AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 463076 10.1155/2013/463076 463076 Research Article Global Exponential Stability of Positive Pseudo-Almost-Periodic Solutions for a Model of Hematopoiesis Meng Junxia Bai Chuanzhi College of Mathematics, Physics and Information Engineering Jiaxing University Jiaxing Zhejiang 314001 China zjxu.edu.cn 2013 20 11 2013 2013 11 08 2013 16 10 2013 2013 Copyright © 2013 Junxia Meng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper presents a new generalized model of hematopoiesis with multiple time-varying delays. The main purpose of this paper is to study the existence and the global exponential stability of the positive pseudo almost periodic solutions, which are more general and complicated than periodic and almost periodic solutions. Under suitable assumptions, and by using fixed point theorem, sufficient conditions are given to ensure that all solutions of this model converge exponentially to the positive pseudo almost periodic solution for the considered model. These results improve and extend some known relevant results.

1. Introduction

As we all know, many phenomena in nature have oscillatory character and their mathematical models have led to the introduction of certain classes of functions to describe them. For example, the pseudo almost periodic functions are the natural generalization of the concept of almost periodicity. These are functions on the real numbers set that can be represented uniquely in the form f=h+φ, where h (the principal term) is an almost periodic function and φ (the ergodic perturbation) a continuous function whose mean vanishes at infinity. Note that there exists abundant literature on the topic (see, e.g., ). In a classic study of population dynamics, the following delay differential equation model (1)x(t)=-a(t)x(t)+i=1mbi(t)1+xn(t-τi(t)), where n is a positive constant and (2)a,bi,τi:(0,+)are  continuous  functions  for  i=1,2,,m, has been used by [7, 8] to describe the dynamics of hematopoiesis (blood cell production). As we known, (1) belongs to a class of biological systems and it (or its analogue equation) has attracted more attention to the problem of almost periodic solutions because of its extensively realistic significance. For example, some criteria ensuring the existence and stability of positive almost periodic solutions were established in  and the references cited therein. However, it is very difficult to study the global stability of positive pseudo almost periodic solution for (1). So far, no attention has been paid to the conditions for the global exponential stability on positive pseudo almost periodic solution of model (1) in terms of its coefficients. On the other hand, since the exponential convergent rate can be unveiled, the global exponential stability plays a key role in characterizing the behavior of dynamical system (see ). Thus, it is worthwhile to continue to investigate the existence and global exponential stability of positive pseudo almost periodic solutions of (1).

Motivated by the above discussions, in this paper, we consider the existence, uniqueness, and global exponential stability of positive pseudo almost periodic solutions of (1). Here in this paper, a new approach will be developed to obtain a delay-independent condition for the global exponential stability of the positive pseudo almost periodic solutions of (1), and the exponential convergent rate can be unveiled.

Throughout this paper, for i=1,2,,m, it will be assumed that a,bi,τi:(0,+) are continuous functions, and (3)a-=infta(t)>0,a+=supta(t),bi-=inftbi(t)>0,bi+=suptbi(t),r=max1im{suptτi(t)}>0. Let + denote nonnegative real number space, let C=C([-r,  0],) be the continuous functions space equipped with the usual supremum norm ·, and let C+=C([-r,0],+). If x(t) is defined on [-r+t0,σ) with t0,σ, then we define xtC where xt(θ)=x(t+θ) for all θ[-r,0].

Due to the biological interpretation of model (1), only positive solutions are meaningful and therefore admissible. Thus we just consider admissible initial conditions. (4)xt0=φ,φC+,φ(0)>0. We write xt(t0,φ)(x(t;t0,φ)) for an admissible solution of the admissible initial value problem (1) and (4). Also, let [t0,η(φ)) be the maximal right interval of existence of xt(t0,φ).

2. Preliminary Results

In this section, some lemmas and definitions will be presented, which are of importance in proving our main results in Section 3.

In this paper, BC(,) denotes the set of bounded continued functions from to . Note that (BC(,),·) is a Banach space where · denotes the sup norm f:=supt|f(t)|.

Definition 1 (see [<xref ref-type="bibr" rid="B16">16</xref>, <xref ref-type="bibr" rid="B17">17</xref>]).

Let u(t)BC(,). u(t) is said to be almost periodic on if, for any ε>0, the set T(u,ε)={δ:|u(t+δ)-u(t)|<ε  for  allt} is relatively dense; that is, for any ε>0, it is possible to find a real number l=l(ε)>0, and for any interval with length l(ε), there exists a number δ=δ(ε) in this interval such that |u(t+δ)-u(t)|<ε, for all t.

We denote by AP(,) the set of the almost periodic functions from to . Besides, the concept of pseudo almost periodicity (pap) was introduced by Zhang in the early nineties. It is a natural generalization of the classical almost periodicity. Precisely, define the class of functions PAP0(,) as follows: (5){fBC(,)limT+12T-TT|f(t)|dt=0}. A function fBC(,) is called pseudo almost periodic if it can be expressed as (6)f=h+φ, where hAP(,) and φPAP0(,). The collection of such functions will be denoted by PAP(,). The functions h and φ in the above definition are, respectively, called the almost periodic component and the ergodic perturbation of the pseudo almost periodic function f. The decomposition given in definition above is unique. Observe that (PAP(,),·) is a Banach space and AP(,) is a proper subspace of PAP(,) since the function ϕ(t)=cosπt+cost+1/(1+t2) is pseudo almost periodic function but not almost periodic. It should be mentioned that pseudo almost periodic functions possess many interesting properties; we will need only a few of them and for the proofs we shall refer to .

Lemma 2.

Let x1(·),σ(·)AP(,), σ(·)BC(,) and x2(·)PAP0(,). Then

x1(t-σ(t))AP(,);

x2(t-σ(t))PAP0(,), if (1-σ(t))-=inft(1-σ(t))>0.

Proof.

( 1 ) For any ε>0, from the uniform continuity of x1(·), we can choose a constant (7)0<δ=δ(ε)<ε2 such that (8)|x1(t)-x1(t′′)|<ε2,t,t′′,|t-t′′|<δ. From the theory of almost periodic functions in [16, 17], it follows that for δ>0, it is possible to find a real number l=l(δ)=l(δ(ε))>0, and for any interval with length l, there exists a number τ=τ(ε) in this interval such that (9)|σ(t+τ)-σ(t)|<δ,|x1(t+τ)-x1(t)|<δ<ε2,|σ(t+τ)-σ(t)||(t+τ)-(t)||σ(t+τ)-σ(t)|t. Combing (8) and (9), we obtain (10)|x1(t+τ-σ(t+τ))-x1(t-σ(t))||x1(t+τ-σ(t+τ))-x1(t+τ-σ(t))|  +|x1(t+τ-σ(t))-x1(t-σ(t))|<ε2+ε2=ε,t. which yields x1(t-σ(t))AP(,).

( 2 ) Set s=t-σ(t); we get (11)012T-TT|x2(t-σ(t))|dt=12T-(T-σ(-T))T-σ(T)|x2(s)|11-σ(t)ds1(1-σ(t))-T+σ+T12(T+σ+)-(T+σ+)T+σ+|x2(s)|ds,1(1-σ(t))-T+σ+T12(T+σ+)whereσ+=suptσ(t), which implies that x2(t-σ(t))PAP0(,).

Remark 3.

Set x(·)=x1(·)+x2(·) with x1(·)AP(,) and x2(·)PAP0(,). It follows from Lemma 2 that (12)x(t-σ(t))PAP(,),if  (1-σ(t))->0,σ(·)AP(,),σ(·)BC(,).

Definition 4 (see [<xref ref-type="bibr" rid="B16">16</xref>, <xref ref-type="bibr" rid="B17">17</xref>]).

Let xRn and let Q(t) be an n×n continuous matrix defined on R. The linear system (13)x(t)=Q(t)x(t) is said to admit an exponential dichotomy on R if there exist positive constants k, and α, projection P, and the fundamental solution matrix X(t) of (13) satisfying (14)X(t)PX-1(s)ke-α(t-s)for  ts,X(t)(I-P)X-1(s)ke-α(s-t)for  ts.

Lemma 5 (see [<xref ref-type="bibr" rid="B6">6</xref>, <xref ref-type="bibr" rid="B16">16</xref>]).

Assume that Q(t) is an almost periodic matrix function and g(t)PAP(Rn). If the linear system (13) admits an exponential dichotomy, then pseudo almost periodic system (15)x(t)=Q(t)x+g(t) has a unique pseudo almost periodic solution x(t), and (16)x(t)=-tX(t)PX-1(s)g(s)ds-t+X(t)(I-P)X-1(s)g(s)ds.

