# Some Differential Inequalities on Time Scales and Their Applications to Feedback Control Systems

Authors: Yonghong Fan; Yangyang Yu; Linlin Wang
Journal: Discrete Dynamics in Nature and Society
 (2017)

---

## Abstract

This paper deals with feedback control systems on time scales. Firstly, we generalize the semicycle concept to time scales and then establish some differential inequalities on time scales. Secondly, as applications of these inequalities, we study the uniform ultimate boundedness of solutions of these systems. We give a new method to investigate the permanence of ecosystem on time scales. And some known results have been generalized. Finally, an example is given to support the result.


---

## Body

## 1. Introduction

The theory of time scales was introduced by Hilger in his Ph.D. thesis [1] in 1988. The study of dynamic equations on time scales has recently attracted a lot of attention, because it reveals many discrepancies and helps to avoid proving results twice, for differential equations and difference equations, respectively. The time scales calculus has wide applications in various fields like biology, engineering, economics, physics, neural networks, social sciences, and so on [2].On the other hand, the permanence of a system is very important. As we know, permanence is affected by such factors as environment, food supplies, attack rates, and so on. As an example, Schreiber and Patel [3] showed that ecoevolutionary feedbacks can mediate permanence at intermediate trade-offs in the attack rates. So it is worthwhile considering the permanence of biological models. The permanence of biological systems has been discussed by many authors [4–17]. In the process of studying these problems, the comparison method plays a paramount role. Such as in [4, 6], some sufficient conditions of permanence for biological systems were established by the comparison method. Permanence is also applied in physics, chemistry, physiology, diseases, economics, and so on. Such as in [18], sufficient conditions of the permanence for a SEIRS system were obtained.To the best of our knowledge, the permanence of ecosystem on time scales with feedback control was first investigated in [13]. The authors considered the N-species cooperation system with time delays and feedback control(1)xiΔt=rit1-expxit-τiikit+∑j=1,j≠ineijtexpxjt-τij-citexpxit-τii-dityit-hityit-ηi,yiΔt=-aityit+bitexpxit+βitexpxit-σi,t∈Tk,where i=1,2,…,n, with the initial condition (2)xiθ,yiθ≥0areallrd-continuous,θ∈-τ,t0∩T,xit0,yit0>0,i=1,2,…,n,on time scales T (which is p-periodic; i.e., there exits p>0 such that if t∈T; then t+p∈T), where Δ stands for the delta-derivative and 1-μtait>0, i=1,2,…,n, as μt=σt-t, σt is defined in the second section (Definition 8).For system (1), they assumed the following:(H1)
rit, ait, bit, cit, dit, eijt, hit, kit, and βiti,j=1,2,…,n are all rd-continuous, real-valued functions which are bounded above and below by positive constants; τij,ηi,σi∈Tpi,j=1,2,…,n are positive constants. τ=maxmax1≤i,j≤nτij,max1≤i≤nσi,max1≤i≤nηi>0, where Tp can be found in [13].(H2)
-rilcil,ril(ki-1)+ciu∈R+=p∈R,1+μtpt>0forallt∈Ti=1,2,…,n are all positively regressive (R is defined as in the second section (Definition 12)), in which they used the notation (3)fu=supt∈Tft,fl=inft∈Tft,wheref is a bounded function in T.They first give the definition of the permanence of ecosystem (1), and the definition of the permanence of other ecosystems mentioned in this paper is similar.Definition 1 (see [13]).
System (1) is said to be permanent if there exist n-dimensional positive constant vectors x∗, x∗, y∗, and y∗, which are independent on the initial condition, such that, for any solution xit,yitT, i=1,2,…,n, (4)x∗≤limt→∞infxit≤limt→∞supxit≤x∗,i=1,2,…,n,y∗≤limt→∞infyit≤limt→∞supyit≤y∗,i=1,2,…,n.Then the main results in [13] can be rewritten as follows.Theorem 2.
