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Research article Special Issues

Global attractors, extremal stability and periodicity for a delayed population model with survival rate on time scales

  • In this paper, we investigate the existence of global attractors, extreme stability, periodicity and asymptotically periodicity of solutions of 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)(1x(d(t))μ(t)),  tT.

    We present many examples to illustrate our results, considering different time scales.

    Citation: Jaqueline G. Mesquita, Urszula Ostaszewska, Ewa Schmeidel, Małgorzata Zdanowicz. Global attractors, extremal stability and periodicity for a delayed population model with survival rate on time scales[J]. Mathematical Biosciences and Engineering, 2021, 18(5): 6819-6840. doi: 10.3934/mbe.2021339

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  • In this paper, we investigate the existence of global attractors, extreme stability, periodicity and asymptotically periodicity of solutions of 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)(1x(d(t))μ(t)),  tT.

    We present many examples to illustrate our results, considering different time scales.



    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)(1x(d(t))μ(t)),  tT (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:TT is the delay function such that ρα(t)d(t)t for some αN. This model is equivalent to

    x(σ(t))=˜γ(t)x(t)+x(d(t))er(t)μ(t)(1x(d(t))μ(t)),   tT

    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:nN0}, T=qN0={qn:nN0}, 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)(1x(t)k(t)),   tZ+. (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)(1x(τ(t))k(t)),  tZ+. (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:TT is defined by σ(t)=inf{sT:s>t} and the backward jump operator ρ:TT by ρ(t)=sup{sT: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 σσn 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:TR at a point tTκ, 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=(tδ,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 sU.

    We say that a function f is delta (or Hilger) differentiable on Tκ provided fΔ(t) exists for all tTκ. The function fΔ:TκR 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  inftTμ(t)>0. (2.1)

    By [15,Theorem 1.16], for any function f:TR, its derivative is given by

    fΔ(t)=f(σ(t))f(t)μ(t)   for all tTκ.

    We consider the delayed population model of the form

    {xΔ(t)=γ(t)x(t)+x(d(t))μ(t)er(t)μ(t)(1x(d(t))k(t)), tt0x(t0)=x0, (2.2)

    with γ:T(,0), r,k:T(0,) and d:TT such that

    ρα(t)d(t)t  for some αN. (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 limtd(t)=.

    By solution of equation (2.2) with initial value x0, we mean function x:TR 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)(1x(ρ2(t0))k(t0))=(1+μ(t0)γ(t0))x(t0)+x(t0)er(t0)μ(t0)(1x(t0)k(t0))x(t2)=(1+μ(t1)γ(t1))x(t1)+x(ρ3(t1))er(t1)μ(t1)(1x(ρ3(t1))k(t1))=(1+μ(t1)γ(t1))x(t1)+x(t0)er(t1)μ(t1)(1x(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)(1x(d(t))k(t)), ttβx(t0)=x0, x(t1)=x1, , x(tβ1)=xβ1 (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

    inftT(1+μ(t)γ(t))=γ0  and  suptT(1+μ(t)γ(t))=γ1.

    (A2) There exist constants ri,ki in (0,) for i=0,1, such that

    inftTr(t)μ(t)=r0,  suptTr(t)μ(t)=r1,  inftTk(t)=k0  and  suptTk(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)=uer0r1uk0  and  U(u)=uer1r0uk1  for  u0

    and the constants M and m be given by

    M=U(k1r0)=k1r0er11  and  m=L(M1γ1).

    (A4) For δ1, we set a constant B as follows

    B=min(L(δM1γ1),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 SR 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 tt0.

    Definition 3.2. A set SR 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

    minsS|x(t)s|<ε  for all  tT(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))}nN, so we can reformulate the above definitions as follows.

    A set SR is said to be invariant relative to (2.2) if for any positive value x(t0) belonging to S, the solution {x(σn(t0))}nN of (2.2) satisfies

    x(σn(t0))S  for all  nN.

    A set SR 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))}nN of (2.2) satisfies

    minsS|x(σn(t0))s|<ε  for all  nN(ε,x(t0)).

