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Research article

Integrated decision-making methods based on 2-tuple linguistic m-polar fuzzy information

  • Received: 02 April 2022 Revised: 26 May 2022 Accepted: 27 May 2022 Published: 07 June 2022
  • MSC : 03E72, 20F10

  • The 2-tuple linguistic m-polar fuzzy sets (2TLmFSs) are acknowledged to represent the multi-polar information owing to the practical structure of m-polar fuzzy sets with the help of linguistic terms. The TOPSIS and ELECTRE series are efficient and widely used methods for solving multi-attribute decision-making problems. This paper aim to augment the literature on multi-attribute group decision making focusing on the the strategic approaches of TOPSIS and ELECTRE-I methods for the 2TLmFSs. In the 2TLmF-TOPSIS method, the relative closeness index is used to rank the alternatives. For the construction of concordance and discordance sets, the superiority and inferiority of alternatives over each other are accessed by using the score and accuracy functions. In the 2TLmF ELECTRE-I, selection of the best alternative is made by the means of an outranking decision graph. At the final step of the 2TLmF ELECTRE-I method, a supplementary approach is developed for the linear ranking of alternatives based on the concordance and discordance outranking indices. The structure of the proposed techniques are illustrated by using a system flow diagram. Finally, two case studies are used to demonstrate the correctness, transparency, and effectiveness of the proposed methods for selecting highway construction project manager and the best textile industry.

    Citation: Muhammad Akram, Uzma Noreen, Mohammed M. Ali Al-Shamiri, Dragan Pamucar. Integrated decision-making methods based on 2-tuple linguistic m-polar fuzzy information[J]. AIMS Mathematics, 2022, 7(8): 14557-14594. doi: 10.3934/math.2022802

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  • The 2-tuple linguistic m-polar fuzzy sets (2TLmFSs) are acknowledged to represent the multi-polar information owing to the practical structure of m-polar fuzzy sets with the help of linguistic terms. The TOPSIS and ELECTRE series are efficient and widely used methods for solving multi-attribute decision-making problems. This paper aim to augment the literature on multi-attribute group decision making focusing on the the strategic approaches of TOPSIS and ELECTRE-I methods for the 2TLmFSs. In the 2TLmF-TOPSIS method, the relative closeness index is used to rank the alternatives. For the construction of concordance and discordance sets, the superiority and inferiority of alternatives over each other are accessed by using the score and accuracy functions. In the 2TLmF ELECTRE-I, selection of the best alternative is made by the means of an outranking decision graph. At the final step of the 2TLmF ELECTRE-I method, a supplementary approach is developed for the linear ranking of alternatives based on the concordance and discordance outranking indices. The structure of the proposed techniques are illustrated by using a system flow diagram. Finally, two case studies are used to demonstrate the correctness, transparency, and effectiveness of the proposed methods for selecting highway construction project manager and the best textile industry.



    Hematopoiesis is a continuous physiological process of blood cell formation that occurs primarily in the bone marrow [1]. In this procedure, as a common origin, hematopoietic stem cells are responsible for the lifetime generation of the three major cell lines: erythrocytes (red blood cells), leukocytes (white blood cells), and platelets. These stem cells have the extraordinary ability to renew, self-replicate, and differentiate into other functional cells. Blood cell numbers in the hematopoietic system maintain a dynamic balance through complex regulatory mechanisms [2]. The concentration of blood cells can be measured in venous blood samples [3]. Under normal physiological conditions, cell concentrations are maintained within the determinable normal range. However, blood cell counts appear to have wide abnormal fluctuations under pathological conditions, leading to a variety of blood diseases [4,5]. For instance, an increased red blood cell count indicates polycythemia vera, cardiovascular disease and stress. A decreased red blood cell count is an indicator of anemia, chronic renal failure and acute hemorrhaging. Therefore, exploring the dynamic changes in blood cell counts is of crucial importance for human life and health.

    Conceptually simple differential equations are believed to be fundamental for modeling a large number of different physiological processes [6]. Delay is typically incorporated into such models to provide a more accurate description for the dynamical systems [7,8]. Delay differential equations are used to describe the development systems that depend on not only the current state but also the historical state and have been widely applied into many fields such as biology, engineering and epidemiology [9,10]. To understand the dynamics of blood cell production, the behaviors of blood cell number have been widely employed by delay differential models. In 1977, first-order nonlinear autonomous differential equations, called by Mackey-Glass equations, with time delay were proposed in [11]. Blood cells change over time with the increase or decrease in number. Hence, different feedback control mechanisms, responsive to the body's demand for cells, are operated in these models. They showed the dynamic behaviors including limit cycle, oscillations and chaotic solutions of models. Lasota [12] studied a delay feedback model in which the nonlinear feedback has an exponential term. By using the approach of ergodic theory, chaotic behaviour of this biological model was obtained.

    Considering the important effects of environmentally driven cyclical factors on organisms, Berezansky et al. [13] studied a continuous hematopoiesis model with periodic coefficient and time delay. The feedback control mechanism was implemented by a monotonically decreasing function. In their other work [14], a unimodal feedback control mechanism was taken into account to show the different behaviors of blood cell number. More comprehensive hematopoiesis models were presented by considering the different maturation times required for different blood cells in hematopoietic system. Continuous hematopoiesis models with multiple feedback loops dominated by different time delays (maturation times) were used in Liu et al. [15] and Wu et al. [16]. On the other hand, Yao [17] and the present author [18] explored such models from a discrete perspective, wherein the discrete models are more effective than continuous ones. All these results suggest the existence or global attractivity of positive periodic solutions for hematopoiesis models under some suitable conditions. The research of positive periodic solutions is of great significance both in theoretical interest and real applications [19,20,21,22].

    Impulses maybe generated in evolutionary system due to transient changes in state experienced at certain moments. In the hematopoietic process, these instantaneous and abrupt changes are caused by external interference such as received radiation, medication or other forms of blood cell pressure [23,24]. The blood cell model driven by impulse effects allows discrete rapid changes in cell number and is more accurate and effective in capturing external disturbances [25]. For relevant literatures about impulse in other research areas, we refer to [26,27,28,29,30,31,32,33,34].

    The mechanism by which blood cells increase and decrease have gradually elucidated by clinical studies. On the other hand, it is also important to identify the essence of hematopoietic phenomena by studying relatively simple mathematical models based on various facts. From that point of view, models of blood cells production have been investigated by various studies. Existing studies on continuous models of blood cell dynamics conducted with impulsive effects can be referred to [23,35,36,37,38]. By contrast, relatively few studies on discrete dynamical models have considered the impulse terms. Therefore, in this paper, we formulate a new discrete blood cell production model considering the key impacts of impulse to study the dynamics of blood cell numbers in hematopoietic system.

    The present paper is organized as follows: In Section 2, we investigate the multiplicity of positive periodic solutions of discrete impulsive blood cell production model. Section 3 constructs a solution transformation and a solution representation for the positive periodic solution of the model. Section 4 discusses the multiple positive periodic solutions, which are found by using the Krasnosel'skii fixed point theorem. Section 5 provides a numerical example and its simulations to illustrate our main result.

    In this paper, we focus on the positive periodic solutions of a hematopoietic system with impulse effects. We begin with the Lasota equation [12], which describes the dynamics of the formation of blood cells based on experimental results

    Δx(k)=αx(k)+βxn(kτ)ex(kτ).

    In this model, x denotes the number of blood cells in blood circulation. The first term on the right-hand side represents the destruction of blood cells and α is the death rate of blood cells. The second term on the right-hand side represents the influx of blood cells into blood circulation, in which β and n are positive constants. The production process of blood cells is inherently nonlinear in nature, and the parameter n is intended to capture this nonlinearity that has nonmonotonic character. The delay τ is the time required for blood cells to attain maturity from immature cells made in bone marrow to mature ones.

    Cyclical fluctuations in the environment have an important effect on many living systems and they give the organisms inherent periodicity. However, constant coefficients and constant time delay could not reflect such periodic environmental influence on blood cells. Hence, it is reasonable and realistic to consider the environmental cyclicity by assuming that mortality, productivity, and maturation during hematopoietic processes change periodically.

    Incorporating periodicity and impulsive effects into the hematopoietic system, we discuss the following blood cell production model:

    {Δx(k)=α(k)x(k)+mi=1βi(k)xn(kτi(k))ex(kτi(k)),kZ+andkkj,Δx(kj)=γjx(kj),jN. (2.1)

    Here, Δ is the forward difference operator defined by Δx(k)=x(k+1)x(k) for kZ; m1 is a natural number and n>1 is a real constant; α:Z(0,1), βi:Z(0,), and τi:ZZ+(1im) are ω-periodic; sequence of real numbers {γj}jN(γj>1) and increasing sequence of natural numbers {kj}jN are ω-periodic, i.e., there exists a qN such that kj+q=kj+ω and γj+q=γj. For this model, we assume that

    (H) 0kj<ω(1+γj)rkj,0r<ω(1α(r))<1.

    The initial condition of system (2.1) is

    x(s)=ϕ(s)>0,sZ[¯τ,0], (2.2)

    where ¯τ=max1im{max0k<ωτi(k)}Z+ and Z[¯τ,0]={¯τ,¯τ+1,,0}. We note that 0<α(k)<1 for kZ and γj>1 for jN. The solution of (2.1) denoted by x(;ϕ) is obviously positive. The purpose of this study is to present a sufficient condition that ensures that system (2.1) has at least two positive ω-periodic solutions. The main result is the following:

    Theorem 2.1. Assume that (H) holds. Let β_i=minkkj,0k<ωβi(k) for each i=1,2,,m and γj=min{γj,0} for jN. If

    cmi=1β_i0kj<ω(1+γj)ρnnn1>enρ (2.3)

    holds, then (2.1) has at least two positive ω-periodic solutions, where

    c=rkj,0r<ω(1α(r))10kj<ω(1+γj)rkj,0r<ω(1α(r))andρ=rkj,0r<ω(1α(r)).

    In this section, we start with the well-known Krasnosel'skii fixed point theorem [39] which is an efficient method for searching periodic solutions of nonlinear differential or difference equations.

    Lemma 3.1. (Krasnosel'skii fixed point theorem) Let (X,||||) be a Banach space, and let PX be a cone in X. Suppose that Ω1 and Ω2 are open bounded subsets of X with θΩ1¯Ω1Ω2. Let Φ:PP be a completely continuous operator on P such that, either

    (i) ||Φx||||x|| for xPΩ1 and ||Φx||||x|| for xPΩ2, or

    (ii) ||Φx||||x|| for xPΩ1 and ||Φx||||x|| for xPΩ2.

    Then, Φ has a fixed point in P(¯Ω2Ω1).

    For i=1,2,,m, let

    ˜α(k)={α(k),kkj,0,k=kj,jN,and˜βi(k)={βi(k),kkj,0,k=kj,jN.

    Under the initial condition y(s)=ϕ(s)>0 for sZ[¯τ,0], we consider the following nonimpulsive difference equation with time-delays:

    Δy(k)=˜α(k)y(k)+0kj<k(1+γj)1mi=1˜βi(k)yn(kτi(k))0kj<kτi(k)(1+γj)ney(kτi(k))0kj<kτi(k)(1+γj), (3.1)

    and we perform the transformation between the solutions of (2.1) and (3.1).

    Lemma 3.2. (i) If y(k) is a solution of (3.1), then x(k)=y(k)0kj<k(1+γj) is a solution of (2.1); (ii) If x(k) is a solution of (2.1), then y(k)=x(k)0kj<k(1+γj)1 is a solution of (3.1).

    Proof. We first prove part (i). For any kkj and jN, we obtain

    x(k+1)(1α(k))x(k)mi=1βi(k)xn(kτi(k))ex(kτi(k))=y(k+1)0kj<k+1(1+γj)(1˜α(k))y(k)0kj<k(1+γj)mi=1˜βi(k)yn(kτi(k))0kj<kτi(k)(1+γj)ney(kτi(k))0kj<kτi(k)(1+γj)=0kj<k(1+γj)(y(k+1)(1˜α(k))y(k)0kj<k(1+γj)1mi=1˜βi(k)yn(kτi(k))0kj<kτi(k)(1+γj)ney(kτi(k))0kj<kτi(k)(1+γj))=0.

    It follows from (3.1) that y(kj+1)=y(kj). Then,

    x(kj+1)=y(kj+1)0kt<kj+1(1+γt)=(1+γj)y(kj)0kt<kj(1+γt)=(1+γj)x(kj).