Assume that (H1) and (H2) hold,ril-(diu+hiu)ui∗>ril(ki-1+ciu)(i=1,2,…,n) and riu-rilcil>0(i=1,2,…,n), where ui∗ can be found in [13]. Then system (1) is permanent.IfT=R, one refers the readers to [14, 17]. Chen and Xie [14] considered the following system:(5)dzitdt=zitrit1-zitkit+∑j=1,j≠ineijt∫-Tij0Kijszjt+sds-citzit-dityit-hit∫-τi0Hisyit+sds,dyitdt=-aityit+bitzit+βit∫-ηi0Giszit+sds,t∈R,where Kijs, His, and Gis are all nonnegative continuous functions such that ∫-Tij0‍Kijsds=1, ∫-τi0‍Hisds=1, and ∫-ηi0‍Giszids=1, i,j=1,2,…,n. It is easy to see that the integrations have no influence on the permanence of system (5). They cancelled the additional limitations of [17] and obtained the following.Theorem 3.
Assume that (H1) holds. Then system (5) is permanent.IfT=Z, one refers the readers to [15, 16]. Let Nit=expxit; they discussed the discrete form of cooperative system (1):(6)Nit+1=Nitexprit1-Nit-τiikit+∑j=1,j≠ineijtNjt-τij-citNit-τii-dityit-hityit-ηi,yit+1=1-aityit+bitNit+βitNit-σi,t∈Z.In [16], Li and Zhang used the techniques in [11] instead of the comparison theorem in [15] and obtained the following.Theorem 4.
Assume that (H1) holds. Then system (6) is permanent.Obviously, in Theorem2, if T=R (or T=Z), the conditions are stronger than that in Theorem 3 (or Theorem 4).This paper deals with system (1) with the initial condition(7)xiθ∈R,yiθ≥0areallrd-continuous,θ∈-τ,t0∩T,yit0>0,i=1,2,…,n,and one assumes that t-τij,t-ηi,t-σi∈T, for t∈T, which implies that system (1) is valid; supT=∞, which ensures the ultimate boundedness of the solutions; and there exists a positive constant l such that(8)μt≤l;one can see that (8) includes the condition: T is p -periodic (in [13]). In fact, if T is p-periodic, then sup{μt∣t∈T}≤p.Our aim is to unify Theorems3 and 4 completely by cancelling the additional limitations of Theorem 2.For simplicity, one first considers a special case (logistic system) of system (1)(9)xΔt=rt1-expxtkt-ctyt,yΔt=-atyt+btexpxt,t∈Tk,with the initial condition (10)xt0∈R,yt0≥0,t0∈T; assume that 1-μtat>0 and(H3)
a,c,k,r,b:T→R+≔x∣x>0 are all bounded, nonnegative, rd-continuous, and real-valued functions.Letzt=expxt; if T=R, then system (9) reduces to the continuous system(11)dztdt=rtzt1-ztkt-ctyt,dytdt=-atyt+btzt,t∈R.In this continuous case, one refers the readers to [7, 19, 20]. They considered the system with feedback control in periodic case or boundedness case and obtained permanence, stability, and existence of periodic solutions for the system.LetNn=expxn; if T=Z, then system (9) is reformulated as(12)Nn+1=Nnexprn1-Nnkn-cnyn,Δyn=-anyn+bnNn,n∈Z;this system or its other forms attracted much interest; one refers the readers to [10–12, 21]. They also considered this system in periodic case or boundedness case and obtained permanence and existence of periodic solutions for this system. Chen [10] discussed the permanence of system (12). Using the comparison theorem, he obtained the following.Theorem 5.
Assume (H3) and(H4)
cuM2<1, where M2=buM1/al and M1=kuexpru-1/ru
hold. Then system (12) is permanent.But for system (12) itself, (H4) may not be necessary. Fan and Wang [11] used a method instead of the comparison theorem and obtained the following.Theorem 6.
Assume (H3) holds. Then system (12) is permanent.One generalizes the method of semicycle [11] to time scales instead of the comparison theorem. By establishing some differential inequalities on time scales, one proves that solutions of systems (1) and (9) are uniformly ultimate bounded. Our result is a unification and extension of Theorems 3 and 4.The remainder of this paper is arranged as follows. In Section2, we give some basic properties about time scales and generalize the semicycle concept to time scales. Two differential inequalities on time scales are proved in Section 3. Section 4 is devoted to uniform ultimate boundedness of solutions of systems (1) and (9). An example is presented in Section 5 to support the result. Conclusions are presented in Section 6.


