‖f(t,y(t),y(τt))−f(t,x(t),x(τt))‖≤L(‖y(t)−x(t)‖+‖y(τt)−x(τt)‖), | (4.1) |
for t∈[a,T)q.
(H2)
f(t,0,0)=[0,0,...,0]⏟nT. | (4.2) |
(H3)
[Γq(α)Tα(1−q)α+Γq(α−β)Tα−β(1−q)α−β]max{‖B0‖+‖B1‖+2LΓq(α), ‖A0‖Γq(α−β)}<1. | (4.3) |
Definition 4.1. The system (1.5) is finite-time stable w.r.t.{δ,ϵ,Te}, with δ<ϵ, if and only if max{‖ϕ‖,‖ψ‖}<δ implies ‖x(t)‖<ϵ, ∀t∈[a,Te]q=[a,Te]∩[a,T)q.
Theorem 4.1. Assume that (H1) and (H3) hold. Then the problem (1.5) has a unique solution.
Proof. First we have to prove that x:Tτa→Rm is a solution of system (1.5) if and only if
x(t)=ϕ(a)+ψ(a)(t−a)−A0(t−a)α−βqΓq(α−β+1)ϕ(a)+A0Γq(α−β)∫ta(t−qs)α−β−1qx(s)∇qs+1Γq(α)∫ta(t−qs)α−1q[B0x(s)+B1x(τs)+f(s,x(s),x(τs))]∇qs,t∈[a,T)q,x(t)=ϕ(t),∇qx(t)=ψ(t),t∈Iτ. | (4.4) |
For t∈Iτ, it is clear that x(t)=ϕ(t) with ∇qx(t)=ψ(t) is the solution of (1.5). For t∈[a,T)q, we apply q∇αa on both sides of (4.4) to obtain
q∇αax(t)=ϕ(a)(t−a)−αqΓq(1−α)+ψ(a)(t−a)1−αqΓq(2−α)−ϕ(a)A0(t−a)−βqΓq(1−β)+A0q∇βax(t)+B0x(t)+B1x(τt)+f(t,x(t),x(τt)), | (4.5) |
where (q∇αaq∇−αax)(t)=x(t) and (q∇αaq∇−(α−β)ax)(t)=q∇βax(t) (by Lemma 2.1) have been used. By using (2.12) and (2.14), we get
qCαax(t)−A0qCβax(t)=B0x(t)+B1x(τt)+f(t,x(t),x(τt)),t∈[a,T)q. |
Conversely, from system (1.5), we can see that x(t)=ϕ(t) and ∇qx(t)=ψ(t) for t∈Iτ. For t∈[a,T)q, we apply q∇−αa on both sides of (1.5) to get
q∇−αa[qCαax(t)−A0qCβax(t)]=1Γq(α)∫ta(t−qs)α−1q[B0x(s)+B1x(τs)+f(s,x(s),x(τs))]∇qs. |
According to Lemma 2.2, we obtain
x(t)=ϕ(a)+ψ(a)(t−a)−A0(t−a)α−βqΓq(α−β+1)ϕ(a)+A0Γq(α−β)∫ta(t−qs)α−β−1qx(s)∇qs+1Γq(α)∫ta(t−qs)α−1q[B0x(s)+B1x(τs)+f(s,x(s),x(τs))]∇qs,t∈[a,T)q. |
Secondly, we will prove the uniqueness of solution to system (1.5). Let x and y be two solutions of system (1.5). Denote z by z(t)=x(t)−y(t). Obviously, z(t)=0 for t∈Iτ, which implies that system (1.5) has a unique solution for t∈Iτ.
For t∈[a,T)q, one has
z(t)=A0Γq(α−β)∫ta(t−qs)α−β−1qz(s)∇qs+1Γq(α)∫ta(t−qs)α−1q[B0z(s)+B1z(τs)+f(s,x(s),x(τs))−f(s,y(s),y(τs))]∇qs. | (4.6) |
If t∈Jτ={a,q−1a,...,τ−1a}, then τt∈Iτ and z(τt)=0. Hence,
z(t)=A0Γq(α−β)∫ta(t−qs)α−β−1qz(s)∇qs+1Γq(α)∫ta(t−qs)α−1q[B0z(s)+f(s,x(s),x(τs))−f(s,y(s),y(τs))]∇qs, |
which implies that
‖z(t)‖≤‖A0‖Γq(α−β)∫ta(t−qs)α−β−1q‖z(s)‖∇qs+1Γq(α)∫ta(t−qs)α−1q[‖B0‖‖z(s)‖+‖f(s,x(s),x(τs))−f(s,y(s),y(τs))‖]∇qs≤‖A0‖Γq(α−β)∫ta(t−qs)α−β−1q‖z(s)‖∇qs+1Γq(α)∫ta(t−qs)α−1q[‖B0‖‖z(s)‖+L(‖z(s)‖+‖z(τs)‖)]∇qs(by (H1))=‖A0‖Γq(α−β)∫ta(t−qs)α−β−1q‖z(s)‖∇qs+‖B0‖+LΓq(α)∫ta(t−qs)α−1q‖z(s)‖∇qs. | (4.7) |
By applying Corollary 3.2 and (H3), we get
‖z(t)‖≤0⋅∞∑n=0wn1n∑k=0CknΓq(α)n−kΓq(α−β)kΓq((n−k)α+k(α−β)+1)(t−a)(n−k)α+k(α−β)q=0, | (4.8) |
where w1=max{‖A0‖Γ(α−β),‖B0‖+LΓ(α)}. This implies x(t)=y(t) for t∈Jτ.
For t∈[τ−1a,T)q, we obtain
z(t)=A0Γq(α−β)∫ta(t−qs)α−β−1qz(s)∇qs+1Γq(α)∫ta(t−qs)α−1q[B0z(s)+f(s,x(s),x(τs))−f(s,y(s),y(τs))]∇qs+1Γq(α)∫ta(t−qs)α−1qB1z(τs)∇qs. | (4.9) |
Therefore,
‖z(t)‖=‖A0‖Γq(α−β)∫ta(t−qs)α−β−1q‖z(s)‖∇qs+1Γq(α)∫ta(t−qs)α−1q[‖B0‖‖z(s)‖+‖f(s,x(s),x(τs))−f(s,y(s),y(τs))‖]∇qs+1Γq(α)∫ta(t−qs)α−1q‖B1‖‖z(τs)‖∇qs≤‖A0‖Γq(α−β)∫ta(t−qs)α−β−1q‖z(s)‖∇qs+‖B0‖+LΓq(α)∫ta(t−qs)α−1q‖z(s)‖∇qs+‖B1‖+LΓq(α)∫ta(t−qs)α−1q‖z(τs)‖∇qs. | (4.10) |
Let z∗(t)=maxθ∈[a,t]q{‖z(θ)‖,‖z(τθ)‖} for t∈[τ−1a,T)q, where [a,t]q=[a,t]∩Ta, it is obvious that z∗(t) is a increasing function. From (4.10), we obtain that
z∗(t)≤‖A0‖Γq(α−β)∫ta(t−qs)α−β−1qz∗(s)∇qs+‖B0‖+LΓq(α)∫ta(t−qs)α−1qz∗(s)∇qs+‖B1‖+LΓq(α)∫ta(t−qs)α−1qz∗(s)∇qs=‖A0‖Γq(α−β)∫ta(t−qs)α−β−1qz∗(s)∇qs+‖B0‖+‖B1‖+2LΓq(α)∫ta(t−qs)α−1qz∗(s)∇qs. | (4.11) |
By applying Corollary 3.2 and (H3) again, we get
‖z(t)‖≤z∗(t)≤0⋅∞∑n=0wn2n∑k=0CknΓq(α)n−kΓq(α−β)kΓq((n−k)α+k(α−β)+1)(t−a)(n−k)α+k(α−β)q=0, |
where w2=max{‖A0‖Γ(α−β),‖B0‖+‖B1‖+2LΓ(α)}. Thus, we end up with x(t)=y(t) for t∈[τ−1a,T)q. The proof is completed.
Theorem 4.2. Assume that the conditions (H1), (H2) and (H3) hold. Then the system (1.5) is finite-time stable if the following condition is satisfied:
(1+(t−a)+‖A0‖(t−a)α−βqΓq(α−β+1))∞∑n=0wn2n∑k=0CknΓq(α)n−kΓq(α−β)kΓq((n−k)α+k(α−β)+1)(t−a)(n−k)α+k(α−β)q<εδ, | (4.12) |
where w2=max{‖B0‖+‖B1‖+2LΓq(α),‖A0‖Γq(α−β)}.
Proof. Applying left q-fractional integral on both sides of (1.5), we obtain
q∇−αa(qCαax(t))−A0q∇−αa(qCβax(t))=qΔ−αa(B0x(t)+B1x(τt)+f(t,x(t),x(τt))). | (4.13) |
By (4.12) and utilizing Lemma 2.2 we have
x(t)=ϕ(a)+ψ(a)(t−a)−A0(t−a)α−βqΓq(α−β+1)ϕ(a)+A0Γq(α−β)∫ta(t−qs)α−β−1qx(s)∇qs+1Γq(α)∫ta(t−qs)α−1q[B0x(s)+B1x(τs)+f(s,x(s),x(τs))]∇qs. |
Thus, by (H1) and (H2), we get
‖x(t)‖≤‖ϕ‖+‖ψ‖(t−a)+‖A0‖‖ϕ‖(t−a)α−βqΓq(α−β+1)+‖A0‖Γq(α−β)∫ta(t−qs)α−β−1q‖x(s)‖∇qs+1Γq(α)∫ta(t−qs)α−1q[‖B0‖‖x(s)‖+‖B1‖‖x(τs)‖+‖f(s,x(s),x(τs))‖]∇qs≤‖ϕ‖+‖ψ‖(t−a)+‖A0‖‖ϕ‖(t−a)α−βqΓq(α−β+1)+‖A0‖Γq(α−β)∫ta(t−qs)α−β−1q‖x(s)‖∇qs+1Γq(α)∫ta(t−qs)α−1q[(‖B0‖+L)‖x(s)‖+(‖B1‖+L)‖x(τs)‖]∇qs. | (4.14) |
Let g(t)=‖ϕ‖+‖ψ‖(t−a)+‖A0‖‖ϕ‖(t−a)α−βqΓq(α−β+1), then g is a nondecreasing function.