    Thus, x(k)=y(k)0kj<k(1+γj) is a solution of (2.1).

    Now, we prove part (ii). For any kkj and jN, one has

    Δy(k)=y(k+1)y(k)=x(k+1)0kj<k+1(1+γj)1x(k)0kj<k(1+γj)1Δy(k)=(x(k+1)x(k))0kj<k(1+γj)1=(α(k)x(k)+mi=1βi(k)xn(kτi(k))ex(kτi(k)))0kj<k(1+γj)1=˜α(k)y(k)+0kj<k(1+γj)1mi=1˜βi(k)yn(kτi(k))0kj<kτi(k)(1+γj)ney(kτi(k))0kj<kτi(k)(1+γj).

    Moreover, from (2.1), we have

    x(kj+1)=(1+γj)x(kj)forjN.

    Hence, it follows the definition of y that

    y(kj+1)=x(kj+1)0kt<kj+1(1+γt)1=(1+γj)x(kj)0kt<kj+1(1+γt)1=x(kj)0kt<kj(1+γt)1=y(kj)

    for jN. That is, y(k) satisfies (3.1). Therefore, y(k)=x(k)0kj<k(1+γj)1 is a solution of (3.1). The proof is complete.

    We define X={x:x(k)=x(k+ω)RforkZ}. Then, X is a finite dimensional Banach space with the norm ||x||=max0k<ω|x(k)|. Let x be an element of X satisfying x(k)>0 for kZ and then we set ˆx(k)=x(k) for kZ[¯τ,)def=[¯τ,)Z and ˆϕ(k)=x(k) for kZ[¯τ,0]. Denote

    H(k,s)=s+1r<k+ω(1˜α(r))10kj<ω(1+γj)0r<ω(1˜α(r)),ksk+ω1withkZ.

    Then, the following representation of a positive ω-periodic solution of (2.1) can be obtained.

    Lemma 3.3. Assume that (H) holds. Then a positive ω-periodic sequence ˆx is the solution x(;ˆϕ) of (2.1) if and only if the original sequence xX satisfies

    x(k)>0andx(k)=k+ω1s=k(H(k,s)skj<k+ω(1+γj)mi=1˜βi(s)xn(sτi(s))ex(sτi(s))). (3.2)

    Proof. (Necessity) Suppose that the positive ω-periodic sequence ˆx is the solution x(;ˆϕ) of (2.1). Then it is obvious that x(k)>0 for kZ. We rewrite (3.1) as

    y(k+1)(1˜α(k))y(k)=0kj<k(1+γj)1mi=1˜βi(k)yn(kτi(k))0kj<kτi(k)(1+γj)ney(kτi(k))0kj<kτi(k)(1+γj).

    Due to Lemma 3.2, we multiply both sides of the above equality by 0r<k+11/(1˜α(r)) to obtain

    x(k+1)0kj<k+1(1+γj)10r<k+111˜α(r)x(k)0kj<k(1+γj)10r<k11˜α(r)=0kj<k(1+γj)1mi=1˜βi(k)xn(kτi(k))ex(kτi(k))0r<k+111˜α(r).

    Summing up both sides from k to k+ω1 results in

    x(k+ω)0kj<k+ω(1+γj)10r<k+ω11˜α(r)x(k)0kj<k(1+γj)10r<k11˜α(r)=k+ω1s=k(0kj<s(1+γj)1mi=1˜βi(s)xn(sτi(s))ex(sτi(s))0r<s+111˜α(r)),

    which leads to

    x(k)(1kkj<k+ω(1+γj)kr<k+ω(1˜α(r)))=k+ω1s=k(skj<k+ω(1+γj)mi=1˜βi(s)xn(sτi(s))ex(sτi(s))s+1r<k+ω(1˜α(r))).

    Using the periodicity of ˜α and γj, we have

    x(k)=110kj<ω(1+γj)0r<ω(1˜α(r))×k+ω1s=k(skj<k+ω(1+γj)mi=1˜βi(s)xn(sτi(s))ex(sτi(s))s+1r<k+ω(1˜α(r)))=k+ω1s=k(H(k,s)skj<k+ω(1+γj)mi=1˜βi(s)xn(sτi(s))ex(sτi(s)))

    for kZ. Thus, x satisfies (3.2).