## 2. Preliminaries

First we give some properties about time scales before our main result (see [1, 2]).Definition 7 (see [1, 2]).
A time scaleT is an arbitrary nonempty closed subset of the real number R.Definition 8 (see [1, 2]).
Fort∈T we define the forward jump operator σ:T→T and the backward jump operator ρ:T→T, by (13)σt≔infs∈T:s>t,ρt≔sups∈T:s<t, respectively.Throughout the paper we make an assumption thata≤b, a,b are points in T.Definition 9 (see [2]).
Define the intervala,b in T by (14)a,b≔t∈T:a≤t≤b,and as usual, we assume t∈T, for all t>T, T∈T is a constant.We note thatfσt=fσt, for t∈T, where fσ,f:T→R.Lemma 10 (see [2]).
IfT=hZ=hk:k∈Z, where h>0, then (15)∫abftΔt=∑k=a/hb/h-1fhkh,ifa<b,0,ifa=b,-∑k=b/ha/h-1fhkh,ifa>b.Lemma 11 ((existence of antiderivatives) (see [2])).
Every rd-continuous function has an antiderivative. In particular ift0∈T, then (16)Ft≔∫t0tfτΔτ,fort∈T,is an antiderivative of f.Definition 12 (see [2]).
We say that a functionp:T→R is regressive provided (17)1+μtpt≠0forallt∈Tkholds, where throughout the paper (18)Tk=T∖ρsupT,supT,ifsupT<∞,T,ifsupT=∞.The set of all regressive and rd-continuous functions f:T→R will be denoted in this paper by (19)R=RT=RT,R.Definition 13 (see [2]).
Ifp∈R, we define the exponential function by (20)ept,s=exp∫stξμτpτΔτ,fors,t∈T,where the cylinder transformation ξhz=1/hlog⁡1+zh, for h>0.
Forh=0, we define ξ0z=z. In this case, T=R, μt=0, and ept,s=exp∫st‍pτΔτ.
WhenT=hZ=hk:k∈Z, we know that μhk=hk+1-hk=h>0; then for s<t, (21)ept,s=exp∫st1hloge1-haτΔτ=exp∑k=s/ht/h-1loge1-hahk=∏k=s/ht/h-11-hahk.Definition 14 (see [2]).
Ifp∈R, the function ⊖p is defined by (22)⊖pt=-pt1+μtpt,fort∈Tk.Lemma 15 (see [2]).
Supposep∈R; then(1)
⊖⊖p=p;(2)
epσt,s=1+μtptept,s;(3)
1/ept,s=e⊖pt,s;(4)
1/ep·,sΔ=-p·/epσ·,s;(5)
ep·,sΔ=p·ep·,s.Similar to the definition of semicycle in discrete case [22], one gives the following.Definition 16.
Letm be a constant and :T→R; a positive semicycle relative to m of f is defined as follows: it consists of a “string" of terms (23)ft,t∈s,t,s,t∈T all greater than or equal to m; and a negative semicycle relative to m of f is defined as follows: it is a “string" of terms (24)ft,t∈p,q,p,q∈Tall less than or equal to m.