Lemma 6 (see [<xref ref-type="bibr" rid="B16">16</xref>, <xref ref-type="bibr" rid="B17">17</xref>]).

Let ci(t) be an almost periodic function on R and (17)M[ci]=limT+1Ttt+Tci(s)ds>0,i=1,2,,n. Then the linear system (18)x(t)=diag(-c1(t),-c2(t),,-cn(t))x(t) admits an exponential dichotomy on R.

Lemma 7 (see [<xref ref-type="bibr" rid="B11">11</xref>, Lemma 2.3]).

Every solution x(t;t0,φ) of (1) and (4) is positive and bounded on [t0,η(φ)), and η(φ)=+.

Lemma 8.

Suppose that there exist two positive constants κ and M such that (19)M>κ,suptR{-a(t)M+i=1mbi(t)}<0,inftR{-a(t)κ+i=1mbi(t)1+Mn}>0. Then, there exists tφ>t0 such that (20)κ<x(t;t0,φ)<M,ttφ.

Proof.

This Lemma can be proven in a similar way to that in Lemma 2.2 of . But for convenience of reading, we give the proof as follows. Let x(t)=x(t;t0,φ). We first claim that there exists t#[t0,+) such that (21)x(t#)<M. Otherwise, (22)x(t)M,t[t0,+), Which, together with (19), implies that (23)x(t)=-a(t)x(t)+i=1mbi(t)1+xn(t-τi(t))-a(t)M+i=1mbi(t)1+Mn-a(t)M+i=1mbi(t)suptR{-a(t)M+i=1mbi(t)}<0,tt0+r. This yields that (24)x(t)=x(t0+r)+t0+rtx(s)dsx(t0+r)+suptR{-a(t)M+i=1mbi(t)}×(t-(t0+r)),      tt0+r. Thus (25)limt+x(t)=-, which contradicts the fact that x(t) is positive and bounded on [t0,+). Hence, (21) holds. In the sequel, we prove that (26)x(t)<M,t[t#,+). Suppose, for the sake of contradiction, there exists t~(t#,+) such that (27)x(t~)=M,x(t)<M,t[t#,t~). Calculating the derivative of x(t), together with (19), (1), and (27), implies that (28)0x(t~)=-a(t~)x(t~)+i=1mbi(t~)1+xn(t~-τi(t~))-a(t~)M+i=1mbi(t~)<0, which is a contradiction and implies that (26) holds.

We finally show that l=liminftx(t)>κ. By way of contradiction, we assume that 0lκ. By the fluctuation lemma [18, Lemma A.1.], there exists a sequence {tk}k1 such that (29)tk+,x(tk)liminft+x(t),x(tk)0ask+. Since {xtk} is bounded and equicontinuous, by the Ascoli-Arzelá theorem, there exists a subsequence, still denoted by itself for simplicity of notation, such that (30)xtkφ*(k+)for  some  φ*C+. Moreover, (31)φ*(0)=lφ*(θ)Mfor  θ[-r,0). Without loss of generality, we assume that all a(tk), bi(tk), and τi(tk) are convergent to a*, bi*, and τi*, respectively. This can be achieved because of almost periodicity. It follows from (32)x(tk)=-a(tk)x(tk)+i=1mbi(tk)1+xn(tk-τi(tk)) that (taking limits) (33)0=-a*l+i=1mbi*1+(φ*(-τi*))n-a*l+i=1mbi*1+Mn-a*κ+i=1mbi*1+MninftR{-a(t)κ+i=1mbi(t)1+Mn}>0, is a contradiction. This proves that l>κ. Hence, from (26), we can choose tφ>t0 such that (34)κ<x(t;t0,φ)<M,ttφ. This ends the proof of Lemma 8.

3. Main Results Theorem 9.

Suppose that (35)a,τiAP(,),τi(·)BC(,),biPAP(,),inft(1-τi(t))>0,i=1,2,,m, and there exist two positive constants κ and M satisfying (19) and (36)suptR{-a(t)+i=1mbi(t)n4κ}<0. Then, there exists a unique positive pseudo almost periodic solution of (1) in the region B*={φφPAP(,),κφ(t)M,forallt}.

Proof.

Consider Υ:[0;1] defined by (37)Υ(u)=suptR{-a(t)+i=1mbi(t)n4κeu},u[0,1]. Then, we have (38)Υ(0)=suptR{-a(t)+i=1mbi(t)n4κ}<0, which implies that there exists a constant ς(0,1] such that (39)Υ(ς)=suptR{-a(t)+i=1mbi(t)n4κeς}<0.

For any ϕPAP(,), from (35), Remark 3, and the composition theorem of pseudo almost periodic functions , we have (40)i=1mbi(t)1+ϕn(t-τi(t))PAP(,). We next consider an auxiliary equation: (41)x(t)=-a(t)x(t)+i=1mbi(t)1+ϕn(t-τi(t)). Notice that M[a]>0; it follows from Lemma 6 that the linear equation (42)x(t)=-a(t)x(t) admits an exponential dichotomy on . Thus, by Lemma 5, we obtain that the system (41) has exactly one pseudo almost periodic solution: (43)xϕ(t)=-te-sta(u)du[i=1mbi(s)1+ϕn(t-τi(s))]ds. Define a mapping T:PAP(,)PAP(,) by setting (44)T(ϕ(t))=xϕ(t),ϕPAP(,).

Since B*={φφPAP(,),κφ(t)M,forallt}, it is easy to see that B* is a closed subset of PAP(R,R). For any ϕB*, from (19), we have (45)xϕ(t)-te-sta(u)du[i=1mbi(s)]ds-te-sta(u)dua(s)Mds=M,tR,(46)  xϕ(t)-te-sta(u)du[i=1mbi(s)1+Mn]ds-te-sta(u)dua(s)κds=κ,tR. This implies that the mapping T is a self-mapping from B* to B*. Now, we prove that the mapping T is a contraction mapping on B*. In fact, for φ,ψB*, we get (47)T(φ)-T(ψ)=suptR|T(φ)(t)-T(ψ)(t)|=suptR|-te-sta(u)dui=1mbi(s)[11+φn(t-τi(s))-te-sta(u)dui=1m-te-sta(u)du-11+ψn(s-τi(s))]ds-te-sta(u)dui=1mbi(s)[11+φn(t-τi(s))-te-sta(u)du|. In view of (39), (45), (46), and (47), from the inequality (48)|11+xn-11+yn|=|-nθn-1(1+θn)2||x-y|nθn-1(2θn)2|x-y|n4κ|x-y|, where x,y[κ,M] and θ lies between x and y, we have (49)T(φ)-T(ψ)suptR-te-sta(u)dui=1mbi(s)n4κ×|φ(s-τi(s))-ψ(s-τi(s))|dssuptR-te-sta(u)dua(s)e-ς×|φ(s-τi(s))-ψ(s-τi(s))|dse-ςφ-ψ. Noting that e-ς<1, it is clear that the mapping T is a contraction on B*. Using Theorem 0.3.1 of , we obtain that the mapping T possesses a unique fixed point φ*B*, Tφ*=φ*. By (41), φ* satisfies (1). So φ* is a positive pseudo almost periodic solution of (1) in B*. The proof of Theorem 9 is now complete.

Theorem 10.

Under the assumptions of Theorem 9, (1) has at least one positive pseudo almost periodic solution x*(t). Moreover, x*(t) is globally exponentially stable; that is, there exist constants Kφ,x*, tφ,x*, and λ>0 such that (50)|x(t;t0,φ)-x*(t)|<Kφ,x*e-λt,t>tφ,x*.

Proof.

By Theorem 9, (1) has a positive pseudo almost periodic solution; say x*(t). It suffices to show that x*(t) is globally exponentially stable. Define a continuous function Γ(u) by setting (51)Γ(u)=suptR{-[a(t)-u]+i=1mbi(t)n4κeru},u[0,1]. Then, we have (52)Γ(0)=suptR{-a(t)+i=1mbi(t)n4κ}<0, which implies that there exist two constants η>0 and λ(0,1] such that (53)Γ(λ)=suptR{-[a(t)-λ]+i=1mbi(t)n4κeλr}<-η<0.

Let x(t)=x(t;t0,φ) and y(t)=x(t)-x*(t), where t[t0-r,+). Then (54)y(t)=-a(t)y(t)+i=1mbi(t)[11+xn(t-τi(t))=-a(t)y(t)+i=1mbi(t)-a(t)y-11+x*n(t-τi(t))].

It follows from Lemma 8 that there exists tφ,x*>t0 such that (55)κx(t),x*(t)M,t[tφ,x*-r,+).

We consider the Lyapunov functional (56)V(t)=|y(t)|eλt. Calculating the upper left derivative of V(t) along the solution y(t) of (54), we have (57)D-(V(t))-a(t)|y(t)|eλt+i=1mbi(t)×|11+xn(t-τi(t))-11+x*n(t-τi(t))|eλt+λ|y(t)|eλt=[-(a(t)-λ)|y(t)|+i=1mbi(t)×|11+xn(t-τi(t))-11+x*n(t-τi(t))|i=1m]eλt,|11+xn(t-τi(t))-12111+x*    n(tτi(t))|t>tφ,x*.

We claim that (58)V(t)=|y(t)|eλt<eλtφ,x*(maxt[t0-r,  tφ,x*]|x(t)-x*(t)|+1):=Kφ,x*,t>tφ,x*. Contrarily, there must exists t*>tφ,x* such that (59)V(t*)=Kφ,x*,V(t)<Kφ,x*,t[t0-r,t*). Together with (48), (57), and (59), we obtain (60)0D-(V(t*))[-(a(t*)-λ)|y(t*)|+i=1mbi(t*)×|11+xn(t*-τi(t*))-(a(t*))-11+x*n(t*-τi(t*))|i=1m]eλt*-(a(t*)-λ)|y(t*)|eλt*+i=1mbi(t*)×n4κeλτi(t*)eλ(t*-τi(t*))|y(t*-τi(t*))|{-(a(t*)-λ)+i=1mbi(t*)n4κeλr}Kφ,x*. Thus, (61)0-(a(t*)-λ)+i=1mbi(t*)n4κeλr, which contradicts (53). Hence, (58) holds. It follows that (62)|y(t)|<Kφ,x*e-λtt>tφ,x*. This completes the proof of Theorem 10.

4. An Example

In this section, we present an example to check the validity of the results we obtained in the previous sections.

Example 1. Consider the following model of hematopoiesis with multiple time-varying delays:(63)x(t)=-1.3x(t)+12(2+12|cos2t|+110011+t2)×11+x(t-2e(1/10)cost)+12×(2+12|sint|+110011+t2)11+x(t-2e(1/10)sint). Obviously (64)a+=a-=1.3,b1-=b2-=1,b1+=b2+=1.26,n=1,r=2e1/10. Let κ=0.5 and M=2. Then (65)-a-M+b1++b2+=-0.08<0,-a+κ+b1-+b2-1+M=160>0,-a-+(b1++b2+)n4κ=-1.3+2.52×12=-0.04<0, which imply that (63) satisfies the assumptions of Theorem 10. Therefore, (63) has a unique positive pseudo almost periodic solution x*(t), which is globally exponentially stable with the exponential convergent rate λ0.01. The numerical simulation in Figure 1 strongly supports the conclusion.

Numerical solution x(t) of (63) for initial value φ(t)0.2,1,1.9.

Remark 11.

We remark that the results in  give no opinions about global exponential convergence for the positive pseudo almost periodic solution. Thus, the results in  and the references therein cannot be applied to prove the global exponential stability of positive pseudo almost periodic solution for (63). This implies that the results of this paper are new and they complement previously known results.

Conflict of Interests

The author declares no conflict of interests. She also declares that she has no financial or personal relationships with other people or organizations that can inappropriately influence her work. There are no professional or other personal interests of any nature or kind in any product, service, and/or company that could be construed as influencing the position presented in, or the review of, this paper.

Acknowledgments

The author would like to express the sincere appreciation to the editor and reviewer for their helpful comments in improving the presentation and quality of the paper. In particular, the author expresses the sincere gratitude to Professor Zhibin Chen and Professor Bingwen Liu for the helpful discussion when this revision work was being carried out. This work was supported by the National Natural Science Foundation of China (Grant no. 11201184), the Natural Scientific Research Fund of Zhejiang Provincial of P.R. China (Grant nos. Y6110436 and LY12A01018), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of P.R. China (Grant no. Z201122436).

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