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Positive periodic solutions for neutral multi-delay logarithmic population model
Journal of Inequalities and Applications volume 2012, Article number: 10 (2012)
Abstract
Based on an abstract continuous theorem of k-set contractive operator and some analysis skill, a new result is obtained for the existence of positive periodic solutions to a neutral multi-delay logarithmic population model. Some sufficient conditions obtained in this article for the existence of positive periodic solutions to a neutral multi-delay logarithmic population model are easy to check. Furthermore, our main result also weakens the condition in the existing results. An example is used to illustrate the applicability of the main result.
MSC 2010: 34C25; 34D40.
1 Introduction
In recent years, there has been considerable interest in the existence of periodic solutions of functional differential equations (see, for example, [1–7]). It is well known that the environments of most natural populations change with time and that such changes induce variation in the growth characteristics of populations. Among many population models, the neutral logarithmic population model has recently attracted the attention of many mathematicians and biologists.
Let ω > 0 be a constant, C ω = {x : x ∈ C(R, R), x(t + ω) = x(t)}, with the norm defined by , and , with the norm defined by ||x||0 = max{|x|0, |x'|0}, then are both Banach spaces. Let .
Lu and Ge [8] studied the existence of positive periodic solutions for neutral logarithmic population model with multiple delays. Based on an abstract continuous theorem of k-set contractive operator, Luo and Luo [9] investigate the following periodic neutral multi-delay logarithmic population model:
where r(t), a j (t), b i (t), σ j (t), τ i (t) are all in C ω with , σ j (t) > 0 and τ i (t) > 0, ∀t ∈ [0, ω], ∀j ∈ {1, 2, ..., n}, ∀i ∈ {1, 2, ..., m}. Furthermore, b i (t) ∈ C1(R, R), σ j (t) ∈ C1(R, R), τ i (t) ∈ C2 (R, R) and , ∀j ∈ {1, 2, ..., n}, ∀i ∈ {1, 2, ..., m}.
Since has a unique inverse. Let μ j (t) be the inverse of t - σ j (t). Similarly, t - τ i (t) has a unique inverse, denoted by γ i (t).
For convenience, denote .
Luo and Luo [9] obtain the following sufficient condition on the existence of positive periodic solutions for neutral logarithmic population model with multiple delays.
Theorem A. Assume the following conditions hold:
( H 1') There exists a constant θ > 0 such that |Γ(t)| > θ, ∀t ∈ [0, ω].
( H 2') and .
Then Equation (1) has at least an ω-positive periodic solution.
The purpose of this article is to further consider the existence of positive periodic solutions to a neutral multi-delay logarithmic population model (1). We will present some new sufficient conditions for the existence of positive periodic solutions to a neutral multi-delay logarithmic population model. In this article, we will replace the assumption (H 1'): |Γ(t)| > θ in Theorem A by different assumption Γ(t) > 0, ∀t ∈ [0, ω], (or Γ(t) < 0, ∀t ∈ [0, ω]). Obviously, it is more easy to check Γ(t) > 0, ∀t ∈ [0, ω], than to find a constant θ > 0 such that |Γ(t)| > θ, ∀t ∈ [0, ω]. At the same time, the assumption (H 2') in Theorem A will be greatly weakened. in Theorem A is replaced by in this article.
2 Main lemmas
Under the transformation N(t) = ex(t), then Equation (1) can be rewritten in the following form:
where .
It is easy to see that in this case the existence of positive periodic solution of Equation (1) is equivalent to the existence of periodic solution of Equation (2). In order to investigate the existence of periodic solution of Equation (2), we need some definitions and lemmas.
Definition 1. Let E be a Banach space, S ⊂ E be a bounded subset, denote α E (S) = inf {δ > 0| there is a finite number of subsets S i ⊂ S such thatand
then α E is called non-compactness measure of S or Kuratowski distance (see[1]), where diamS i denotes the diameter of set S i .
Definition 2. Let E1and E2be Banach spaces, D ⊂ E1, A : D → E2be a continuous and bounded operator. If there exists a constant k ≥ 0 satisfyingfor any bounded set S ⊂ D, then A is called k-set contractive operator on D.
Definition 3. Let X, Y be normed vector spaces, L : DomL ⊂ X → Y be a linear mapping. This mapping L will be called a Fredholm mapping of index 0 if dimKerL = codimImL < ∞ and ImL is closed in Y[3].
Assume that L : DomL ⊂ X → Y is a Fredholm operator with index 0, from [3], we know that sup{δ > 0|δα X (B) ≤ α Y (L(B))} exists for any bounded set B ⊂ DomL, so we can define
Now let L : X → Y be a Fredholm operator with index 0, X and Y be Banach spaces, Ω ⊂ X be an open and bounded set, and let be a k-set contractive operator with k < l(L). By using the homotopy invariance of k-set contractive operator's topological degree D[(L, N), Ω], Petryshyn and Yu [10] proved the following result.
Lemma 1. [10]Assume that L : X → Y is a Fredholm operator with index 0, r ∈ Y is a fixed point, is a k-set contractive operator with k < l(L), where Ω ⊂ X is bounded, open, and symmetric about 0 ∈ Ω. Furthermore, we also assume that
(R 1)
(R 2)
where[·,·] is a bilinear form on Y × X, and Q is the projection of Y onto Coker, where Coker is the cokernel of the operator L. Then there existssatisfying Lx = Nx + r.
In the rest of this article, we set
and
then Equation (2) is equivalent to the equation
where r = r(t). Clearly, Equation (2) has an ω-periodic solution if and only if Equation (5) has a solution .
Lemma 2. [7]The differential operator L is a Fredholm operator with index 0, and satisfies l(L) ≥ 1.
Lemma 3. [9]If, then N : Ω → C ω is a k-set contractive operator.
Lemma 4. [8]Suppose and τ'(t) < 1, ∀t ∈ [0, ω]. Then the function t - τ (t) has a inverse μ(t) satisfying μ ∈ C(R, R) with μ(a + ω) = μ(a) + ω.
Lemma 5. [11]Let x(t) be continuous differentiable T-periodic function (T > 0). Then for any t* ∈ (-∞, +∞)
3 Main results
Let μ j (t) be the inverse of t - σ j (t), γ j (t) be the inverse of t - τ i (t) and .
Theorem 1. Assume the following conditions hold:
(H 1) If Γ(t) > 0, ∀t ∈ [0, ω] (or Γ(t) < 0, ∀t ∈ [0, ω]);
(H 2) and.
Then Equation (1) has at least an ω-positive periodic solution.
Proof. Suppose that x(t) is an arbitrary ω-periodic solution of the following operator equation
where L and N are defined by Equations (3) and (4), respectively. Then x(t) satisfies
Integrating both sides of Equation (7) over [0, ω] gives
i.e.,
Let t - σ j (t) = s, i.e., t = μ j (s). Lemma 4 implies that
Lemma 4 implies μ j (0 + ω) = μ j (0) + ω, γ i (0 + ω) = γ i (0) + ω, ∀j ∈ {1, ..., n}, i ∈{1, ..., m}.
Noting that σ j (0) = σ j (ω), τ i (0) = τ i (ω), then
Noting that Γ(t) > 0, we have
Furthermore
Similarly
Combining (13) and (14) with (9) yields
Since Γ(t) > 0, it follows from the extended integral mean value theorem that there exists η ∈ [0, ω] satisfying
i.e.,
By Lemma 5, we obtain
So
Multiplying both sides of Equation (7) by x'(t) and integrating them over [0, ω], we have
Cauchy-Schwarz inequality implies
Meanwhile
Substituting Equations (18) and (21) into (20) gives
From the assumption it follows from Equation (22) that there exists constant M > 0 such that
Then
Again from (7), we get
Condition implies that
Let and . Then . Equations (24) and (26) imply that all conditions of Lemma 1 except (R2) hold. Next, we prove that the condition (R2) of Lemma 1 is also satisfied. We define a bounded bilinear form [·,· ] on as follows:
Define Q : Y → CokerL by
Obviously,
Without loss of generality, we may assume that x = M3. Thus
Therefore, by Lemma 1, Equation (1) has at least an ω-positive periodic solution. □
Since , then . So . From Theorem 1, we have
Corollary 1. Assume that the following conditions hold
(H 1') .
(H 2') and, i = 1, ..., m.
Then Equation (1) has at least an ω-positive periodic solution.
4 Example
Example 1 is given to illustrate the effectiveness of our new sufficient conditions, also to demonstrate the difference between the proposed result in this paper and the result in [9].
Example 1. Consider the following equation:
where.
Let ω = 2π. Corresponding to Equation (1), we have n = m = 1, , , σ1(t) = τ1(t) = π So, . Thus
The conditions in Theorem 1 in this article are satisfied. Hence Equation (29) has at least an 2π-positive periodic solution. However, the conditionin Theorem A(Theorem 3.1 in[9]) is not satisfied. Since
Theorem 3.1 in[9]can not be applied to this example. Let. Although the conditionin Theorem A (Theorem 3.1 in[9]) is satisfied, it is more complex to check the condition |Γ(t)| > θ, ∀t ∈ [0, ω] in Theorem A than to test Γ(t) > 0, ∀t ∈ [0, ω]. This example illustrates the advantages of the proposed results in this paper over the existing ones.
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Acknowledgements
The authors are grateful to the referees for their valuable comments which have led to improvement of the presentation. This study was partly supported by the Zhong Nan Da Xue Qian Yan Yan Jiu Ji Hua under grant No. 2010QZZD015, Hunan Scientific Plan under grant No. 2011FJ6037, NSFC under grant No. 61070190 and NFSS under grant 10BJL020.
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Tang, ML., Tang, XH. Positive periodic solutions for neutral multi-delay logarithmic population model. J Inequal Appl 2012, 10 (2012). https://doi.org/10.1186/1029-242X-2012-10
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DOI: https://doi.org/10.1186/1029-242X-2012-10