    Equation (2.2) is said to be extremely stable if for any two positive solutions {x(σn(t0))}nN and {y(σn(t0))}nN 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))=n1k=0(1+μ(σk(t))γ(σk(t)))x(t)+n1k=0[n1j=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)(1uk(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)(1x(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)))=n1k=0(1+μ(σk+1(t))γ(σk+1(t)))x(σ(t))+n1k=0[n1j=k+1(1+μ(σj+1(t))γ(σj+1(t)))]g(σk+1(t),x(d(σk+1(t))))=nk=1(1+μ(σk(t))γ(σk(t)))x(σ(t))+nk=1[nj=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))=nk=1(1+μ(σk(t))γ(σk(t)))[(1+μ(t)γ(t))x(t)+g(t,x(d(t)))]+nk=1[nj=k+2(1+μ(σj(t))γ(σj(t)))]g(σk(t),x(d(σk(t))))=nk=0(1+μ(σk(t))γ(σk(t)))x(t)+nk=0[nj=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  u0. (3.3)

    In consequence,

    g(v,u)M  for  v[t0,)T  and  u0. (3.4)

    From (3.1) and (3.4), due to assumption (A1), we have

    x(σn(t0))γn1x(t0)+Mn1k=0γnk11=γn1x(t0)+M1γn11γ1 (3.5)

    for any nN. Note that for x(t0)(0,δM1γ1] with δ1, we get from (3.5)

    x(σn(t0))γn1δM1γ1+MMγn11γ1=γn1(δMM)+M1γ1δMM+M1γ1=δM1γ1. (3.6)

    Using the notations from (A3)–(A4) and by properties of function L, we obtain

    L(u)B   for  u[B1γ0,δM1γ1],

    where B is defined in (A4). By inequality (3.3), we get

    g(v,u)B  for  v[t0,)T  and  u[B1γ0,δM1γ1]. (3.7)

    Assuming x(t0)B1γ0, it follows by Lemma 3.5, inequality (3.7) and assumption (A1), the following inequality

    x(σn(t0))(n1k=0γ0)x(t0)+(n1k=0(n1j=k+1γ0)g(σk(t0),x(d(σk(t0)))))n1k=0γ0B1γ0+n1k=0γn1k0B=γn0B1γ0+B(1γn01γ0)=B1γ0 (3.8)

    for nN. 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 [B1γ0,δM1γ1] is invariant relative to (2.2) for positive values of x(t0), where δ1 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 [˜m1γ0,M1γ1] 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)M1γ1|<ε  and  γn0|x(t0)˜m1γ0|<ε  for any  nN(ε,x(t0)).

    It implies

    γn1|x(t0)|<ε+|γn1M1γ1|=ε+Mγn11γ1.

    Applying the above to (3.5), we obtain

    x(σn(t0))<ε+Mγn11γ1+M1γ1Mγn11γ1=ε+M1γ1  for any  nN(ε,x(t0)).

    In analogous way, we get

    x(σn(t0))>˜m1γ0ε  for  nN(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(1γ0)>1, (3.9)

    then

    lim supt|1r(t)μ(t)x(d(t))k(t)|er(t)μ(t)(1x(d(t))k(t))max{|1r0˜mk1(1γ0)|,|1r1Mk0(1γ1)|}.

    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

    ˜m1γ0ε<x(d(t))<M1γ1+ε  for  tT. (3.10)

    It implies the following estimates

    1r(t)μ(t)x(d(t))k(t)<1r0k1(˜m1γ0ε) (3.11)

    and

    1r(t)μ(t)x(d(t))k(t)>1r1k0(M1γ1+ε) (3.12)

    for tT such that tT. Since ε is arbitrary, we can write

    lim supt|1r(t)μ(t)x(d(t))k(t)|max{|1r0˜mk1(1γ0)|,|1r1Mk0(1γ1)|}.

    Now, it remains to show that

    lim supter(t)μ(t)(1x(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(1γ0)εk0>1εk0  for any tT.