## 3. Differential Inequalities

By using the semicycle concept and the comparison theorem, we give some differential inequalities as follows.Lemma 17.
Assume thatA,B>0 and x10=x0>0, and further suppose that x1t satisfies the following:
(i)(25)x1Δt≤B-Aexpx1t,fort≥0;then(26)limt→∞supx1t≤Bl+lnBA.
(ii)(27)x1Δt≥B-Aexpx1t,fort≥0,and there exists a constant M, such that limt→∞⁡supx1t<M; then (28)limt→∞infx1t≥B-AexpMl+lnBA.Proof.
(i) According to the oscillating property ofx1t, we divided the proof into two cases. Firstly we assume that x1t does not oscillate ultimately about lnB/A. If there exists a constant T1>0, such that x1t≤lnB/A, for t>T1, then (29)x1t≤lnBA≤Bl+lnBA;thus (26) holds true. If there exists a constant T1>0, such that x1t>lnB/A, for t>T1, then x1Δt<0, which means that x1t is eventually decreasing; therefore, limt→∞⁡x1t exists. Then (25) yields limt→∞⁡x1t=lnB/A, which leads to (30)limt→∞supx1t=lnBA≤Bl+lnBA.
Finally, we assume thatx1t oscillates about lnB/A, by (25), we know that x1t≥lnB/A implies x1Δt≤0. Thus, by semicycle concept, we assume that x1t≥lnB/A, for t∈sα,tα, α∈L, where L is an index set, and the interval sα,tα satisfies the following:(a)
For anyα,β∈L, if α≠β, sα,tα∩sβ,tβ=⌀.(b)
Ifsα is left-scattered, then x1ρsα<lnB/A.(c)
Ifsα is left-dense, then there exists a hollow left neighbourhood Uo-sα of sα such that x1t<lnB/A, for t∈Uo-sα.(d)
Iftα is right-scattered, then x1σtα<lnB/A.(e)
Iftα is right-dense, then there exists a hollow right neighbourhood Uo+tα of tα such that x1t<lnB/A, for t∈Uo+tα.
Notice thatlimt→∞⁡supx1t=limα→∞⁡supx1sα. If sα is left-scattered, by integrating (25) over the set ρsα,sα, we have (31)x1sα-x1ρsα≤∫ρsαsαB-Aexpx1tΔt≤∫ρsαsαBΔt,by (b), it follows that(32)x1sα≤lnBA+∫ρsαsαBΔt,and from (8), we know sα-ρsα≤l; then x1sα≤Bl+lnB/A.
Ifsα is left-dense, we choose t1∈Uo-sα, such that sα-t1≤l. By integrating (25) over the set t1,sα, we have (33)x1sα-x1t1≤∫t1sαB-Aexpx1tΔt≤∫t1sαBΔt,notice that x1t1<lnB/A, and it follows that(34)x1sα≤lnBA+∫t1sαBΔt≤Bl+lnBA.
Then(35)limt→∞supx1t=limα→∞supx1sα≤Bl+lnBA,which completes the proof of (i).
(ii) The proof is similar to (i) of Lemma17.Lemma 18.
Assume thatCt,Dt>0 are bounded and rd-continuous functions, 1-C(t)μt>0, and Cl>0; further suppose that y1t satisfies the following:
(iii)(36)y1Δt+Cty1t≤Dt,fort≥T0;then there exists a constant T2>T0, such that, for t>T2, one has (37)y1t≤y1T2e-Ct,T2+DtCl.
Particularly, ifDt is bounded above ultimately with respect to K1, then (38)limt→∞supy1t≤K1Cl.
(iv)(39)y1Δt+Cty1t≥Dt,fort≥T0;there exists a constant T3>T0, such that for, t>T3, one has (40)y1t≥y1T3-DT3Cue-Ct,T3+DT3Cu.
Particularly, ifDt is bounded below ultimately with respect to k1, then (41)limt→∞infy1t≥k1Cu.Proof.
(iii) From1-C(t)μt>0, we have -C∈R. Note that 0≤e-Cσt,T0≤e-Ct,T0→0, when t→∞. Hence, there exists a T2>T0, such that e-Cσt,T2≤1, as t>T2. We use the product rule to calculate (42)y1e⊖-Ct,T2Δt=y1Δte⊖-Cσt,T2+y1t⊖-Cte⊖-Ct,T2=y1Δte⊖-Cσt,T2+y1t⊖-Ct1+σt-t⊖-Cte⊖-Cσt,T2=y1Δt-⊖⊖-Cty1te⊖-Cσt,T2=y1Δt+Cty1te⊖-Cσt,T2,noticing that Dt is rd-continuous, from Lemma 11, (43)Dt=∫T2tdτΔτ,fort∈T,and by integrating both sides over the set T2,t, we have (44)y1te⊖-Ct,T2-y1T2=∫T2ty1Δτ+Cτy1τe⊖-Cστ,T2Δτ≤∫T2tDτe⊖-Cστ,T2Δτ=∫T2tDτ1e-Cστ,T2Δτ=∫T2tDτCτCτe-Cστ,T2Δτ≤1Cl∫T2tDτCτe-Cστ,T2Δτ≤1ClDte-Ct,T2T2t-∫T2tdτe-Cστ,T2Δτ=1ClDte-Ct,T2-DT2-∫T2tdτe-Cστ,T2Δτ≤1ClDte-Ct,T2-DT2-∫T2tdτΔτ=1ClDte-Ct,T2-Dt≤1ClDte-Ct,T2;then (45)y1t≤y1T2e-Ct,T2+e-Ct,T21ClDte-Ct,T2=y1T2e-Ct,T2+DtCl.Thus (46)limt→∞supy1t≤K1Cl.
(iv) From (iii) of Lemma18, we know that there exists a constant T3>T0, such that DT3=k1 and Dt=∫T3t‍dτΔτ; then for t>T3, (47)y1te⊖-Ct,T3-y1T3=∫T3ty1Δτ+Cτy1τe⊖-Cστ,T3Δτ;then (48)y1t≥y1T3e-Ct,T3+e-Ct,T3∫T3tDτe⊖-Cστ,T3Δτ=y1T3e-Ct,T3+e-Ct,T3∫T3tDτCτCτe-Cστ,T3Δτ≥y1T3e-Ct,T3+e-Ct,T31Cu∫T3tDτCτe-Cστ,T3Δτ=y1T3e-Ct,T3+e-Ct,T31CuDte-Ct,T3T3t-∫T3tdτe-Cστ,T3Δτ=y1T3e-Ct,T3+e-Ct,T31CuDte-Ct,T3-DT3-∫T3tdτe-Cστ,T3Δτ≥y1T3e-Ct,T3+DtCu-e-Ct,T3CuDT3-1Cu∫T3te-Ct,T3e-Cστ,T3dτΔτ≥y1T3e-Ct,T3+DtCu-e-Ct,T3CuDT3-1Cu∫T3tdτΔτ≥y1T3e-Ct,T3+DtCu+1Cu-e-Ct,T3CuDT3-DtCu≥y1T3-DT3Cue-Ct,T3+DT3Cu.Thus, (49)limt→∞infy1t≥k1Cu.
Then the proof of Lemma18 is completed.


