
Under the uncertain market demand and quality level, a total profit model of green closed-loop supply chain system (GCL-SCS) considering corporate environmental responsibility (CER) and government differential weight subsidy (GDWS) is constructed. Based on incentive-compatibility theory, the optimal subsidy allocation policy and green investment level were explored. Fuzzy chance-constrained programming (FCCP) is used to clarify the uncertainty factors of this model; while genetic algorithm (GA) and CPLEX are used to find and compare a calculating example's approximate optimal solution about this model. The main calculating results indicate that: (1) Enterprises can make optimal recycling, production and sales strategies according to different potential demand; (2) Without government subsidy, enterprises' higher green investment level will reduce their average gross profit, increase the quality level of recycled products and decrease the recycling rate, hence reduce their environmental protection willingness; (3) Based on incentive-compatibility theory, when government subsidy weight is set as 0.34~0.41 for consumers, enterprises' higher green investment level will enhance their average gross profit, reduce the quality level of recycled products and increase the recycling rate, which will improve their environmental protection willingness; (4) Under uncertain environment, the combination of reasonable government subsidy policy and enterprises green investment can make up for the defect of enterprises green investment alone, maximize utilities of government and enterprises, and optimize the green closed loop supply chain.
Citation: Jianquan Guo, Guanlan Wang, Mitsuo Gen. Green closed-loop supply chain optimization strategy considering CER and incentive-compatibility theory under uncertainty[J]. Mathematical Biosciences and Engineering, 2022, 19(9): 9520-9549. doi: 10.3934/mbe.2022443
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Under the uncertain market demand and quality level, a total profit model of green closed-loop supply chain system (GCL-SCS) considering corporate environmental responsibility (CER) and government differential weight subsidy (GDWS) is constructed. Based on incentive-compatibility theory, the optimal subsidy allocation policy and green investment level were explored. Fuzzy chance-constrained programming (FCCP) is used to clarify the uncertainty factors of this model; while genetic algorithm (GA) and CPLEX are used to find and compare a calculating example's approximate optimal solution about this model. The main calculating results indicate that: (1) Enterprises can make optimal recycling, production and sales strategies according to different potential demand; (2) Without government subsidy, enterprises' higher green investment level will reduce their average gross profit, increase the quality level of recycled products and decrease the recycling rate, hence reduce their environmental protection willingness; (3) Based on incentive-compatibility theory, when government subsidy weight is set as 0.34~0.41 for consumers, enterprises' higher green investment level will enhance their average gross profit, reduce the quality level of recycled products and increase the recycling rate, which will improve their environmental protection willingness; (4) Under uncertain environment, the combination of reasonable government subsidy policy and enterprises green investment can make up for the defect of enterprises green investment alone, maximize utilities of government and enterprises, and optimize the green closed loop supply chain.
In this paper, we are interested to investigate the delayed population model with survival rate on isolated time scales given by
xΔ(t)=γ(t)x(t)+x(d(t))μ(t)er(t)μ(t)(1−x(d(t))μ(t)), t∈T | (1.1) |
where γ:T→(−∞,0), r,k:T→(0,∞) describe, respectively, the intrinsic growth rate and the carrying capacity of the habitat, and d:T→T is the delay function such that ρα(t)⩽d(t)⩽t for some α∈N. This model is equivalent to
x(σ(t))=˜γ(t)x(t)+x(d(t))er(t)μ(t)(1−x(d(t))μ(t)), t∈T |
where the function ˜γ(t)=1+μ(t)γ(t) belongs to (0,1). This is a generalization of the model considered in [1] for any isolated time scales. Clearly, in the particular case T=Z, our model reaches the one found in [1].
A quick look at the formulation of the model described by equation (1.1) may seem different, since in its formulation appears the graininess function in the denominator of the second term on the right–hand side of the equation. However, in [2], the authors show that this formulation is necessary when we are dealing with quantum calculus (which is also encompassed here), since depending on the formulation of the model and the assumptions, one cannot even ensure the existence of ω–periodic solutions without considering this term for the quantum case (see [2] for details). But it is important to mention that our model reaches the model for the case T=Z considered in the literature, showing that this formulation is appropriate and unifies all the cases.
We point out that our model is valid for all isolated time scales, which includes many important examples such as T=Z, T=N20={n2:n∈N0}, T=qN0={qn:n∈N0}, q>1. This last one is known as quantum scale and it has been investigated by many authors [3], mainly concerning the ω–periodicity (see [4] and [5]). This quantum scale has several applications in many fields of physics such as cosmic strings and black holes [6], conformal quantum mechanics, nuclear and high energy physics, fractional quantum Hall effect, and high-Tc superconductors [7]. Thermostatistics of q-bosons and q-fermions can be established using basic numbers and employing the quantum calculus [8]. On the other hand, it worths mentioning the importance of time scales to describe population models, since it allows to consider a variety of scenery and many possibilities in the behavior of different populations (see, for instance, [9]). Also, the population models for quantum calculus play important role, bringing relevant applications (see [10] and [11]).
The formulation of this model for its analogue for T=Z without delays was investigated by many authors. See the references [12], [13] and [14] for instance. In particular, in [14], the authors investigated the extreme stability of the following discrete logistic equation
x(t+1)=x(t)er(t)(1−x(t)k(t)), t∈Z+. | (1.2) |
In [1], the author considered a version of the model with delays
x(t+1)=γ(t)x(t)+x(τ(t))er(t)(1−x(τ(t))k(t)), t∈Z+. | (1.3) |
The formulation considered here in this present paper generalizes (1.2) and (1.3). We are interested to investigate the asymptotic behavior of the solutions of (1.1) on isolated time scales, including global attractor, extremely stability, asymptotic periodicity and periodicity.
This paper is divided as follows. In the second section, we present some preliminary results on theory of time scale and explain the delayed model that will be investigated. In the third section, we investigate the stability of equation (1.1). The fourth section is devoted to study the extremal stability of (1.1) and to present some examples to illustrate our main results. Finally, the goal of last section is to investigate the periodicity and asymptotically periodicity of solutions of (1.1), and to present examples.
In this section, our goal is to recall some basic definitions and results from time scale theory. For more details, we refer [15] and [16].
A time scale T is any closed and nonempty subset of R endowed with the topology inherited from R.
Definition 2.1. The forward jump operator σ:T→T is defined by σ(t)=inf{s∈T:s>t} and the backward jump operator ρ:T→T by ρ(t)=sup{s∈T:s>t}, provided inf∅=supT and sup∅=infT.
If σ(t)>t, then t is called right–scattered. Otherwise, t is called right–dense. Similarly, if ρ<t, then t is said to be left–scattered, while if ρ(t)=t, then t is called left–dense.
From now on, we only consider isolated time scales, i.e., all points are right–scattered and all points are left–scattered.
Moreover we denote the composition σ∘…∘σ⏟n times by σn. The same notation we use for the composition of operator ρ.
Definition 2.2. The graininess function μ:T→[0,∞) is defined by μ(t)=σ(t)−t.
The delta (or Hilger) derivative of f:T→R at a point t∈Tκ, where
Tκ={T∖(ρ(supT),supT],ifsupT<∞T,ifsupT=∞ |
is defined in the following way:
Definition 2.3 ([15]). The delta derivative of function f at a point t, denoted by fΔ(t), is the number (provided it exists) with the property that given any ε>0, there is a neighborhood U of t (i.e., U=(t−δ,t+δ)∩T for some δ>0) such that
|(f(σ(t))−f(s))−fΔ(t)(σ(t)−s)|⩽ε|σ(t)−s| for all s∈U. |
We say that a function f is delta (or Hilger) differentiable on Tκ provided fΔ(t) exists for all t∈Tκ. The function fΔ:Tκ→R is then called the (delta) derivative of f on Tκ.
Throughout this paper, we assume that T is an isolated time scale such that
supT=∞, infT=t0 and inft∈Tμ(t)>0. | (2.1) |
By [15,Theorem 1.16], for any function f:T→R, its derivative is given by
fΔ(t)=f(σ(t))−f(t)μ(t) for all t∈Tκ. |
We consider the delayed population model of the form
{xΔ(t)=γ(t)x(t)+x(d(t))μ(t)er(t)μ(t)(1−x(d(t))k(t)), t⩾t0x(t0)=x0, | (2.2) |
with γ:T→(−∞,0), r,k:T→(0,∞) and d:T→T such that
ρα(t)⩽d(t)⩽t for some α∈N. | (2.3) |
The functions r and k describe, respectively, the intrinsic growth rate and the carrying capacity of the habitat. The delay is introduced to this model by the function d. From (2.3), it is clear that limt→∞d(t)=∞.
By solution of equation (2.2) with initial value x0, we mean function x:T→R which satisfies (2.2) for t>t0 and x(t0)=x0.
Our aim is to study the stability, existence of a global attractor and the extreme stability, as well as periodicity and asymptotically periodicity of (2.2), according to the notion of periodicity for isolated time scales given in [17] by Bohner et al.
Remark 2.4. Let us emphasize that solution of equation (2.2) depends on only one initial value x(t0), since the delay function d can be expressed in terms of iterations of the backward jump operator ρ.
Example 2.5. Suppose that the time scale T={t0,t1,t2,…} satisfies condition (2.1). Consider a delay function of the form
d(ti)={ρ2(ti)if i is evenρ3(ti)if i is odd. |
By equation (2.2), we obtain
x(t1)=(1+μ(t0)γ(t0))x(t0)+x(ρ2(t0))er(t0)μ(t0)(1−x(ρ2(t0))k(t0))=(1+μ(t0)γ(t0))x(t0)+x(t0)er(t0)μ(t0)(1−x(t0)k(t0))x(t2)=(1+μ(t1)γ(t1))x(t1)+x(ρ3(t1))er(t1)μ(t1)(1−x(ρ3(t1))k(t1))=(1+μ(t1)γ(t1))x(t1)+x(t0)er(t1)μ(t1)(1−x(t0)k(t1)) |
and so on.
Remark 2.6. We can also consider the model of population of the form
xΔ(t)=γ(t)x(t)+x(d(t))μ(t)er(t)μ(t)(1−x(d(t))k(t)), t⩾tβx(t0)=x0, x(t1)=x1, …, x(tβ−1)=xβ−1 | (2.4) |
i.e., with β initial values, where β depends on the delay function d.