    In consequence,

    lim inftx(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{|1r0˜mk1(1γ0)|,|1r1Mk0(1γ1)|}<1γ1, (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 tT, we have by Lemma 3.5

    x(σn(t))=n1k=0(1+μ(σk(t))γ(σk(t)))x(t)+n1k=0[n1j=k+1(1+μ(σj(t))γ(σj(t)))]g(σk(t),x(d(σk(t))))

    and

    y(σn(t))=n1k=0(1+μ(σk(t))γ(σk(t)))y(t)+n1k=0[n1j=k+1(1+μ(σj(t))γ(σj(t)))]g(σk(t),y(d(σk(t))))

    for any tT and nN. In consequence, we have

    x(σn(t))y(σn(t))=n1k=0(1+μ(σk(t))γ(σk(t)))(x(t)y(t))+n1i=0[n1j=i+1(1+μ(σj(t))γ(σj(t)))]{x(d(σi(t)))e˜r(σi(t))(1x(d(σi(t)))k(σi(t)))y(d(σi(t)))e˜r(σi(t))(1y(d(σi(t)))k(σi(t)))}  for all nN,

    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)|+
    +n1i=0γni11|1˜r(σi(t))η(d(σi(t)))k(σi(t))|e˜r(σi(t))(1η(d(σi(t)))k(σi(t)))|x(d(σi(t)))y(d(σi(t)))| (3.15)

    for all  nN, where η(d(σi(t))) is between x(d(σi(t))) and y(d(σi(t))) for i=0,1,,n1. By condition (3.14), there exists real number M1 such that

    max{|1r0˜mk1(1γ0)|,|1r1Mk0(1γ1)|}<M1<1γ1. (3.16)

    Hence, by Lemma 3.8 and by the definition of η, there exists t2T such that for tt2, we have

    |1˜r(σi(t))η(d(σi(t)))k(σi(t))|e˜r(σi(t))(1η(d(σi(t)))k(σi(t)))M1   for all iN. (3.17)

    On the other hand, for any tT, due to Theorems 3.6 and 3.7, the sequence

    {x(σn(t))y(σn(t))}nN

    is bounded. Hence, there exists a0 such that

    lim supn|x(σn(t))y(σn(t))|=a. (3.18)

    In conclusion, for every ε>0, there exists t3T such that

    |x(σn(t))y(σn(t))|<a+ε  for all  nN  and  t3tT. (3.19)

    It is convenient to choose t3 such that t3t2, since it implies condition (3.17) also holds. Combining inequalities (3.15), (3.17) and (3.19), we obtain for tt3 and for all nN

    |x(σn(t))y(σn(t))|γn1|x(t)y(t)|+1γn11γ1M1(a+ε). (3.20)

    Taking lim sup when n+ on both sides of the above inequality, we get

    a11γ1M1(a+ε).

    Since inequality (3.16) is satisfied, we have

    aM1ε1γ1M1.

    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 limnσn(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:nN0,k{0,15,25,35,45,1}},

    where N0 is the set of nonnegative integers. Then t0=0 and mintTμ(t)=15. Consider equation (2.2) with d=ρ2,

    γ(t)={0.4 if  t{3n+1:nN0}3.4 if  t{3n+k:nN0,k{0,15,25,35,45}},
    r(t)={0.425 if  t{3n+1:nN0}0.405 if  t{3n+k:nN0,k{0,15,25,35,45}},

    and

    k(t)={10 if  t{3n+1:nN0}9+t[t] if  t{3n+k:nN0,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)=k1r0er1110.62602;m=L(M1γ1)8.02958;r0k0r18.57647;

    we get ˜m=min{m,r0k0r1}=m. By Theorem 3.7, interval [˜m1γ0,M1γ1][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(1γ0)1.00370
    Figure 1.  Example 3.11 - the part of plot of two chosen solutions of (2.2) for n=120 points.

    and

    max{|1r0˜mk1(1γ0)|,|1r1Mk0(1γ1)|}0.47584<0.68=1γ1.

    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.

    Figure 2.  Example 3.11 - difference of solutions x and y.

    Example 3.12. Let T=qN, where q=1.1, and consider equation (2.2) with

    d(t)={ρ2(t) if t{q2n:nN}ρ(t) if t{q2n1:nN},
    γ(t)={0.75t(q1) if t{q2n1:nN}0.65t(q1) if t{q2n:nN},
    r(t)={0.35t(q1) if t{q2n1:nN}0.45t(q1) if t{q2n:nN},

    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

    M20.21769,  m14.74683  and  ˜m=11.7.

    By Theorem 3.7, interval [˜m1γ0,M1γ1][15.6,31.10414] is a global attractor of (2.2). We check that

    ˜mk1(1γ0)=1.04>1,

    and

    max{|1r0˜mk1(1γ0)|,|1r1Mk0(1γ1)|}=0.532<0.65=1γ1.

    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.

    Figure 3.  Example 3.12 - the part of plot of two chosen solutions of (2.2) for n=120 points.
    Figure 4.  Example 3.12 - the plot of xy for n=120 points.

    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:TR is called ωperiodic 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 ωperiodicity which can be found in [17].

    Lemma 3.14. A function f:TR is ωperiodic 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, ωperiodicity condition given in Lemma 3.14 takes the known form

    f(t+ω)=f(t)  for all  tZ.

    When T=2N0, then one can check that function f is ωperiodic if

    2ωf(2ωt)=f(t)  for all  t2N0,

    reaching the ω–periodicity for the quantum case. See [4,5,10,11].

    Definition 3.16. A function f:TR is said to be asymptotically ωperiodic (or asymptotically ωperiodic for tt1) if there exist two functions p,q:TR such that

    f(t)=p(t)+q(t),

    where p(t) is ωperiodic (or ωperiodic for tt1) 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:TR is such that

    suptT|x(t)|[˜m1γ0,M1γ1]

    and

    ˜mk1(1γ0)1, (3.21)

    then for all tT, the inequality

    |1r(t)μ(t)x(d(t))k(t)|er(t)μ(t)(1x(d(t))k(t))max{|1r0˜mk1(1γ0)|,|1r1Mk0(1γ1)|}

    holds.

    Lemma 3.18. Suppose conditions (A1)(A3), (A5), (3.14) and (3.21) hold. If x:TR is an asymptotically ωperiodic function, r,k:TR are ωperiodic functions and there exists t1T such that for any ωperiodic function p:TR, pd is also ωperiodic for all tt1, then g(t,x(d(t))) defined by (3.2) is an asymptotically ωperiodic function for tt1.

    Proof. Since x is asymptotically ωperiodic, it can be decomposed by

    x(t)=p(t)+q(t),

    where p is ωperiodic and limtq(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)(1p(d(t))k(t))||1r(t)μ(t)ξk(t)|er(t)μ(t)(1ξk(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 ωperiodicity 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))(1p(d(ν(t)))k(ν(t)))=μ(ν(t))p(d(ν(t)))eμ(ν(t))r(ν(t))(1p(d(ν(t)))μ(ν(t))k(ν(t))μ(ν(t)))=μ(t)p(d(t))eμ(t)r(t)(1p(d(t))μ(t)k(t)μ(t))=μ(t)p(d(t))eμ(t)r(t)(1p(d(t))k(t))

    for tt1 since by assumption there exists t1T such that pd is ωperiodic for tt1. Thus, by Lemma 3.14, the function p(d(t))er(t)μ(t)(1p(d(t))k(t)) is ωperiodic for tt1, 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:TR are ω–periodic functions and for any ω–periodic function p:TR, pd is also an ω–periodic function. If x:TR 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:TR is an asymptotically ωperiodic function and γ:TR is an ωperiodic function, then function μγx is an asymptotically ωperiodic function.

    Proof. Since x is an asymptotically ω–periodic function, it can be decomposed by

    x(t)=p(t)+q(t),

    where p is ωperiodic and limtq(t)=0, which implies

    μγx=μγp+μγq.

