Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012, Article ID 135075, 15 pages
http://dx.doi.org/10.1155/2012/135075
Research Article

On the Positive Almost Periodic Solutions of a Class of Nonlinear Lotka-Volterra Type System with Feedback Control

Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, China

Received 12 November 2011; Accepted 28 December 2011

Academic Editor: Junjie Wei

Copyright © 2012 Ni Hua et al. 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.

Abstract

With the help of the variable substitution and applying the fixed point theorem, we derive the sufficient conditions which guarantee the existence of the positive almost periodic solutions for a class of Lotka-Volterra type system. The main results improve and generalize the former corresponding results.

1. Introduction

Denote   is a positive constant or , the norm of a bounded continuous function space is defined as , where .

We call an almost periodic function is positive if and only if each component has its positive infimum. Denote is a continuous almost periodic function on = is a continuous almost periodic function on is a sequence of real numbers,for more almost periodic monographs, see references [1, 2].

In [3], Teng first studied the existence of the almost periodic solutions for the following scalar equation: where , and , then based on (1.1) and combined Schauder fixed point theorem, he studied the existence of the almost periodic solution for a class of Lotka-Volterra system where are continuous almost periodic functions, is a continuous almost periodic function in uniformly with respect to , and . The author of [4] inherited in the method and ideas in [3], and promoted its conclusions to the following system with feedback control where is the control variable, .

The authors of [5] used transformation techniques and fixed point theorem and studied a time-delay system with feedback control which is much wider then the system (1.3), In the case of non-Lipschitz condition, they gave a sufficient condition of the existence of the almost periodic solutions for the system (1.4).

However, in the real world, the competition between species is not always shown by the linear relationship, while shown by a certain degree of nonlinearity, therefore, studying the following system becomes more realistic and necessary, over this paper, we study the system which is more extensive than the system (1.4) as follows: where is the control variable, is a positive constant, , by using Schauder’s fixed point theorem, we get the sufficient conditions of the existence of the almost periodic solution for the system (1.5), and by using the contraction mapping principle, we give the conclusion of the existence of a unique almost periodic solution for the system (1.5) in the one-dimensional case, some new results are obtained.

2. Some Related Lemmas

Lemma 2.1 (see [6]). For the equation where and are continuous, if  , then (2.1) exists a unique almost periodic solution , and can be written as follows where , is the real part of .

Lemma 2.2 (see [2]). Suppose ,, then the following conditions are equivalent.(1); (2) for any such that ;(3) exists which implies exists (any sense);(4), implies (any sense);(5), implies there is so that (any sense). Consider the equation where and are continuous on .

Lemma 2.3. If (2.3) satisfies one of the following conditions
(1)(2),

then (2.3) exists a unique positive almost periodic solution , and .

Proof. Let , then (2.3) can be changed as follows: by the condition , we have , also by the condition , according to Lemma 2.1, it follows that (2.4) exists a unique positive almost periodic solution , and can be written as follows:
Next, we prove that . If there is a sequence such that are convergent uniformly on , then there exists a sequence such that is convergent uniformly on , and is also an almost periodic solution of (2.4), by the uniqueness of the almost periodic solution of (2.4), we can get , by Lemma 2.2, it follows . Since , (2.3) exists a unique positive almost periodic solution , and it can be written as follows by (2.6), we can easily get .
If the condition (2) holds, similarly, we can prove that (2.3) exists a unique positive almost periodic solution , it can be written as follows and . This is the end of the proof of Lemma 2.3.

Lemma 2.4 (see [2]). Suppose that an almost periodic sequence is convergent uniformly on any compact set of ,   is an almost periodic function, and , then is convergent uniformly on .

3. The Conclusion of the N-dimentional System

Consider the following equation where .

By Lemma 2.3, it follows if one of the following conditions holds()()

then (3.1) exists a unique positive almost periodic solution , we can easily get if holds, then if () holds, then Now, consider (1.5), suppose() are continuous almost periodic functions, is continuous for all variables, and is almost periodic in uniformly with respect to ,   is continuous for all variables, and is almost periodic in uniformly with respect to ;(),  for all ;(),  for all ;(); ().

Following the front of the defined, suppose that or hold, and denoted the unique positive almost periodic solution of the following equation by where For any , we  define

then is a Banach space with the model of construct a bounded closed convex set as follows:() For any and , ,() For any and ,,.

Theorem 3.1. If , , , or , , , hold, then (1.5) exists at least a positive almost periodic solution in .