Throughout this paper, we consider the following general assumptions on equation (2.2):
(A1) There exist γ0 and γ1 in (0,1) such that
inft∈T(1+μ(t)γ(t))=γ0 and supt∈T(1+μ(t)γ(t))=γ1. |
(A2) There exist constants ri,ki in (0,∞) for i=0,1, such that
inft∈Tr(t)μ(t)=r0, supt∈Tr(t)μ(t)=r1, inft∈Tk(t)=k0 and supt∈Tk(t)=k1. |
In sequel, we introduce the following notation which will be important to our purposes:
(A3) Let the functions L and U be defined as follows
L(u)=uer0−r1uk0 and U(u)=uer1−r0uk1 for u⩾0 |
and the constants M and m be given by
M=U(k1r0)=k1r0er1−1 and m=L(M1−γ1). |
(A4) For δ⩾1, we set a constant B as follows
B=min(L(δM1−γ1),r0k0r1). |
(A5) Let constant ˜m be given by
˜m=min{m,r0k0r1}. |
It is not difficult to check that M is the maximum value of the function U and r0k0r1 is the fixed point of the function L.
In this section, our goal is to investigate the stability of (2.2). We start by recalling some important definitions.
Definition 3.1. A set S⊂R is said to be invariant relative to (2.2) if for any positive value x(t0) such that x(t0)∈S, the solution x of (2.2) satisfies x(t)∈S for all t⩾t0.
Definition 3.2. A set S⊂R is said to be a global attractor of (2.2) if for any ε>0 and positive value of x(t0), there exists an element T(ε,x(t0))∈T such that the solution x of (2.2) satisfies
mins∈S|x(t)−s|<ε for all t⩾T(ε,x(t0)). |
Definition 3.3. Equation (2.2) is said to be extremely stable if for any two positive solutions x and y of (2.2), we have
limt→∞|x(t)−y(t)|=0. |
Remark 3.4. If (2.1) is fulfilled, any function x can be represented as a sequence {x(σn(t0))}n∈N, so we can reformulate the above definitions as follows.
A set S⊂R is said to be invariant relative to (2.2) if for any positive value x(t0) belonging to S, the solution {x(σn(t0))}n∈N of (2.2) satisfies
x(σn(t0))∈S for all n∈N. |
A set S⊂R is said to be a global attractor of (2.2) if for any ε>0 and positive value of x(t0), there exists a natural number N(ε,x(t0)) such that the solution {x(σn(t0))}n∈N of (2.2) satisfies
mins∈S|x(σn(t0))−s|<ε for all n⩾N(ε,x(t0)). |
Equation (2.2) is said to be extremely stable if for any two positive solutions {x(σn(t0))}n∈N and {y(σn(t0))}n∈N of (2.2) we have
limn→∞|x(σn(t0))−y(σn(t0))|=0. |
Lemma 3.5. Any solution x of (2.2) satisfies
x(σn(t))=n−1∏k=0(1+μ(σk(t))γ(σk(t)))x(t)+n−1∑k=0[n−1∏j=k+1(1+μ(σj(t))γ(σj(t)))]g(σk(t),x(d(σk(t)))), | (3.1) |
where
g(v,u)=uer(v)μ(v)(1−uk(v)). | (3.2) |
Proof. Let us prove by induction. Since x is a solution of (2.2), we have
x(σ(t))−x(t)μ(t)=γ(t)x(t)+x(d(t))μ(t)er(t)μ(t)(1−x(d(t))k(t)). |
It implies immediately that (3.1) holds for n=1.
Suppose now that (3.1) holds for n. Let us show that it also happens for n+1. Hence,
x(σn+1(t))=x(σn(σ(t)))=n−1∏k=0(1+μ(σk+1(t))γ(σk+1(t)))x(σ(t))+n−1∑k=0[n−1∏j=k+1(1+μ(σj+1(t))γ(σj+1(t)))]g(σk+1(t),x(d(σk+1(t))))=n∏k=1(1+μ(σk(t))γ(σk(t)))x(σ(t))+n∑k=1[n∏j=k+2(1+μ(σj(t))γ(σj(t)))]g(σk(t),x(d(σk(t)))). |
Using the definition of x(σ(t)) given by the case n=1 and replacing in the above equation, we have
x(σn+1(t))=n∏k=1(1+μ(σk(t))γ(σk(t)))[(1+μ(t)γ(t))x(t)+g(t,x(d(t)))]+n∑k=1[n∏j=k+2(1+μ(σj(t))γ(σj(t)))]g(σk(t),x(d(σk(t))))=n∏k=0(1+μ(σk(t))γ(σk(t)))x(t)+n∑k=0[n∏j=k+1(1+μ(σj(t))γ(σj(t)))]g(σk(t),x(d(σk(t)))), |
getting the desired result.
From assumption (A1) and (3.1), it follows that positive value of x(t0) implies solution x of (2.2) takes positive values only.