    By (A1), μγ is a bounded function, which implies that there exists limtμ(t)γ(t)q(t)=0. Thus, it remains to show that function μγp is ωperiodic. 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:TR is an asymptotically ωperiodic for tt1 and γ:TR is an ωperiodic function, then μγx is an asymptotically ωperiodic function for tt1.

    To guarantee the existence of an asymptotically ωperiodic 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+GyΩ for any x,yΩ,

    (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 ωperiodic functions and there exists t1T such that for any ωperiodic function p:TR, pd is also ωperiodic for tt1 then there exists tT such that equation (2.2) has a unique ωperiodic (for tt1) solution x and all other solutions are asymptotically ωperiodic.

    Proof. Let B(T) denote a Banach space of the form

    B(T):={x={x(t)}tt0:suptT|x(t)|<}

    equipped with the norm defined by x=suptT|x(t)|. It is not difficult to show that the set

    B(T)ap:={xB(T):x is asymptotically ω–periodic for tt1}

    with the supremum norm defined above is also a Banach space. Let us introduce the following subset of B(T)ap

    Ωap:={xB(T)ap:˜m1γ0xM1γ1}.

    Observe that Ωap is a bounded, convex and closed subset in B(T)ap. Let us define two operators F,G:ΩapB(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,yΩap 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))))γ1M1γ1+M=M1γ1 (3.23)

    and

    (Fx)(t)+(Gy)(t) γ0˜m1γ0+˜m = ˜m1γ0. (3.24)

    Clearly, (3.23) and (3.24) also remain valid for t=t0. In consequence,

    ˜m1γ0Fx+GyM1γ1.

    By Lemma 3.18 and Corollary 3.21, we obtain that function Fx+Gy is asymptotically ωperiodic for tt1, hence Fx+GyΩap.

    The next step is to show that F is a contraction. Taking any x,yΩap, 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 tT, we get

    |(Fx)(t)(Fy)(t)|γ1xy+|1r(ρ(t))μ(ρ(t))ξ(d(ρ(t)))k(ρ(t))|er(ρ(t))μ(ρ(t))(1ξ(d(ρ(t)))k(ρ(t)))xy,

    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{|1r0˜mk1(1γ0)|,|1r1Mk0(1γ1)|}1γ1ε0.

    Finally, by the arbitrariness of ε0, condition (3.14) and Lemma 3.17 lead to estimate

    FxFy(1γ1)xy  for all tT,

    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 ˜xΩap such that

    ˜x(t)=(F˜x)(t)+(G˜x)(t) for all tT.

    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 ωperiodic (for tt1) solution of (2.2). Thus, ˜x has the following decomposition

    ˜x(t)=˜p(t)+˜q(t), (3.26)

    where ˜p(t) is ωperiodic for tt1 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 tt1

    (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))||1r(t)μ(t)ξk(t)|er(t)μ(t)(1ξk(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 ωperiodic (for tt1) solution of (2.2).

    To prove the uniqueness, assume ˜y is another ωperiodic (for tt1) 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 tt1.

    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 limtq(t)=0, hence x is an asymptotically ωperiodic 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

    T={3n+k:nN0,k{0,15,25,35,45,1}},

    where N0 is the set of nonnegative integers. Consider equation (2.2) with d=ρ5 and functions γ,r,k defined as in Example 3.11.

    It is easy to check that functions γ, r and k are 6periodic, since for all tT we have μ(t)=μ(σ6(t)). This property of graininess function implies that for any 6periodic function p holds p=p(σ6). In consequence, the composition of any 6periodic function p with the delay function d, i.e., pd is 6periodic for tσ5(t0). Since all assumptions of Theorem 3.23 are satisfied equation (2.2) admits unique ωperiodic (for tσ5(t0)) solution and all other solutions are asymptotically ωperiodic (see Figure 5).

    Figure 5.  Example 3.24 - the part of plot of solution (with initial value x(0)=1) of (2.2) for n=60 points.

    In general, assuming d=ρj, we have that if p is 6periodic function, then pd is 6periodic for tσj(t0).

    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 ω-periodic solution and all other solutions are asymptotically ω-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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