Proof. If (), (), () hold, for any , consider the following equation by the condition (), it follows , by Lemma 2.3,it follows that (3.8) exists a unique positive almost periodic solution, and it can be written as follows and , since , thus by (3.2) and (3.9), we can get , on the other hand, since , it follows .
Consider the following equation by Lemma 2.1, (3.10) exists a unique positive almost periodic solution and it can be written as follows: also by the condition , the first formula of (3.5) and (3.12), we have , and , also , thus , therefore, we can get , hence we can define
Now we are committed to prove the continuity of in . Suppose , and when , define , then is a compact set of , since , there exist the positive numbers , such that when similarly, since , there exist the positive numbers and , such that when ,  for any
Therefore, by the mean value theorem Where, Also by the mean value theorem, we can obtain the following: where, is between the positive almost periodic function and , thus By (3.9), (3.12), and (3.14), it follows that thus we have Hence In addition, taking into account that are continuous uniformly on , when , it follows , thus when , we have therefore, is continuous.

Then to prove is relatively compact in . By the boundedness of and (3.8), (3.10), we can obtain that there exists a positive number such that , thus () is uniformly bounded and equicontinuous on , by the theorem of Ascoli-arzela, for any sequence in , there exist a subsequences (also denoted by ) such that is convergent uniformly in any compact set of , also combined with Lemma 2.4,is convergent uniformly on , that is to say is relatively compact in . According to Schauder's fixed point theorem, exists as a fixed point in , that is to say (1.5) exists at least a positive almost periodic solution in . Similarly, when (), (), (), () hold, we can prove that (1.5) exists at least a positive almost periodic solution in .

Remark 3.2. When , then (1.5) turns into (1.4), obviously, Theorem 3.1 is the generalization of Theorems 2.1 and 2.2 in [5];

Remark 3.3. In Theorem 2.2 of [5], it requires the condition, in this paper, we do not require , this can be seen in the weak conditions we get the similar results, thus Theorem 3.1 extends the results of Theorem 2.2  of the paper [5].

4. The Conclusions Of The One-dimentional System

Here we will discuss the system (1.5) in the one-dimensional case where are continuous almost periodic functions, is almost periodic in t uniformly with respect to (, and is almost periodic in t uniformly with respect to , and . For convenience, we introduce the following notations. For a continuous bounded function , denote ,(); ()()

for , are positive constants; (),  where,.

In the next paper, we suppose that initial values of system (4.1) satisfy , for the solution of the system (4.1), we only consider its positive initial value, that is to say It is not difficult for us to see the solution of (4.1) with the positive initial value is always positive.

Theorem 4.1. If hold, then (4.1) exists a unique positive almost periodic solution.

Proof. Let , then (4.1) can be changed into the following equation:
For, define then is a Banach space with the model of , for all, consider the following equation: by Lemma 2.1, it follows that (4.5) exists a unique positive almost periodic solution , it can be written as follows: similar to the proof of Lemma 2.3 we can obtain in addition, Note that the condition holds, thus we have , hence Note that , if , then , thus we can get if , then , therefore No matter what or , we always have Also we have Consider the following equation: By Lemma 2.1 we can see that (4.14) has a unique positive almost periodic solution , it can be written as follows Similar to the proof of Lemma 2.3, we can get Note that the condition holds, it follows that, hence Now, define a mapping From the above discussion we can see , thus , for given any , we have By the mean value theorem (where of the following formula is between and ), the above formula can be changed as follows: notice that the condition holds, it follows by (4.21), is a contraction mapping, thus exists a unique fixed point in , this fixed point is the only positive almost periodic solution of (4.3), and . note that ,  , thus (4.1) exists a positive almost periodic solution , and . This completes the proof of Theorem 4.1.

Acknowledgments

This paper is supported by the National Nature Science Foundation of China (70871056) and Jiangsu Province Innovation Project of Graduate Education (1221190037).

References

  1. A. M. Fink, Almost Periodic Differential Equations, vol. 377 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1974.
  2. C. Y. He, Almost Periodic Differential Equations, Higher Education Press, Beijing, China, 1992.
  3. Z. D. Teng, “On the positive almost periodic solutions of a class of Lotka-Volterra type systems with delays,” Journal of Mathematical Analysis and Applications, vol. 249, no. 2, pp. 433–444, 2000. View at Publisher · View at Google Scholar
  4. C. Wang and J. Shi, “Positive almost periodic solutions of a class of Lotka-Volterra type competitive system with delays and feedback controls,” Applied Mathematics and Computation, vol. 193, no. 1, pp. 240–252, 2007. View at Publisher · View at Google Scholar
  5. Y. Xie and X. G. Li, “On the positive almost periodic solutions of a class of Lotka-Volterra type system with feedback control,” Annals of Mathematics, vol. 30, no. 2, pp. 161–168, 2009. View at Google Scholar
  6. W. A. Coppel, Dichotomies in Stability Theory, vol. 629 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1978.