Almost Periodic Solutions of a Discrete Model via Quasi-uniform Asymptotic Stability

Fang Lini N’gbo N’gbo Xia Yonghui

(Department of Mathematics,Zhejiang Normal University,Jinhua 321004,China)

Abstract This paper concerns the existence and uniqueness of almost periodic solutions to a discrete system.Because in general,the conditions to guarantee the quasi-uniform asymptotic stability is relatively weaker,we employ the quasi-uniform asymptotic stability to derive the sufficient conditions for the existence of positive almost periodic solutions to the concerned system. The effectiveness of our results are verified by a numerical example.

Key words Discrete system Almost periodic solution Asymptotical stability

1 Introduction

The literature on periodic and almost periodic solutions of differential equations is rather well furnished. The discovery of relationships between stability theory and almost periodic solutions of differential equations,has awaken great interest among researchers[1-3,7]. Subsequently,applications to real world problems are continuously investigated[8-11].

Recently, in mathematical modeling, discrete time models are preferred over the continuous time models, especially when the processes evolve in stages. As it is directly applicable to various fields(economics, populations dynamics, species interactions, mathematical biology, ecology etc), the theory of difference equation is much more substantiated than the corresponding theory of differential equation[15].

In real world applications, the assumption of periodicity for the parameters to mimic the periodic enviroment is more reasonable. However, in cases of models where periods are hardly identified, it is more suitable to consider almost periodicity. Recent important advances on the theory of almost periodic solutions of difference equations are to be mentioned,one can refer to[16-21,27-30,33-37]. For instance,Bezandry [16, 17] studied the existence of almost periodic solutions to non-autonomous higher order difference equations. Cao,et al. [24]and Edgardo[25]discussed the theory of almost periodic stability for fractional difference equations. Hamaya[29]and Romanovskii[34]investigated the bifurcation and exponential dichotomy of almost periodic solutions of difference equations.

In addition,many scholars have obtained a series of interesting conclusions about the application of almost periodic solutions of difference equations. Slyusarchuk[4-6]considered the existence conditions of almost periodic solutions of difference equations in a metric spaces. Yuan and Hong[12,13]studied the almost periodic solutions for the differential equations with piecewise constant argument. Xia [14]first discussed the relationships between the existence and globally quasi-uniformly asymptotic stability of almost periodic solutions of difference systems. Jehad[22],Yao[26]and Hamaya[31,32]obtained the existence of almost periodic solutions to Nicholson’s blowflies and Volterra discrete models.

Inspired by the aforementioned work,we extend the differential equation of boy-after-girl model to discrete time model of the form as follows:

The main purpose of this paper is to consider the existence of almost periodic solutions of the above model based on the the global quasi-uniform asymptotic stability.

The rest of our paper is articulated as follows: in the next section,we give definitions and properties of almost periodic solutions of a discrete system. In section 3,we establish the existence of almost periodic solutions for the difference system. In section 4, an example is given to illustrate the feasibility and effectiveness of our results.

2 Preliminaries

Zhou and Zheng[23]established the following differential boy-after-girl model:

wherex(t)is a function describing the estrangedness between the lovers at timet,f1is the natural growth rate of the estrangedness,the constantKis the capacity of the estrangedness,y(t)is a function denoting the academic record at timet,the constantEis the mean of the academic record,a1represents the growth rate of the boy’s academic record,k1andk2are proportional coefficients describing the relationship between the estrangedness and academic record,V1is the expected value,mrepresents the coefficient of competition,nis the studying parameter,lw> 0 is the probability that the girl will be pursued by another boys. For the sake of analysis,takinga=f1,b=k1m,c=k1n,d=k1(V1+lw-m-n),e=a1-k2(V1+lw-m-n),f=k2n,g=k2m,then system(2.1)can be transformed to

In order to discuss the existence of almost periodic solution of this model, assume thata(t),b(t),c(t),d(t),e(t),f(t),g(t)are positive almost periodic functions.

We denote by Rkthe real Euclideank-space. The lettersn,m,l,i,j,kdenote integers in the sequence.Z is the set of integers, Z+is the set of positive integers and Z-is the set of negative integers. We set Z[a,b]=a,a+1,··· ,b-1,b,wherea,b ∈Z.

For a bounded sequencefdefined on Z,we definef+andf-as follows:

Definition 2.1([19]) A sequencef: Z→Rkis said to be almost periodic if theε-translation set off

is relatively dense in Z for allε>0,that is,for any givenε>0,there exists an integerl(ε)>0 such that each interval of lengthl(ε)contains an integerτinE(ε,x).τis called theε-translation number off.

The following lemmas are essential for proving the main results. They were provided in [19] for arbitrary time scale T. When T=Z,we have:

Lemma 2.1([19]) Iff:Z→R is an almost periodic sequence,thenf(n)is bounded.

Lemma 2.2([19]) Ifg,f:Z→R is an almost periodic sequence,thenf ·gandf+gare almost periodic sequences.

Definition 2.2 ([14]) Assume thatφ(n) andx(n,n0,xn0) are solutions of (2.2), wherex(n,n0,xn0) is a solution of (2.2) with initial valuex(n0) =xn0. If for anyε> 0,s> 0, there exist positive integersT(s,ε)>0 andM(s)>0 such that whenn0∈Z[n1,+∞),|xn0-φ(n)|

then,we callφ(n)is globally quasi-uniformly asymptotically stable on Z[n1,+∞).

Definition 2.3([14]) If equation(2.2)satisfies the conditions as follows:

(i)f(n,x)is almost periodic innuniformly forx ∈K,whereKis a compact set inD；

(ii)for everyg ∈H(f),the hull equationx(n+1)=g(n,x)has a unique solutionφto the initial value problems inK,

then we say that equation(2.2)satisfies the standard assumptions.

Lemma 2.4 ([14]) Under the standard assumptions of Definition 2.3, if (2.2) has a globally quasi-uniformly asymptotically stable bounded solution on Z[n1,+∞), then (2.2) has a unique almost periodic solution on Z which is globally quasi-uniformly asymptotically stable on Z.

In particular, we consider system(2.2)under the assumption thatf(n,x)isω-periodic, i.e.f(n+ω,x)=f(n,x).Then we have the following lemma.

Lemma 2.5 ([14]) Under the standard assumptions of Definition 2.3, ifφ(n) is a globally quasi-uniformly asymptotically stable bounded solution of system(2.2)on Z[n1,+∞),thenφ(n)is the unique periodic solution with periodωof(2.2)on Z.

3 Existence of almost periodic solutions

For the continuous relevant properties of almost periodic function,we refer to Fink[1]. Suppose that all parameters are almost periodic functions. Refer to [14], we can obtain the discrete system by using differential equations with piecewise constant arguments. Assume that the average growth rates in(2.1)change at regular intervals of time, and we can incorporate this aspect into(2.1)to obtain the following modified system,which can be regarded as the semi-discretization of(2.1):

where[t]is the integer part oft,t ∈(-∞,+∞).

The equation of type (3.1) is called a differential equation with piecewise constant parameters.On any interval of the form [n,n+ 1), we integrate (3.1) and obtain forn ≤ t ≤ n+ 1,n=··· ,-2,-1,0,1,2,··· ,

exists ann ∈Z(n0,+∞)such thatxi(n)≥Di(i=1,2),then

Theorem 3.1([14]) If all the assumptions in Proposition 3.3 are satisfied,then system(1.1)has a unique almost periodic solution which is globally quasi-uniformly asymptotically stable on Z.

Theorem 3.2([14]) Leta(n),b(n),c(n),d(n),e(n),f(n),g(n)be positiveω-periodic sequences(ω ∈Z). If all the assumptions in Proposition 3.3 are satisfied,then system(1.1)has a uniqueω-periodic solution which is globally quasi-uniformly asymptotically stable on Z.

4 An example

Consider the following system

It is easy to verify that system (4.1) satisfies all the assumptions. Thus, system (4.1) admits the almost periodic solution that is quasi-uniformly asymptotically stable. The numerical simulation of (4.1) as in Figure 1 shows that the existence of almost periodic solutions,and the effectiveness of our results. While the numerical simulation of(4.1)as in Figure 2 shows that an almost periodic solution is quasi-uniformly asymptotically stable(onlyy(n)is plotted for better observation). This means that if all the assumptions in Proposition 3.3 are satisfied, the estrangedness between the lovers and the academic achievement are almost periodic which is quasi-uniformly asymptotically stable.

Figure 1 Dynamics of system(4.1)with(x(0),y(0))=(0.5,1)and n ∈Z[0,40)

Figure 2 Dynamics of system(4.1)with initial values(x(0),y(0))=(0.5,1),(1.5,2),(2.5,3),(0.1,0.1)and n ∈Z[0,30)