    (Sufficiency) Direct computation shows that

    H(k+ω,s+ω)=s+ω+1r<k+2ω(1˜α(r))10kj<ω(1+γj)0r<ω(1˜α(r))=s+1r<k+ω(1˜a(r+ω))10kj<ω(1+γj)0r<ω(1˜α(r))=s+1r<k+ω(1˜a(r))10kj<ω(1+γj)0r<ω(1˜α(r))=H(k,s).

    Hence, H is ω-periodic with respect to both variables k and s. Then, we can prove that an element x of X having the expression (3.2) is ω-periodic. In fact,

    x(k+ω)=k+2ω1s=k+ω(H(k+ω,s)skj<k+2ω(1+γj)mi=1˜βi(s)xn(sτi(s))ex(sτi(s)))=k+ω1s=k(H(k+ω,s+ω)s+ωkj<k+2ω(1+γj)mi=1˜βi(s+ω)xn(s+ωτi(s+ω))ex(s+ωτi(s+ω)))=k+ω1s=k(H(k,s)skj<k+ω(1+γj)mi=1˜βi(s)xn(sτi(s))ex(sτi(s)))=x(k).

    Now, we construct ˆx and ˆϕ from the element x of X given by (3.2). To prove that ˆx is the positive ω-periodic solution x(;ˆϕ) of (2.1), it suffices to show that x satisfies (2.1). For kkj, since ˜βi=βi(i=1,2,,m) for kZ, we have

    x(k+1)=k+ωs=k+1(H(k+1,s)skj<k+ω+1(1+γj)mi=1βi(s)xn(sτi(s))ex(sτi(s)))=k+ω1s=k+1(H(k+1,s)skj<k+ω+1(1+γj)mi=1βi(s)xn(sτi(s))ex(sτi(s)))+H(k+1,k+ω)k+ωkj<k+ω+1(1+γj)mi=1βi(k+ω)xn(k+ωτi(k+ω))ex(k+ωτi(k+ω))=k+ω1s=k+1(skj<k+ω+1(1+γj)s+1r<k+ω+1(1α(r))10kj<ω(1+γj)0r<ω(1α(r))mi=1βi(s)xn(sτi(s))ex(sτi(s)))+k+ω+1r<k+ω+1(1α(r))10kj<ω(1+γj)0r<ω(1α(r))mi=1βi(k)xn(kτi(k))ex(kτi(k))=k+ω1s=k+1(skj<k+ω+1(1+γj)s+1r<k+ω+1(1α(r))10kj<ω(1+γj)0r<ω(1α(r))mi=1βi(s)xn(sτi(s))ex(sτi(s)))+110kj<ω(1+γj)0r<ω(1α(r))mi=1βi(k)xn(kτi(k))ex(kτi(k)).

    Moreover,

    (1α(k))x(k)=k+ω1s=k((1α(k))H(k,s)skj<k+ω(1+γj)mi=1βi(s)xn(sτi(s))ex(sτi(s)))=k+ω1s=k+1((1α(k))H(k,s)skj<k+ω(1+γj)mi=1βi(s)xn(sτi(s))ex(sτi(s)))+(1α(k))H(k,k)kkj<k+ω(1+γj)mi=1βi(k)xn(kτi(k))ex(kτi(k))=k+ω1s=k+1((1α(k+ω))skj<k+ω(1+γj)s+1r<k+ω(1α(r))10kj<ω(1+γj)0r<ω(1α(r))mi=1βi(s)xn(sτi(s))ex(sτi(s)))+(1α(k))0kj<ω(1+γj)k+1r<k+ω(1α(r))10kj<ω(1+γj)0r<ω(1α(r))mi=1βi(k)xn(kτi(k))ex(kτi(k))=k+ω1s=k+1(skj<k+ω+1(1+γj)s+1r<k+ω+1(1α(r))10kj<ω(1+γj)0r<ω(1α(r))mi=1βi(s)xn(sτi(s))ex(sτi(s)))+0kj<ω(1+γj)0r<ω(1α(r))10kj<ω(1+γj)0r<ω(1α(r))mi=1βi(k)xn(kτi(k))ex(kτi(k)).