## 4. Applications

Before giving our main result, we list the definition of uniform ultimate boundedness.Definition 19.
The solutionx1t,x2tT of system (9) is said to be uniformly ultimate bounded if there exist two constants λ1 and λ2 such that, for any initial condition x10,x20T∈R2, (50)λ1≤limt→∞infxit≤limt→∞supxit≤λ2,i=1,2,where λ1 and λ2 are independent on x10,x20T.The definition of the uniform ultimate boundedness of solutions ofN-species system is similar.Then we give the applications of these differential inequalities in this section.Theorem 20.
Assume that (H3) holds,al,kl,rl>0. Then the solutions for system (9) are uniformly ultimate bounded.Proof.
From1-μtat>0, we have -a∈R. From (9) and yt0≥0, we can see (51)yt=yt0e-at,t0+e-at,t0∫t0tbτexpxτe⊖-aστ,t0Δτ≥0;then (52)xΔt≤rt-rtktexp⁡xt≤ru-rlkuexp⁡xt;from Lemma 17, we have(53)limt→∞supxt≤rul+lnrukurl≔M1.
From (9) and (53), there exists a constant T2>t0, such that, for t>T2, (54)yΔt+atyt=btexpxt≤buexpM1;from Lemma 18, we have(55)limt→∞supyt≤buexpM1al≔M2.
From (9) and Lemma 18, (56)yΔt+atyt=btexpxt≤buexpxt;then (57)yt≤yT2e-at,T2+buexpxtal.Note that 0≤yT2e-at,T2→0, when t→∞. Hence, there exists T4, such that cuyT2e-aT4,T2≤1/2, for t>T4. Then we get(58)yt≤yT2e-aT4,T2+buexpxtal,for t>T4.
Note thatyT2e-aT4,T2≔K, from (9) and (58), when t>T4, (59)xΔt≥rt1-1ktexpxt-ctK+buexpxtal=rt1-1ktexpxt-rtctK+buexpxtal=rt1-ctK-rtkt+rtctbualexpxt≥rl1-cuK-rukl+rucubualexpxt;noticing that rl-rlcuK≥rl/2>0, from Lemma 17, we have(60)limt→∞infxt≥rl1-cuK-rukl+rucubualexpM1l+lnalklrl1-cuKalru+klrucubu≔m1.
From (9) and (60), there exists a constant T5>T4, such that, for t>T5, (61)yΔt+atyt=btexpxt≥blexpm1;from Lemma 18, we have(62)limt→∞infyt≥blexpm1au≔m2.
From (53), (55), (60), and (62), we obtain (63)m1≤limt→∞infxt≤limt→∞supxt≤M1,m2≤limt→∞infyt≤limt→∞supyt≤M2;then the solutions for system (9) are uniformly ultimately bounded.
This completes the proof of Theorem20.Theorem 21.
Assume the assumption(H5)
rit, ait, bit, cit, dit, eijt, hit, kit, and βiti,j=1,2,…,n are all bounded, nonnegative, rd-continuous, and real-valued functions; τij,ηi,σii,j=1,2,…,n are positive constants. τ=maxmax1≤i,j≤nτij,max1≤i≤nσi,max1≤i≤nηi>0
holds;rilcil,ail,kil>0. Then the solutions of system (1) are uniformly ultimate bounded.Proof.
By similar analysis as that in Theorem20 and the initial condition (7), we have yit≥0. From (1), we can see xiΔt≤rit≤riu, by integrating both sides over the set t-τii,t,(64)xit-xit-τii=∫t-τiitxiΔtΔt≤∫t-τiitriuΔt=τiiriu,so expxit-τii≥expxit-τiiriu; then the first equation of (1) implies (65)xiΔt≤rit-ritcitexp⁡xit-τii≤riu-rilcilexp⁡xit-τii≤riu-rilcilexp⁡τiiriuexpxit;from Lemma 17, we have(66)limt→∞supxit≤riul+τiiriulnriurilcil≔Mi1.