Set ˉx(t)=maxθ∈[a,t]q{‖x(θ)‖,‖x(τθ)‖}, then by (4.14) we get
ˉx(t)≤g(t)+‖A0‖Γq(α−β)∫ta(t−qs)α−β−1qˉx(s)∇qs+‖B0‖+‖B1‖+2LΓq(α)∫ta(t−qs)α−1qˉx(s)∇qs=g(t)+(‖B0‖+‖B1‖+2L)q∇−αaˉx(t)+‖A0‖q∇−(α−β)aˉx(t). | (4.15) |
Applying the result of Corollary 3.2, we have
‖x(t)‖≤ˉx(t)≤g(t)∞∑n=0wn2n∑k=0CknΓq(α)n−kΓq(α−β)kΓq((n−k)α+k(α−β)+1)(t−a)(n−k)α+k(α−β)q≤δ(1+(t−a)+‖A0‖(t−a)α−βqΓq(α−β+1))∞∑n=0wn2n∑k=0CknΓq(α)n−kΓq(α−β)kΓq((n−k)α+k(α−β)+1)(t−a)(n−k)α+k(α−β)q<ε. | (4.16) |
Therefore, the system (1.5) is finite-time stable. The proof is completed.
If x∈Rn, then ‖x‖=∑ni=1|xi|. If A∈Rn×n, then the induced norm ‖⋅‖ is defined as ‖A‖=max1≤j≤n∑ni=1|aij|.
Example 5.1. Consider the nonlinear delay q-fractional differential difference system
{qC1.8ax(t)−(00.620.560)qC0.8ax(t)=(00.080.1090)x(t)+(0.15000.12)x(τt)+f(t,x(t),x(τt)),t∈[a,T)q,x(t)=ϕ(t),∇qx(t)=ψ(t),t∈Iτ, | (5.1) |
where α=1.8, β=0.8, q=0.6, a=q5=0.65, T=q−1=0.6−1, τ=q3=0.63, x(t)=[x1(t),x2(t)]T∈R2,
f(t,x(t),x(τt))=14[sinx1(t),sinx2(τt)]T−15[arctanx1(τt),arctanx2(τt)]T, |
and
ϕ(t)=[0.05,0.035]T,ψ(t)=[0.04,0.045]T,t∈Iτ={0.69,0.68,0.67,0.66,0.65}. |
Obviously, ‖ϕ‖=‖ψ‖=0.0085<0.1=δ, ϵ=1. We can see that f satisfies conditions (H1) (L=14) and (H2). We can calculate ‖A0‖=0.62, ‖B0‖=0.109, ‖B1‖=0.15.
When T=0.6−1, it is easy to check that
[Γq(α)Tα(1−q)α+Γq(α−β)Tα−β(1−q)α−β]max{‖B0‖+‖B1‖+2LΓq(α),‖A0‖Γq(α−β)}=0.8992<1, |
that is, (H3) holds. By using Matlab (the pseudo-code to compute different values of Γq(σ), see [32]), when t=1∈[a,T)q,
(1+(t−a)+‖A0‖(t−a)α−βqΓq(α−β+1))∞∑n=0wn2n∑k=0CknΓq(α)n−kΓq(α−β)kΓq((n−k)α+k(α−β)+1)(t−a)(n−k)α+k(α−β)q≈8.4593<10=ϵδ. |
Thus, we obtain Te=1.
In this paper, we introduced and proved new generalizations for q-fractional Gronwall inequality. We examined the validity and applicability of our results by considering the existence and uniqueness of solutions of nonlinear delay q-fractional difference damped system. Moreover, a novel and easy to verify sufficient conditions have been provided in this paper which are easy 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. Motivated by previous works [33,34], the possible applications of fractional q-difference in the field of stability theory will be considered in the future.
The authors are grateful to the anonymous referees for valuable comments and suggestions that helped to improve the quality of the paper. This work is supported by Natural Science Foundation of China (11571136).
The authors declare that there is no conflicts of interest.
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