    Then, we obtain

    x(k+1)(1α(k))x(k)=110kj<ω(1+γj)0r<ω(1α(r))mi=1βi(k)xn(kτi(k))ex(kτi(k))0kj<ω(1+γj)0r<ω(1α(r))10kj<ω(1+γj)0r<ω(1α(r))mi=1βi(k)xn(kτi(k))ex(kτi(k))=mi=1βi(k)xn(kτi(k))ex(kτi(k)).

    For k=kj, one has

    x(kj+1)=kj+ωs=kj+1(H(kj+1,s)skt<kj+ω+1(1+γt)mi=1˜βi(s)xn(sτi(s))ex(sτi(s)))=kj+ω1s=kj+1(H(kj+1,s)skt<kj+ω+1(1+γt)mi=1˜βi(s)xn(sτi(s))ex(sτi(s)))+H(kj+1,kj+ω)kj+ωkt<kj+ω+1(1+γt)mi=1˜βi(k+ω)xn(k+ωτi(k+ω))ex(k+ωτi(k+ω))=kj+ω1s=kj+1(skt<kj+ω+1(1+γt)s+1r<kj+ω+1(1˜a(r))10kt<ω(1+γt)0r<ω(1˜a(r))mi=1˜βi(s)xn(sτi(s))ex(sτi(s))),

    and

    x(kj)=kj+ω1s=kj(H(kj,s)skt<kj+ω(1+γt)mi=1˜βi(s)xn(sτi(s))ex(sτi(s)))=kj+ω1s=kj+1(H(kj,s)skt<kj+ω(1+γt)mi=1˜βi(s)xn(sτi(s))ex(sτi(s)))+H(kj,kj)(1+γj)kjkj<kj+ω(1+γj)mi=1˜βi(k)xn(kτi(k))ex(kτi(k))=kj+ω1s=kj+1(skt<kj+ω(1+γt)s+1r<kj+ω(1˜α(r))10kt<ω(1+γt)0r<ω(1˜α(r))mi=1˜βi(s)xn(sτi(s))ex(sτi(s))).

    Because of the periodicity of γj, we see that x(kj+1)=(1+γj)x(kj). Hence, x satisfies system (2.1). The proof is complete.

    We define a cone in Banach space X by P={xX:x(k)ρ||x||forkZ}. Let

    Φx(k)=k+ω1s=k(H(k,s)skj<k+ω(1+γj)mi=1˜βi(s)xn(sτi(s))ex(sτi(s)))forxP. (3.3)

    Then, it is easy to obtain the following property of the operator Φ.

    Lemma 3.4. Assume that (H) holds. Then the operator Φ:PP is completely continuous.

    Proof. In view of the proof of sufficiency of Lemma 3.3, it follows from (3.3) that Φx(k+ω)=Φx(k). Hence, ΦxX. Since 0<˜α<1 for kZ, we can estimate that

    H(k,s)k+1r<k+ω(1˜α(r))10kj<ω(1+γj)0r<ω(1˜α(r))kr<k+ω(1˜α(r))10kj<ω(1+γj)0r<ω(1˜α(r))=rkj,0r<ω(1α(r))10kj<ω(1+γj)rkj,0r<ω(α(r))=c.

    Moreover,

    H(k,s)k+ωr<k+ω(1˜α(r))10kj<ω(1+γj)0r<ω(1˜α(r))=110kj<ω(1+γj)0r<ω(1˜α(r))=110kj<ω(1+γj)rkj,0r<ω(1α(r))d.

    The above two inequalities lead to

    ck+ω1s=k(skj<k+ω(1+γj)mi=1˜βi(s)xn(sτi(s))ex(sτi(s)))Φx(k)dk+ω1s=k(skj<k+ω(1+γj)mi=1˜βi(s)xn(sτi(s))ex(sτi(s))). (3.4)

    Hence,

    Φdk+ω1s=k(skj<k+ω(1+γj)mi=1˜βi(s)xn(sτi(s))ex(sτi(s)))dcΦx(k).

    Therefore,

    Φx(k)rkj,0r<ω(1α(r))Φ=ρΦ.

    Thus, ΦxP. The straightforward calculations can show that Φ is a completely continuous operator. The proof is complete.

