Abstract and Applied Analysis

Volume 2012 (2012), Article ID 423416, 16 pages

doi: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

Hui Fang

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., [1–5]). 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 [2–5]. 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).

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