From (1) and (66), there exists a constant T2>t0, such that, for t>T2, (67)yiΔt+aityit=bitexpxit+βitexpxit-σi≤biu+βiuexpMi1;from Lemma 18, we have(68)limt→∞supyit≤biu+βiuexpMi1ail≔Mi2.From (1), (66), and (68), we have (69)xiΔt≥ril-riuexpMi1kil-riuciuexpMi1-diu+hiuMi2≔ki2;by integrating both sides over the set t-σi,t, we have xit-xit-σi≥σiki2; thus from (1) and Lemma 18, (70)yiΔt+aityit=bitexpxit+βitexpxit-σi≤biu+βiuexpσiki2expxit;then (71)yit≤yiT2e-ait,T2+biu+βiuexpσiki2expxit.Note that 0≤yiT2e-ait,T2→0, when t→∞. Hence, there exists T6 such that (72)diu+hiuyiT2e-aiT6,T2≤ril2,for t>T6. Then we get(73)yit≤yiT2e-aiT6,T2+biu+βiuexpσiki2expxit,for t>T6.
LetKi=yiT2e-aiT6,T2 and ki3=biu+βiu/expσik2; noticing that xit-xit-ηi≥ηiki2 and xit-xit-τii≥τiiki2, from (1) and (73), when t>T6,(74)
x
i
Δ
t
=
r
i
t
1
-
e
x
p
x
i
t
-
τ
i
i
k
i
t
+
∑
j
=
1
,,
j
≠
i
n
e
i
j
t
e
x
p
x
j
t
-
τ
i
j
-
c
i
t
e
x
p
x
i
t
-
τ
i
i
-
d
i
t
y
i
t
-
h
i
t
y
i
t
-
η
i
≥
r
i
t
1
-
e
x
p
x
i
t
-
τ
i
i
k
i
t
-
c
i
t
e
x
p
x
i
t
-
τ
i
i
-
d
i
t
K
i
+
k
i
3
e
x
p
x
i
t
-
h
i
t
K
i
+
k
i
3
e
x
p
x
i
t
-
η
i
≥
r
i
t
1
-
e
x
p
x
i
t
-
τ
i
i
k
i
t
-
c
i
t
e
x
p
x
i
t
-
τ
i
i
-
d
i
t
K
i
+
k
i
3
e
x
p
x
i
t
-
h
i
t
K
i
+
k
i
3
e
x
p
η
i
k
i
2
e
x
p
x
i
t
≥
r
i
l
-
d
i
u
+
h
i
u
K
i
-
d
i
u
+
h
i
u
k
i
3
+
k
i
3
e
x
p
η
i
k
i
2
e
x
p
x
i
t
-
r
i
u
k
i
-
l
+
c
i
u
e
x
p
τ
i
i
k
i
2
e
x
p
x
i
t
≥
r
i
l
-
d
i
u
+
h
i
u
K
i
-
d
i
u
+
h
i
u
k
i
3
+
k
i
3
e
x
p
η
i
k
i
2
+
r
i
u
k
i
-
l
+
c
i
u
e
x
p
τ
i
i
k
i
2
e
x
p
x
i
t
,noticing that ril-diu+hiuKi≥ril/2>0, denote diu+hiuki3+ki3/expηiki2+riuki-l+ciu/expτiiki2≔ki4; from Lemma 17, we have(75)limt→∞infxit≥ril2-ki4expMi1l+lnril2ki4≔mi1.
From (1) and (75), there exists a constant T7>T6, such that, for t>T7, (76)yiΔt+aityit=bitexpxit+βitexpxit-σi≥bil+βilexpmi1;from Lemma 18, we have(77)limt→∞infyit≥bil+βilexpmi1aiu≔mi2.
From (66), (68), (75), and (77), we obtain (78)mi1≤limt→∞infxit≤limt→∞supxit≤Mi1,mi2≤limt→∞infyit≤limt→∞supyit≤Mi2;then the solutions for system (1) are uniformly ultimate bounded.
Therefore, the proof of Theorem21 is completed.Noticing the differences of Definitions1 and 19, due to the exponential transformations in Section 1, when solutions of system (1) or (9) are uniformly ultimate bounded, systems (5) and (6) or (11) and (12) are permanent, and the contrary also holds true. Then we have the following remarks.Remark 22.
IfT=Z, then Theorems 20 and 21 reduce to Theorems 6 and 4, respectively.
IfT=R, we could get a new method to solve the continuous result from Theorem 20. Thus, our result is a well unification of continuous and discrete cases.Remark 23.
IfT=R, then the result implies Theorem 3.
It is obvious that the conditions of Theorem21 are weaker than that in Theorem 2, and it includes the known results of [14, 16] on R or Z, respectively. Thus, Theorem 21 is a well unification of Theorems 3 and 4. And our result shows that the additional conditions in [15, 17] are not necessary.