    In this section, we investigate the multiplicity of positive ω-periodic solutions. The main result Theorem 2.1 is proved by the Krasnosel'skii fixed point theorem below.

    Proof. From the solution representation (3.2) in Lemma 3.3, we observe that the positive ω-periodic solution of (2.1) is a fixed point of the completely continuous operator Φ defined by (3.3). It suffices to establish the existence of two fixed points of Φ. Let ¯βi=maxkkj,0k<ωβi(k) for each i=1,2,,m and γ+j=max{γj,0} for j=1,2,. It follows that

    limu0un/ueu=limu0un1/eu=0,limuun/eu=0.

    Choose a constant ε0 satisfying

    0<ε0<1dωmi=1¯βi0kj<ω(1+γ+j),

    where d=1/(10kj<ω(1+γj)rkj,0r<ω(1α(r))). Then one can find an u0 with 0<u0<n and an U with n<U such that

    un/eu<ε0u0,0uu0;un/eu<ε0U,uU. (4.1)

    Let u1=U/ρ>U. The condition (2.3) implies that there exists a sufficiently small σ(0,u1n) such that

    cmi=1β_i0kj<ω(1+γj)nnρnenρ>n+σ. (4.2)

    Then we define four open bounded subsets Ωi(1i4) of X by

    Ω1={xX:||x||<u0},Ω2={xX:||x||<n},Ω3={xX:||x||<n+σ},Ω4={xX:||x||<u1}

    with θΩ1¯Ω1Ω2¯Ω2Ω3¯Ω3Ω4¯Ω4.

    According to the characteristics of the open bounded sets Ωi(1i4), the discussion is divided into four cases.

    Case 1. Suppose that x is an element of PΩ1X. For kZ, we see that 0x(k)<u0 and x is ω-periodic. By (3.4) and (4.1), one has

    (Φx)(k)dk+ω1s=k(skj<k+ω(1+γ+j)mi=1¯βixn(sτi(s))ex(sτi(s)))dmi=1¯βiε0u0(k+ω1s=kkkj<k+ω(1+γ+j))dωmi=1¯βi0kj<ω(1+γ+j)ε0u0<u0

    for kZ. Hence, ||Φx||<u0=||x|| for xPΩ1.

    Case 2. Suppose that x is an element of PΩ2X. For kZ, we see that nρ=ρxx(k)<n and x is ω-periodic. The unimodal property of un/eu shows that minnρunun/eu=nnρn/enρ. Hence, from (2.3) and (3.4), it leads to

    (Φx)(k)ck+ω1s=k(skj<k+ω(1+γj)mi=1˜βi(s)xn(sτi(s))ex(sτi(s)))ck+ω1s=k(kkj<k+ω(1+γj)mi=1˜βi(s)xn(sτi(s))ex(sτi(s)))c0kj<ω(1+γj)nnρnenρk+ω1s=kmi=1˜βi(s)cmi=1β_i0kj<ω(1+γj)nnρn/enρ>n

    for kZ. Whence ||Φx||>n=||x|| for xPΩ2.

    Case 3. Suppose that x is an element of PΩ3X. For kZ, we see that ρ(n+σ)=ρxx(k)n+σ and x is ω-periodic. Note that σ is a sufficiently small positive constant, then

    minρ(n+σ)u(n+σ)uneu=ρn(n+σ)neρ(n+σ)>nnρnenρ.

    It follows from (3.4) and (4.2) that, for kZ,

    (Φx)(k)ck+ω1s=k(kkj<k+ω(1+γj)mi=1˜βi(s)xn(sτi(s))ex(sτi(s)))c0kj<ω(1+γj)ρn(n+σ)neρ(n+σ)k+ω1s=kmi=1˜βi(s)(Φx)(k)cmi=1β_i0kj<ω(1+γj)nnρnenρ>n+σ.

    Hence, ||Φx||>n+σ=||x|| for xPΩ3.

