About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2012 (2012), Article ID 423416, 16 pages
http://dx.doi.org/10.1155/2012/423416
Research Article

Existence of Two Positive Periodic Solutions for a Neutral Multi-Delay Logarithmic Population Model with a Periodic Harvesting Rate

Department of Mathematics, Kunming University of Science and Technology, Yunnan 650500, China

Received 14 July 2012; Accepted 12 November 2012

Academic Editor: Wing-Sum Cheung

Copyright © 2012 Hui Fang. 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

By using an abstract existence result based on a coincidence degree theory for -set contractive mapping, a new result is obtained for the existence of at least two positive periodic solutions for a neutral multidelay logarithmic population model with a periodic harvesting rate. An example is given to illustrate the effectiveness of the result.

1. Introduction

In recent years, many papers have been published on the existence of positive periodic solutions for neutral delay logarithmic population models by using a topological degree theory for -set contractive mapping (see, e.g., [15]). Recently, Xia [6] obtained some new sufficient conditions for the existence and uniqueness of an almost periodic solution of a multispecies logarithmic population model with feedback controls. However, few papers deal with the existence of multiple positive periodic solutions for neutral multidelay logarithmic population models with harvesting. The main difficulty is hard to obtain a priori bounds on solutions for neutral multi-delay models with harvesting.

In this paper, we consider the following neutral multi-delay logarithmic population model of single-species population growth with a periodic harvesting rate where are nonnegative continuous -periodic functions, and denotes the harvesting rate. When , (1.1) was considered by [25]. When , and are constants, (1.1) was considered by [7].

The purpose of this paper is to establish the existence of at least two positive periodic solutions for a neutral multi-delay logarithmic population model (1.1) by using a coincidence degree theory for -set contractions. Motivated by the work of Chen [8], some novel techniques are employed to find a priori bounds on solutions.

2. Preliminaries

We now briefly state the part of the coincidence degree theory for -set contractive mapping developed by Hetzer [9, 10]. For more details, we refer to [11].

Let be a Banach space. For a bounded subset , let denote the (Kuratowski) measure of noncompactness defined by Here, diam denotes the maximum distance between the points in the set .

Let and be Banach spaces with norms and , respectively and a bounded open subset of . A continuous and bounded mapping is called -set contractive if for any bounded , we have Also, for a continuous and bounded map , we define

Let be a linear mapping and be a continuous mapping. The mapping will be called a Fredholm mapping of index zero if and is closed in . If is a Fredholm mapping of index zero, there then exist continuous projectors and such that . If we define as the restriction of to , then is invertible. We denote the inverse of that map by . If is an open bounded subset of , the mapping will be called --set contractive on if is bounded and is -set contractive. Since is isomorphic to , there exists an isomorphism .

Lemma 2.1 ([11, Proposition XI.2.]). Let be a closed Fredholm mapping of index zero and let be -set contractive with Then is a --set contraction with constant .

The following lemma [[11, page 213] will play a key role in this paper.

Lemma 2.2. Let be a Fredholm mapping of index zero and let be --set contractive on . Suppose(i) for every and every ;(ii) for every ;(iii)Brouwer degree . Then has at least one solution in .

3. Main Result

Let denote the linear space of real valued continuous -periodic functions on . The linear space is a Banach space with the usual norm for given by . Let denote the linear space of -periodic functions with the first-order continuous derivative. is a Banach space with norm .

Let and and let be given by . Since , we see that is a bounded (with bound = 1) linear map.

Under the transformation of , (1.1) can be rewritten as Next define a nonlinear map by Now, if for some , then the problem (3.1) has a -periodic solution .

In the following, we denote where is a continuous nonnegative -periodic solution.

From now on, we always assume that),  for all  .(), and where is the inverse function of is the inverse function of .  Let   Let where We first give some technological lemmas.

Set

Lemma 3.1. Assume that are positive constants such that Then there exist such that If one assumes further that then the following inequalities also hold:

Proof. Clearly, if and only if . Therefore, noticing that we have Set Since is monotonically decreasing on .
Therefore, we have that is, which implies Again, noticing that by the monotonicity of the function on the interval and , it is easy to see that the assertion holds.

Set

Lemma 3.2. Assume that ,  and   hold. Then the following assertions hold.
(1) There exist such that
(2) There exist such that
(3) There exist such that
(4)

Proof. Noticing that we have It follows from , (3.25) that Therefore, by Lemma 3.1, the assertions (1)–(3) hold. Furthermore, by (3.26) and the assertions (1)–(3), the assertion (4) also holds.

Lemma 3.3 (see [12]). is a Fredholm map of index and satisfies

Lemma 3.4 (see [2]). Suppose and ,  for all  . Then the function has an inverse function satisfying with .

Lemma 3.5. Assume that ()–() hold. Let , and where . Then is a -set-contractive map.

Proof. The proof is similar to that of Lemma  3.3 in [12], so we omit it.

Theorem 3.6. Assume that ()–() hold. Then (1.1) has at least two positive -periodic solutions.