## 5. An Example

In this section, we give a numerical example to support our main result. LetT=⋃k=0∞12k+3,12k+11; we assume(79)xΔt=0.6+0.4cost1-expxt1+0.5cost-0.01yt,yΔt=-0.3yt+0.8+0.2sintexpxt,t∈T;it is easy to see that the conditions in Theorem 20 are verified. In this case, we know that (80)x12k+3=x12k-1+4×r12k-11-expx12k-1k12k-1-c12k-1y12k-1,y12k+3=y12k-1-4×a12k-1y12k-1+b12k-1expx12k-1;here k=1,2,3,…. By applying numerical analysis from Matlab, we obtain Figure 1. It is obvious that the solutions for system (79) are uniformly ultimate bounded. Our numerical simulation supports our theoretical findings (see Figure 1).Figure 1
For the initial conditionx(0)=3 and y(0)=4. “∗" presents x(t) and “−" presents y(t).


## 6. Conclusion

In this paper, we consider two systems with feedback control on time scales. It is shown that these systems contain the differential and difference systems as special cases. By constructing some differential inequalities on time scales, the uniform ultimate boundedness of solutions for these systems is obtained. Compared with some previous relative results, our results could unify the continuous and discrete cases sufficiently (see Theorems3 and 4). This provides a new method to get the permanence of ecosystems. Our results generalize and extend some known results. And a numerical example shows the verification of our main results.



---

*Source: 9195613-2017-02-28.xml*