By assumption (A2), we get
L(u)⩽g(v,u)⩽U(u) for v∈[t0,∞)T and u⩾0. | (3.3) |
In consequence,
g(v,u)⩽M for v∈[t0,∞)T and u⩾0. | (3.4) |
From (3.1) and (3.4), due to assumption (A1), we have
x(σn(t0))⩽γn1x(t0)+Mn−1∑k=0γn−k−11=γn1x(t0)+M1−γn11−γ1 | (3.5) |
for any n∈N. Note that for x(t0)∈(0,δM1−γ1] with δ⩾1, we get from (3.5)
x(σn(t0))⩽γn1δM1−γ1+M−Mγn11−γ1=γn1(δM−M)+M1−γ1⩽δM−M+M1−γ1=δM1−γ1. | (3.6) |
Using the notations from (A3)–(A4) and by properties of function L, we obtain
L(u)⩾B for u∈[B1−γ0,δM1−γ1], |
where B is defined in (A4). By inequality (3.3), we get
g(v,u)⩾B for v∈[t0,∞)T and u∈[B1−γ0,δM1−γ1]. | (3.7) |
Assuming x(t0)⩾B1−γ0, it follows by Lemma 3.5, inequality (3.7) and assumption (A1), the following inequality
x(σn(t0))⩾(n−1∏k=0γ0)x(t0)+(n−1∑k=0(n−1∏j=k+1γ0)g(σk(t0),x(d(σk(t0)))))⩾n−1∏k=0γ0B1−γ0+n−1∑k=0γn−1−k0B=γn0B1−γ0+B(1−γn01−γ0)=B1−γ0 | (3.8) |
for n∈N. Hence, inequalities (3.6) and (3.8) allow us to formulate the following theorem.
Theorem 3.6. If conditions (A1)–(A4) hold, then the set [B1−γ0,δM1−γ1] is invariant relative to (2.2) for positive values of x(t0), where δ⩾1 and B is given in (A4).
The next theorem brings the statement on a global attractor.
Theorem 3.7. Under assumptions (A1)–(A3) and (A5), the set [˜m1−γ0,M1−γ1] is a global attractor of (2.2) for positive values of x(t0).
Proof. Since γ0 and γ1 belong to (0,1), for any ε>0 and positive value x(t0), there exists an integer N(ε,x(t0))>0 such that
γn1|x(t0)−M1−γ1|<ε and γn0|x(t0)−˜m1−γ0|<ε for any n⩾N(ε,x(t0)). |
It implies
γn1|x(t0)|<ε+|γn1M1−γ1|=ε+Mγn11−γ1. |
Applying the above to (3.5), we obtain
x(σn(t0))<ε+Mγn11−γ1+M1−γ1−Mγn11−γ1=ε+M1−γ1 for any n⩾N(ε,x(t0)). |
In analogous way, we get
x(σn(t0))>˜m1−γ0−ε for n⩾N(ε,x(t0)), |
which concludes the proof.
In this section, we are interested to investigate the extreme stability of equation (2.2).
Lemma 3.8. Let the assumptions (A1)–(A3) and (A5) hold. If x is a solution of (2.2) such that x(t0) is a positive value and
˜mk1(1−γ0)>1, | (3.9) |
then
lim supt→∞|1−r(t)μ(t)x(d(t))k(t)|er(t)μ(t)(1−x(d(t))k(t))⩽max{|1−r0˜mk1(1−γ0)|,|1−r1Mk0(1−γ1)|}. |
Proof. By Theorem 3.7, for any given solution x of (2.2) and for every ε>0, there exists T=T(ε,x(t0))∈T such that
˜m1−γ0−ε<x(d(t))<M1−γ1+ε for t⩾T. | (3.10) |
It implies the following estimates
1−r(t)μ(t)x(d(t))k(t)<1−r0k1(˜m1−γ0−ε) | (3.11) |
and
1−r(t)μ(t)x(d(t))k(t)>1−r1k0(M1−γ1+ε) | (3.12) |
for t⩾T such that t∈T. Since ε is arbitrary, we can write
lim supt→∞|1−r(t)μ(t)x(d(t))k(t)|⩽max{|1−r0˜mk1(1−γ0)|,|1−r1Mk0(1−γ1)|}. |
Now, it remains to show that
lim supt→∞er(t)μ(t)(1−x(d(t))k(t))⩽1 | (3.13) |
to conclude the proof. Observe that the left–hand side of inequality in formula (3.10) combined with (3.9) implies that
x(d(t))k(t)>˜mk1(1−γ0)−εk0>1−εk0 for any t⩾T. |
In consequence,
lim inft→∞x(d(t))k(t)⩾1. |
This ends the proof.
Theorem 3.9. Let assumptions (A1)–(A3) and (A5) hold. If condition (3.9) is satisfied and
max{|1−r0˜mk1(1−γ0)|,|1−r1Mk0(1−γ1)|}<1−γ1, | (3.14) |
then equation (2.2) is extremely stable.
Proof. Let x and y be arbitrary positive solutions of (2.2). Since x and y satisfy (2.2) for all t∈T, we have by Lemma 3.5
x(σn(t))=n−1∏k=0(1+μ(σk(t))γ(σk(t)))x(t)+n−1∑k=0[n−1∏j=k+1(1+μ(σj(t))γ(σj(t)))]g(σk(t),x(d(σk(t)))) |
and
y(σn(t))=n−1∏k=0(1+μ(σk(t))γ(σk(t)))y(t)+n−1∑k=0[n−1∏j=k+1(1+μ(σj(t))γ(σj(t)))]g(σk(t),y(d(σk(t)))) |
for any t∈T and n∈N. In consequence, we have
x(σn(t))−y(σn(t))=n−1∏k=0(1+μ(σk(t))γ(σk(t)))(x(t)−y(t))+n−1∑i=0[n−1∏j=i+1(1+μ(σj(t))γ(σj(t)))]{x(d(σi(t)))e˜r(σi(t))(1−x(d(σi(t)))k(σi(t)))−y(d(σi(t)))e˜r(σi(t))(1−y(d(σi(t)))k(σi(t)))} for all n∈N, |
where ˜r=rμ. Applying assumption (A1) and Mean Value Theorem to the above, we get the following estimate
|x(σn(t))−y(σn(t))|⩽γn1|x(t)−y(t)|+ |
+n−1∑i=0γn−i−11|1−˜r(σi(t))η(d(σi(t)))k(σi(t))|e˜r(σi(t))(1−η(d(σi(t)))k(σi(t)))|x(d(σi(t)))−y(d(σi(t)))| | (3.15) |
for all n∈N, where η(d(σi(t))) is between x(d(σi(t))) and y(d(σi(t))) for i=0,1,…,n−1. By condition (3.14), there exists real number M1 such that
max{|1−r0˜mk1(1−γ0)|,|1−r1Mk0(1−γ1)|}<M1<1−γ1. | (3.16) |
Hence, by Lemma 3.8 and by the definition of η, there exists t2∈T such that for t⩾t2, we have
|1−˜r(σi(t))η(d(σi(t)))k(σi(t))|e˜r(σi(t))(1−η(d(σi(t)))k(σi(t)))⩽M1 for all i∈N. | (3.17) |
On the other hand, for any t∈T, due to Theorems 3.6 and 3.7, the sequence
{x(σn(t))−y(σn(t))}n∈N |
is bounded. Hence, there exists a⩾0 such that
lim supn→∞|x(σn(t))−y(σn(t))|=a. | (3.18) |
In conclusion, for every ε>0, there exists t3∈T such that
|x(σn(t))−y(σn(t))|<a+ε for all n∈N and t3⩽t∈T. | (3.19) |
It is convenient to choose t3 such that t3⩾t2, since it implies condition (3.17) also holds. Combining inequalities (3.15), (3.17) and (3.19), we obtain for t⩾t3 and for all n∈N
|x(σn(t))−y(σn(t))|⩽γn1|x(t)−y(t)|+1−γn11−γ1M1(a+ε). | (3.20) |
Taking lim sup when n→+∞ on both sides of the above inequality, we get
a⩽11−γ1M1(a+ε). |
Since inequality (3.16) is satisfied, we have
a⩽M1ε1−γ1−M1. |
By the arbitrariness of ε, we obtain that a=0, obtaining the desired result.
Remark 3.10. Notice that since T is an isolated time scale such that supT=+∞, it is clear that limn→∞σn(t)=+∞. From this, we can infer by the properties of lim sup that (3.18) also holds for t sufficiently large, obtaining (3.19).
In sequel, we present some examples to illustrate the above results.
Example 3.11. Let
T={3n+k:n∈N0,k∈{0,15,25,35,45,1}}, |
where N0 is the set of nonnegative integers. Then t0=0 and mint∈Tμ(t)=15. Consider equation (2.2) with d=ρ2,
γ(t)={−0.4 if t∈{3n+1:n∈N0}−3.4 if t∈{3n+k:n∈N0,k∈{0,15,25,35,45}}, |
r(t)={0.425 if t∈{3n+1:n∈N0}0.405 if t∈{3n+k:n∈N0,k∈{0,15,25,35,45}}, |
and
k(t)={10 if t∈{3n+1:n∈N0}9+t−[t] if t∈{3n+k:n∈N0,k∈{0,15,25,35,45}}. |
Hence
γ0=0.2, γ1=0.32, r0=0.81, r1=0.85, k0=9, k1=10. |
Calculating
{M=U(k1r0)=k1r0er1−1≈10.62602;m=L(M1−γ1)≈8.02958;r0k0r1≈8.57647; |
we get ˜m=min{m,r0k0r1}=m. By Theorem 3.7, interval [˜m1−γ0,M1−γ1]≈[10.03697,15.62651] is a global attractor of (2.2). Figure 1 shows behavior of two solutions x and y with positive initial values x(t0)=1 and y(t0)=20, respectively. The range of the global attractor is illustrated by red dotted lines. Hence,
˜mk1(1−γ0)≈1.00370 |
and
max{|1−r0˜mk1(1−γ0)|,|1−r1Mk0(1−γ1)|}≈0.47584<0.68=1−γ1. |
Therefore, conditions (3.9) and (3.14) are satisfied. Theorem 3.9 implies that (2.2) is extremely stable. In Figure 2, difference of two solutions x and y with initial conditions x(t0)=1 and y(t0)=20 is shown, confirming that (2.2) is extremly stable.
Example 3.12. Let T=qN, where q=1.1, and consider equation (2.2) with
d(t)={ρ2(t) if t∈{q2n:n∈N}ρ(t) if t∈{q2n−1:n∈N}, |
γ(t)={−0.75t(q−1) if t∈{q2n−1:n∈N}−0.65t(q−1) if t∈{q2n:n∈N}, |
r(t)={0.35t(q−1) if t∈{q2n−1:n∈N}0.45t(q−1) if t∈{q2n:n∈N}, |
and
k(t)=14+sin(tπ). |
Then constants introduced by assumption (A1)–(A2) are following
γ0=0.25, γ1=0.35, r0=0.35, r1=0.45, k0=13, k1=15. |
Hence
M≈20.21769, m≈14.74683 and ˜m=11.7. |
By Theorem 3.7, interval [˜m1−γ0,M1−γ1]≈[15.6,31.10414] is a global attractor of (2.2). We check that
˜mk1(1−γ0)=1.04>1, |
and
max{|1−r0˜mk1(1−γ0)|,|1−r1Mk0(1−γ1)|}=0.532<0.65=1−γ1. |
Thus assumptions of Theorem 3.9 are satisfied. Hence equation (2.2) is extremely stable. Figure 3 shows behavior of the solutions x and y with initial values x(t0)=1 and y(t0)=37 (for 120 points from the time scale). Difference of those solutions is illustrated in Figure 4.
In this section, our goal is to investigate the existence of ω–periodic solutions and asymptotically ω–periodic solutions of (3.1), using the new concept of periodicity on isolated time scales introduced in [17].
Let us start by recalling the idea of periodicity on isolated time scales introduced in [17].
Definition 3.13. A function f:T→R is called ω−periodic if
νΔfν=f, |
where ν=σω.
Since condition (2.1) is satisfied, T contains only isolated points and
νΔ(t)=ν(σ(t))−ν(t)μ(t)=σ(ν(t))−ν(t)μ(t)=μ(ν(t))μ(t) |
(see [17]). Therefore, we can formulate the following equivalent condition of ω−periodicity which can be found in [17].
Lemma 3.14. A function f:T→R is ω−periodic if and only if (μf)ν=μf.
Remark 3.15. Observe that for T=Z we have μ(t)=1, ν(t)=t+ω and in this case, ω−periodicity condition given in Lemma 3.14 takes the known form
f(t+ω)=f(t) for all t∈Z. |
When T=2N0, then one can check that function f is ω−periodic if
2ωf(2ωt)=f(t) for all t∈2N0, |
reaching the ω–periodicity for the quantum case. See [4,5,10,11].
Definition 3.16. A function f:T→R is said to be asymptotically ω−periodic (or asymptotically ω−periodic for t⩾t1) if there exist two functions p,q:T→R such that
f(t)=p(t)+q(t), |
where p(t) is ω−periodic (or ω−periodic for t⩾t1) and q(t)→0 as t→∞.
As in the previous section, assume ˜m=min{m,r0k0r1}. The next result follows the same way as the proof of Lemma 3.8. Therefore, we omit its proof here.
Lemma 3.17. Assume (A1)–(A3) are satisfied. If x:T→R is such that
supt∈T|x(t)|∈[˜m1−γ0,M1−γ1] |
and
˜mk1(1−γ0)⩾1, | (3.21) |
then for all t∈T, the inequality
|1−r(t)μ(t)x(d(t))k(t)|er(t)μ(t)(1−x(d(t))k(t))⩽max{|1−r0˜mk1(1−γ0)|,|1−r1Mk0(1−γ1)|} |
holds.
Lemma 3.18. Suppose conditions (A1)–(A3), (A5), (3.14) and (3.21) hold. If x:T→R is an asymptotically ω−periodic function, r,k:T→R are ω−periodic functions and there exists t1∈T such that for any ω−periodic function p:T→R, p∘d is also ω−periodic for all t⩾t1, then g(t,x(d(t))) defined by (3.2) is an asymptotically ω−periodic function for t⩾t1.
Proof. Since x is asymptotically ω−periodic, it can be decomposed by
x(t)=p(t)+q(t), |
where p is ω−periodic and limt→∞q(t)=0. Applying Mean Value Theorem, we obtain the following inequality
|g(t,x(d(t)))−g(t,p(d(t)))|=|g(t,x(d(t)))−p(d(t))er(t)μ(t)(1−p(d(t))k(t))|⩽|1−r(t)μ(t)ξk(t)|er(t)μ(t)(1−ξk(t))|x(d(t))−p(d(t))|, | (3.22) |
where ξ is between x(d(t)) and p(d(t)). By Lemma 3.17, inequality (3.14) and asymptotic ω−periodicity of x, the right hand side of (3.22) tends to 0 if t→∞. On the other hand, notice that
μ(ν(t))p(d(ν(t)))eμ(ν(t))r(ν(t))(1−p(d(ν(t)))k(ν(t)))=μ(ν(t))p(d(ν(t)))eμ(ν(t))r(ν(t))(1−p(d(ν(t)))μ(ν(t))k(ν(t))μ(ν(t)))=μ(t)p(d(t))eμ(t)r(t)(1−p(d(t))μ(t)k(t)μ(t))=μ(t)p(d(t))eμ(t)r(t)(1−p(d(t))k(t)) |
for t⩾t1 since by assumption there exists t1∈T such that p∘d is ω−periodic for t⩾t1. Thus, by Lemma 3.14, the function p(d(t))er(t)μ(t)(1−p(d(t))k(t)) is ω−periodic for t⩾t1, proving the lemma.
The proof of the next result follows the same way as the proof of the previous result. Thus, we omit it here.
Corollary 3.19. Suppose r,k:T→R are ω–periodic functions and for any ω–periodic function p:T→R, p∘d is also an ω–periodic function. If x:T→R is an ω–periodic function, then g(t,x(d(t))) is also an ω–periodic function.
Lemma 3.20. Assume (A1) holds. If x:T→R is an asymptotically ω−periodic function and γ:T→R is an ω−periodic function, then function μγx is an asymptotically ω−periodic function.
Proof. Since x is an asymptotically ω–periodic function, it can be decomposed by
x(t)=p(t)+q(t), |
where p is ω−periodic and limt→∞q(t)=0, which implies
μγx=μγp+μγq. |
By (A1), μγ is a bounded function, which implies that there exists limt→∞μ(t)γ(t)q(t)=0. Thus, it remains to show that function μγp is ω−periodic. By Lemma 3.14, we obtain that (μγ)ν=μγ and (μp)ν=μp. It implies the following equality
(μμγp)ν=(μγ)ν(μp)ν=μμγp. |
Applying Lemma 3.14 again, we get the desired result.
In the same manner, we can prove the following result.
Corollary 3.21. Suppose (A1) holds. If x:T→R is an asymptotically ω−periodic for t⩾t1 and γ:T→R is an ω−periodic function, then μγx is an asymptotically ω−periodic function for t⩾t1.
To guarantee the existence of an asymptotically ω−periodic solution of (2.2), we apply the Krasnoselskii Fixed Point Theorem.
Theorem 3.22 ([18]). Let B be a Banach space, let Ω be a bounded, convex and closed subset of B and let F,G be maps of Ω into B such that
(i) Fx+Gy∈Ω for any x,y∈Ω,
(ii) F is a contraction,
(iii) G is completely continuous.
Then operator F+G has a fixed point in Ω.
Theorem 3.23. Let conditions (A1)–(A3), (A5), (3.14) and (3.21) hold. If γ,r and k are ω−periodic functions and there exists t1∈T such that for any ω−periodic function p:T→R, p∘d is also ω−periodic for t⩾t1 then there exists t∗∈T such that equation (2.2) has a unique ω−periodic (for t⩾t1) solution x and all other solutions are asymptotically ω−periodic.
Proof. Let B(T) denote a Banach space of the form
B(T):={x={x(t)}t⩾t0:supt∈T|x(t)|<∞} |
equipped with the norm defined by ‖x‖=supt∈T|x(t)|. It is not difficult to show that the set
B(T)ap:={x∈B(T):x is asymptotically ω–periodic for t⩾t1} |
with the supremum norm defined above is also a Banach space. Let us introduce the following subset of B(T)ap
Ωap:={x∈B(T)ap:˜m1−γ0⩽‖x‖⩽M1−γ1}. |
Observe that Ωap is a bounded, convex and closed subset in B(T)ap. Let us define two operators F,G:Ωap→B(T)ap in the following way
(Fx)(t)={0, if t=t0(1+μ(ρ(t))γ(ρ(t)))x(ρ(t))+g(ρ(t),x(d(ρ(t)))), if t>t0, |
where function g is given by (3.2), and
(Gx)(t)={x(t) if t=t00 if t>t0. |
By (A1), for any x,y∈Ωap and for t>t0, we get
(Fx)(t)+(Gy)(t)=(1+μ(ρ(t))γ(ρ(t)))x(ρ(t))+g(ρ(t),x(d(ρ(t))))⩽γ1M1−γ1+M=M1−γ1 | (3.23) |
and
(Fx)(t)+(Gy)(t)⩾ γ0˜m1−γ0+˜m = ˜m1−γ0. | (3.24) |
Clearly, (3.23) and (3.24) also remain valid for t=t0. In consequence,
˜m1−γ0⩽‖Fx+Gy‖⩽M1−γ1. |
By Lemma 3.18 and Corollary 3.21, we obtain that function Fx+Gy is asymptotically ω−periodic for t⩾t1, hence Fx+Gy∈Ωap.
The next step is to show that F is a contraction. Taking any x,y∈Ωap, we have
|(Fx)(t)−(Fy)(t)|⩽|1+μ(ρ(t))γ(ρ(t))||x(ρ(t))−y(ρ(t))|+|g(ρ(t),x(d(ρ(t))))−g(ρ(t),y(d(ρ(t))))|. |
By condition (A1) and Mean Value Theorem, for any t∈T, we get
|(Fx)(t)−(Fy)(t)|⩽γ1‖x−y‖+|1−r(ρ(t))μ(ρ(t))ξ(d(ρ(t)))k(ρ(t))|er(ρ(t))μ(ρ(t))(1−ξ(d(ρ(t)))k(ρ(t)))‖x−y‖, |
where ξ(d(ρ(t))) is between x(d(ρ(t))) and y(d(ρ(t))). By condition (3.14), we can choose an ε0>0 such that
max{|1−r0˜mk1(1−γ0)|,|1−r1Mk0(1−γ1)|}⩽1−γ1−ε0. |
Finally, by the arbitrariness of ε0, condition (3.14) and Lemma 3.17 lead to estimate
‖Fx−Fy‖⩽(1−γ1)‖x−y‖ for all t∈T, |
which means that F is a contraction.
To be able to use the Krasnoselskii Fixed Point Theorem, it remains to verify that G is completely continuous. It is evident that GΩap is a bounded subset in R and this implies that it is relatively compact. Thus, G is completely continuous.
Theorem 3.22 implies the existence of ˜x∈Ωap such that
˜x(t)=(F˜x)(t)+(G˜x)(t) for all t∈T. |
It can be equivalently rewritten as
˜x(σ(t))=(1+μ(t)γ(t))˜x(t)+g(t,˜x(d(t))) for t>t0. | (3.25) |
This means that ˜x is an asymptotically ω−periodic (for t⩾t1) solution of (2.2). Thus, ˜x has the following decomposition
˜x(t)=˜p(t)+˜q(t), | (3.26) |
where ˜p(t) is ω−periodic for t⩾t1 and limt→∞˜q(t)=0. Combining (3.25) and (3.26), we obtain
(1+μ(t)γ(t))˜x(t)+g(t,˜x(d(t)))=˜p(σ(t))+˜q(σ(t)) |
which implies
(1+μ(t)γ(t))(˜p(t)+˜q(t))+g(t,˜x(d(t)))−g(t,˜p(d(t)))+g(t,˜p(d(t)))=˜p(σ(t))+˜q(σ(t)). |
We claim that
˜p(σ(t))=(1+μ(t)γ(t))˜p(t)+g(t,˜p(d(t))) | (3.27) |
and
˜q(σ(t))=(1+μ(t)γ(t))˜q(t)+g(t,˜x(d(t)))−g(t,˜p(d(t))). |
Firstly, notice that (1+μ(t)γ(t))˜p(t)+g(t,˜p(d(t))) is ω–periodic. Indeed, by Lemma 3.14, we get for t⩾t1
(μ(t)[(1+μ(t)γ(t))˜p(t)+g(t,˜p(d(t)))])ν |
=(μ(t)˜p(t))ν+(μ(t)γ(t))ν(μ(t)˜p(t))ν+(μ(t)g(t,˜p(d(t))))ν |
=μ(t)˜p(t)+μ(t)γ(t)μ(t)˜p(t)+μ(t)g(t,˜p(d(t))) |
=μ(t)(1+γ(t)μ(t))˜p(t)+μ(t)g(t,˜p(d(t))), |
since g(t,˜p(d(t))) is ω–periodic, by Corollary 3.19. On the other hand, proceeding the same way as in (3.22) we obtain by applying Mean Value Theorem
|g(t,˜x(d(t)))−g(t,˜p(d(t)))|=|g(t,˜x(d(t)))−˜p(d(t))er(t)μ(t)(1−˜p(d(t))k(t))|⩽|1−r(t)μ(t)ξk(t)|er(t)μ(t)(1−ξk(t))|˜x(d(t))−˜p(d(t))|, |
where ξ is between x(d(t)) an p(d(t)). Hence, we get
limt→∞|g(t,˜x(d(t)))−g(t,˜p(d(t)))|=0 |
and also, it implies that
limt→∞(1+μ(t)γ(t))˜q(t)+g(t,˜x(d(t)))−g(t,˜p(d(t)))=0, |
since ˜q(t)→0 as t→∞ and 1+μγ is bounded. Therefore, by the uniqueness of decomposition, the claim follows. By the equality (3.27), we obtain ˜p is an ω−periodic (for t⩾t1) solution of (2.2).
To prove the uniqueness, assume ˜y is another ω−periodic (for t⩾t1) solution of (2.2), then by Theorem 3.9, we have
limt→∞|˜p(t)−˜y(t)|=0. |
This clearly forces ˜p(t)=˜y(t) for t⩾t1.
Finally, let x be an arbitrary solution of (2.2), then applying again Theorem 3.9, we have
limt→∞|x(t)−˜x(t)|=0. |
It implies that
x(t)=˜p(t)+q(t), |
with limt→∞q(t)=0, hence x is an asymptotically \omega- periodic solution of (2.2), proving the result.
Let us illustrate the above result returning to the equation considered in Example 3.11.
Example 3.24. Let
{\Bbb T} = \Big\{3n+k: n \in {\Bbb N}_0, k\in \Big\{0, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}, 1\Big\}\Big\}, |
where {\mathbb N}_0 is the set of nonnegative integers. Consider equation (2.2) with d = \rho^5 and functions \gamma, r, k defined as in Example 3.11.
It is easy to check that functions \gamma , r and k are 6- periodic, since for all t\in {\Bbb T} we have \mu(t) = \mu(\sigma^6(t)) . This property of graininess function implies that for any 6- periodic function p holds p = p(\sigma^6) . In consequence, the composition of any 6- periodic function p with the delay function d , i.e., p\circ d is 6- periodic for t\geqslant \sigma^{5}(t_0) . Since all assumptions of Theorem 3.23 are satisfied equation (2.2) admits unique \omega- periodic (for t\geqslant \sigma^{5}(t_0) ) solution and all other solutions are asymptotically \omega- periodic (see Figure 5).
In general, assuming d = \rho^{j} , we have that if p is 6- periodic function, then p\circ d is 6- periodic for t\geqslant \sigma^{j}(t_0) .
In this paper we conducted an analysis of stability, extreme stability and periodicity on all solutions of (2.2). Under certain assumptions, the global attractor of (2.2) for positive initial value is determined. Sufficient conditions for extreme stability of considered equations are given. We also presented conditions under which equation (2.2) has a unique \omega -periodic solution and all other solutions are asymptotically \omega -periodic.
Jaqueline Godoy Mesquita was partially supported by the program ''For women in Sciences from L'Oreal–UNESCO–ABC 2019'' and CNPq 407952/2016-0.
The authors declare that there are no conflict of interest associated with this publication.
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