    Case 4. Suppose that x is an element of PΩ4X. For kZ, we see that ρu1x(k)u1 and x is ω-periodic. By (3.4) and (4.1), one has

    (Φx)(k)dk+ω1s=k(kkj<k+ω(1+γ+j)mi=1¯βixn(sτi(s))ex(sτi(s)))dmi=1¯βiε0U(k+ω1s=kkkj<k+ω(1+γ+j))dωmi=1¯βi0kj<ω(1+γ+j)ε0U<U<u1

    for kZ, which implies that ||Φx||<u1=||x|| for xPΩ4.

    Now we are in the position to conclude that the conditions of the Krasnosel'skii fixed point theorem are satisfied by Ω1, Ω2 and by Ω3, Ω4, respectively. Therefore, the operator given by (3.3) has two fixed points. Among them, one is x1P(¯Ω2Ω1) which satisfies x1(k)ρ||x1|| for kZ and 0<u0||x||n, and the other is x2P(¯Ω4Ω3) which satisfies x2(k)γ||x2|| for kZ and 0<n+σ||x2||u1. Let ˆx1(k)=x1(k) for kZ[¯τ,) and ˆϕ1(k)=x1(k) for kZ[¯τ,0]. Moreover, let ˆx2(k)=x2(k) for kZ[¯τ,) and ˆϕ2(k)=x2(k) for kZ[¯τ,0]. Then, Lemma 3.3 implies that ˆx1 is a positive ω-periodic solution of (2.1) with the initial function ˆϕ1 and ˆx2 is another positive ω-periodic solution of (2.1) with the initial function ˆϕ2. Thus, (2.1) has two different positive ω-periodic solutions ˆx1 and ˆx2 satisfying ˆx1n<n+σˆx2. The proof is complete.

    Let us consider the difference equation

    {Δx(k)=α(k)x(k)+β1(k)x2(k1)ex(k1)+β2(k)x2(k7)ex(k7),kZ+andkkj,Δx(kj)=γjx(kj),kj=2j1,jN. (5.1)

    Here, all coefficients are positive 4-periodic discrete functions defined as follows:

    α(k)={0.5,ifk=0,0.4,ifk=2,β1(k)={9,ifk=0,8,ifk=2,β2(k)={35,ifk=0,40,ifk=2.

    The impulse values are γ1=0.5 for k1=1 and γ2=1.5 for k2=3. We can verify that there are at least two positive 4-periodic solutions of (5.1).

    It is obvious that 2i=1β_i=35+8=43. Since γ1=min{0.5,0}=0.5, γ2=min{1.5,0}=0, it follows that

    0kj<4(1+γj)=(10.5)×1=0.5.

    In view of values of α, we have

    ρ=rkj,0r<4(1α(r))=(10.5)(10.4)=0.3.

    Hence,

    c=ρ1ρ0kj<4(1+γj)=0.310.3(10.5)(1+1.5)=0.48.

    Note that n=2. It can be checked easily that

    c2i=1β_i0kj<4(1+γj)ρnnn1=0.48×43×0.5×o.32×2=1.8576>e0.6.

    Therefore, condition (2.3) is satisfied. Thus, from Theorem 2.1, we see that (5.1) has at least two positive 4-periodic solutions.

    Figure 1a shows that as k gradually increases, the four solutions separately determined by arbitrarily given four initial functions display the same periodic changes. In other words, there exists a positive 4-periodic solution of (5.1). The details of this positive 4-periodic solution are revealed in Figure 1b.

    Figure 1.  Graphs of four arbitrary positive solutions of system (5.1). The numerical simulations show that there is a positive 4-periodic solution of (5.1) and this positive 4-periodic solution is locally asymptotically stable.

    Theorem 2.1 ensures the existence of two positive 4-periodic solutions. In addition to the locally asymptotically stable positive 4-periodic solution described above, system (5.1) has another positive 4-periodic solution. In general, this solution is considered unstable and very difficult to find.

    A discrete blood cell production model subjected to impulse effects is studied in this paper. This model considers sudden changes in the number of blood cells at certain times that cannot be ignored. The sufficient condition that guarantees the multiplicity of positive periodic solutions is established by Krasnoseli-skii fixed point theorem. To be precise, there exist at least two positive periodic solutions of the model under this condition. By using coefficients and impulse values, this sufficient condition can be easily verified.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The paper was supported by Fundamental Research Funds for the Central Universities (Grant No. 41422003).

    The author declares no conflict of interest regarding the publication of this paper.



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