Existence of Two Positive Periodic Solutions for a Neutral Multi-Delay Logarithmic Population Model with a Periodic Harvesting Rate
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.
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  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 .
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 , some novel techniques are employed to find a priori bounds on solutions.
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.
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.
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
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.
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
Lemma 3.3 (see ). is a Fredholm map of index and satisfies
Lemma 3.4 (see ). 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 , 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:
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.
This paper is supported by the National Natural Science Foundation of China (Grant nos. 10971085 and 11061016).
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