Existence and exponential stability of the unique positive almost periodic solution for the Lasota-Wazewska difference model
© Yao; licensee Springer. 2014
Received: 6 January 2014
Accepted: 2 July 2014
Published: 4 August 2014
In this paper, a discrete Lasota-Wazewska model is studied. By using the fixed point theorem of decreasing operator, we obtain sufficient conditions for the existence of a unique positive almost periodic solution. Particularly, we give iterative sequence which converges to the positive almost periodic solution. Moreover, we investigate exponential stability of the positive almost periodic solution by the Lyapunov functional.
which described the survival of red blood cells in animals. Kulenovic and Ladas  investigated the oscillation and global attractivity of the above model (1.1). Moreover, the model (1.1) and some generalized models have been investigated by many authors; see Graef et al. , Kulenovic et al. , Xu and Li , Jiang , Li and Wang .
The assumption that the environment is constant is rarely the case in real life. When the environmental fluctuation is taken into account, a model must be nonautonomous. Due to the various seasonal effects of the environmental factors in a real life situation (e.g., seasonal effects of weather, food supplies, mating habits, harvesting, etc.), it is rational and practical to study the biological system with periodic coefficients or almost periodic coefficients. Many authors [6, 7] have studied nonautonomous differential equations with periodic coefficients of the above model (1.1).
Though most models are described with differential equations, the discrete-time models governed by difference equations are more appropriate than the continuous ones when the size of the population is rarely small or the population has non-overlapping generations. It is also known that the discrete models can provide more efficient computational methods for numerical simulations [8–10]. However, the studies in the past [2–7] were concerned with the continuous case of the above model (1.1).
To our knowledge, studies on the uniqueness and exponential stability of positive almost periodic solutions for discrete models are scarce.
where , , , , , , are bounded almost periodic functions, . Due to biological reasons, we restrict our attention to positive solutions of equation (1.2). The initial condition of equation (1.2) is for .
In the study of biological systems, an important ecological problem is concerned with the existence of positive periodic solutions or positive almost periodic solutions. Recently, many authors investigated the existence of positive periodic solutions by using the Krasnoselskii cone fixed point theorem and Mawhin’s coincidence degree theory [6, 7, 11–13]. Most of the past studies are concerned with the existence of at least one positive periodic solution [6, 11, 12, 14].
Almost periodicity is more practical and more close to the reality in biological systems [15, 16], the recent contributions such as the almost periodic solutions of delay and impulsive differential equations [17–20] have appeared.
However, few papers study the existence and exponential stability of unique positive almost periodic solutions for discrete models. For the existence and uniqueness of a positive almost periodic solution, the method used in most of the past studies is the contraction mapping fixed point theorem.
In this paper, different from the past studies, we aim to obtain sufficient conditions that guarantee the existence of a unique positive almost periodic solution of discrete model (1.2) by using the fixed point theorem of decreasing operator. Particularly, we give an iterative sequence which converges to the positive almost periodic solution. We also obtain sufficient conditions for the exponential stability of the unique positive almost periodic solution by means of the Lyapunov functional. The results of this paper are new and more valuable in applications, which complement the previously obtained results in [2–7].
For any bounded sequence , we define , .
For equation (1.2), we assume that the bounded almost periodic sequences , , satisfy , , .
Definition 1 
A sequence is called an almost periodic sequence if the ε-translation set is a relatively dense set in Z for all ; that is, for any , there exists a constant such that each interval of length contains a number such that for all .
δ is called the ε-translation number of .
, implies ;
, implies .
Every cone induces an ordering in X, we define ‘≤’ with respect to P by if and only if .
Definition 3 A cone P of X is called a normal cone if there exists a positive constant σ such that for any , .
Definition 4 Let P be a cone of X and be an operator. A is called decreasing if implies .
The following fixed point theorem of decreasing operator (see ) is an important tool in our proofs.
Lemma 1 
P is a normal cone of the Banach space X, the operator is decreasing;
, , where ;
- (iii)For , there exists such that
Then A has a unique positive fixed point . Moreover, (), where () for any initial .
In this paper, we will use the above Lemma 1 to investigate the existence of a unique positive almost periodic solution of model (1.2).
Remark In Lemma 1, the operator A does not need continuity and compactness.
Lemma 2 Every solution of equation (1.2) is positive.
Since for , we can deduce that , , …, and for all . The proof is complete. □
Let . For , we define , then X is a Banach space.
Obviously, is the almost periodic solution of equation (1.2) if and only if x is the fixed point of the operator A.
Lemma 3 .
Thus , so we have . The proof is complete. □
3 Existence and uniqueness of a positive almost periodic solution
Let , .
Theorem 1 Assume that , then equation (1.2) has a unique almost periodic positive solution . Moreover, (), where () for any initial .
Proof It is clear that Ω is a normal cone, is a decreasing operator.
which implies .
which implies , here .
Finally, we show that condition (iii) of Lemma 1 is satisfied.
By Lemma 1, we know that the operator A has a unique positive fixed point , (), () for any initial . The proof of Theorem 1 is complete. □
Remark 1 Theorem 1 of this paper not only gives sufficient conditions for the existence of a unique positive almost periodic solution, but also gives the iterative sequence which converges to the positive almost periodic solution .
So, we get .
4 Exponential stability of a positive almost periodic solution
In this section, we study the exponential stability of a positive almost periodic solution.
Theorem 2 Assume that and , then equation (1.2) has a unique exponentially stable almost periodic positive solution.
Proof Since holds, by Theorem 1 we know equation (1.2) has a unique almost periodic positive solution , and . Now we prove that is exponentially stable.
Suppose that is an arbitrary solution of equation (1.2) with the initial function for . Assume that the initial function of the almost periodic positive solution is for .
Consider the function , .
Since , then there exists a constant such that .
Suppose that claim (4.4) is not true, then there must exist such that and for , .
in which ξ lies between and .
which contradicts .
So claim (4.4) is true. Hence for all , .
That is, for all , , which means that is exponentially stable.
The proof of Theorem 2 is complete. □
The author is entirely responsible for this research. The author read and approved the final manuscript.
The author thanks the referees for their valuable comments and suggestions in improving the presentation of the manuscript. This work is supported by Natural Science Foundation of Education Department of Anhui Province (KJ2014A043).
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