Proof. Let for , that is, Therefore, we have By (3.31), we have Integrating this identity leads to By Lemma 3.4, we have where is the inverse function of , and is the inverse function of .
Then From (3.33)–(3.35), we have which implies for some .
Therefore, by , we have By and Lemma 3.1, we have Set then we have which implies It follows from (3.30) that By (3.36) and , we have By this and (3.43), we obtain Meanwhile, Substituting (3.42) and (3.46) into (3.45) gives Since we have Then, Again from (3.30), we get Since we have Choose , such that Then, it is clear that From this and (3.30), we obtain that It follows from (3.55) that which implies By the assertion (1) of Lemma 3.2, we have It follows from (3.56) that which implies By the assertion (3) of Lemma 3.2, we have Hence, it follows from (3.50), (3.59), and (3.62) that Clearly, are independent of . Now, let us consider with . Note that It follows from the assertion (2) of Lemma 3.2 that has two distinct solutions: By the assertion (4) of Lemma 3.2, one can take such that Let Then are bounded open subsets of . Clearly, , where as defined in Lemma 3.5. It follows from Lemma 3.5 that is a -set-contractive map . Therefore, it follows from Lemmas 2.1 and 3.3 that is --set contractive on with .
By the assertion (4) of Lemma 3.2, (3.52), (3.63), (3.65), and (3.66), it is easy to verify that satisfies the assumptions (i) and (ii) in Lemma 2.2  . By the assertion (2) of Lemma 3.2, a direct computation gives Here, is taken as the identity mapping since . So far we have proved that satisfies all the assumptions in Lemma 2.2  . Hence, (3.1) has at least two -periodic solutions and . Since , are different. Let . Then are two different positive -periodic solutions of (1.1). The proof is complete.

Example 3.7. Consider the following equation: where and the constant . Clearly, is satisfied.
Let be the inverse function of , and be the inverse function of . Then we have Hence, is satisfied.
It is easy to see that Therefore, we have So, Hence, is satisfied.
Also, it is easy to see that Therefore, we obtain that Noticing that , we have . Therefore, for some sufficiently small , the following inequalities hold: By (3.76)-(3.77), is also satisfied. Therefore, all necessary conditions of Theorem 3.6 are satisfied. By Theorem 3.6, (3.69) has at least two positive -periodic solutions.

Acknowledgment

This paper is supported by the National Natural Science Foundation of China (Grant nos. 10971085 and 11061016).

References

  1. R. Wang and X. Zhang, “Positive periodic solution for a neutral logarithmic population model with feedback control,” Applied Mathematics and Computation, vol. 217, no. 19, pp. 7692–7702, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. S. Lu and W. Ge, “Existence of positive periodic solutions for neutral logarithmic population model with multiple delays,” Journal of Computational and Applied Mathematics, vol. 166, no. 2, pp. 371–383, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. Q. Wang, Y. Wang, and B. Dai, “Existence and uniqueness of positive periodic solutions for a neutral logarithmic population model,” Applied Mathematics and Computation, vol. 213, no. 1, pp. 137–147, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. M. Tang and X. Tang, “Positive periodic solutions for neutral multi-delay logarithmic population model,” Journal of Inequalities and Applications, vol. 2012, pp. 1–10, 2012. View at Publisher · View at Google Scholar
  5. Y. Luo and Z. Luo, “Existence of positive periodic solutions for neutral multi-delay logarithmic population model,” Applied Mathematics and Computation, vol. 216, no. 4, pp. 1310–1315, 2010. View at Publisher · View at Google Scholar
  6. Y. Xia, “Almost periodic solution of a population model: via spectral radius of matrix,” Bulletin of the Malaysian Mathematical Sciences Society. In press. View at Publisher · View at Google Scholar
  7. Y. K. Li, “On a periodic neutral delay logarithmic population model,” Journal of Systems Science and Mathematical Sciences, vol. 19, no. 1, pp. 34–38, 1999. View at Zentralblatt MATH
  8. Y. Chen, “Multiple periodic solutions of delayed predator-prey systems with type IV functional responses,” Nonlinear Analysis: Real World Applications, vol. 5, no. 1, pp. 45–53, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. G. Hetzer, “Some remarks on Φ+-operators and on the coincidence degree for a Fredholm equation with noncompact nonlinear perturbations,” Annales de la Societé Scientifique de Bruxelles, vol. 89, no. 4, pp. 497–508, 1975.
  10. G. Hetzer, “Some applications of the coincidence degree for set-contractions to functional differential equations of neutral type,” Commentationes Mathematicae Universitatis Carolinae, vol. 16, no. 1, pp. 121–138, 1975. View at Zentralblatt MATH
  11. R. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, Springer, Berlin, Germany, 1977.
  12. Z. Liu and Y. Mao, “Existence theorem for periodic solutions of higher order nonlinear differential equations,” Journal of Mathematical Analysis and Applications, vol. 216, no. 2, pp. 481–490, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH