This Provisional PDF corresponds to the article as it appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Positive periodic solutions for neutral multi-delay logarithmic population model Journal of Inequalities and Applications 2012, 2012:10 doi:10.1186/1029-242X-2012-10 Mei-Lan Tang (csutmlang@163.com) Xian-Hua Tang (tangxh@mail.csu.edu.cn) ISSN 1029-242X Article type Research Submission date 7 March 2011 Acceptance date 16 January 2012 Publication date 16 January 2012 Article URL http://www.journalofinequalitiesandapplications.com/content/2012/1/10 This peer-reviewed article was published immediately upon acceptance. It can be downloaded, printed and distributed freely for any purposes (see copyright notice below). 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Positive periodic solutions for neutral multi-delay logarithmic population model Mei-Lan Tang ∗ and Xian-Hua Tang School of Mathematical Science and Computing Technology, Central South University, Changsha, 410083, China ∗ Corresponding author: csutmlang@163.com Email address: XHT: tangxh@mail.csu.edu.cn 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. Keywords: positive periodic solution; existence; k -set contractive operator; logarithmic population model. 1 MSC 2010: 34C25; 34D40. 1 Introduction In recent years, there has b een 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 |x| 0 = max t∈[0,ω] |x(t)|, and C 1 ω = {x : x ∈ C 1 (R, R), x(t + ω ) = x(t)}, with the norm defined by x 0 = max{|x| 0 , |x  | 0 }, then C ω , C 1 ω are both Banach spaces. Let ¯ h = 1 ω  ω 0 h(t)dt, ∀h ∈ C ω . 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: dN dt = N(t)   r(t) − n  j=1 a j (t) ln N(t − σ j (t)) − m  i=1 b i (t) d dt ln N(t − τ i (t))   (1) where r(t), a j (t), b i (t), σ j (t), τ i (t) are all in C ω with ¯r > 0, σ j (t) > 0 and τ i (t) > 0,∀t ∈ 2 [0, ω], ∀j ∈ {1, 2, . . . , n}, ∀i ∈ { 1, 2, . . . , m}. Furthermore, b i (t) ∈ C 1 (R, R), σ j (t) ∈ C 1 (R, R), τ i (t) ∈ C 2 (R, R) and σ  j (t) < 1, τ  i (t) < 1, ∀j ∈ {1, 2, . . . , n}, ∀i ∈ {1, 2, . . . , m}. Since σ  j (t) < 1, ∀t ∈ [0, ω], t − σ j (t) 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 Γ(t) = n  j=1 a j (µ j (t)) 1−σ  j (µ j (t)) − m  i=1 b  i (γ i (t)) 1−τ  i (γ i (t)) . 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: (H1  ) There exists a constant θ > 0 such that |Γ(t)| > θ, ∀t ∈ [0, ω]. (H2  )  n j=1 |a j | 0 ω +  m i=1 |b i | 0 |1 − τ  i | 1/2 0 < 1 and  m i=1 |b i | 0 |1 − τ  i | 0 < 1. 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 (H1  ): |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 ( H2  ) in Theorem A will be greatly weakened.  n j=1 |a j | 0 ω +  m i=1 |b i | 0 |1− τ  i | 1/2 0 < 1 in Theorem A is replaced by 1 2  n j=1 |a j | 0 ω +  m i=1 |b i | 0 |1 − τ  i | 1/2 0 < 1 in this article. 3 2 Main lemmas Under the transformation N(t) = e x(t) , then Equation (1) can be rewritten in the following form: x  (t) = r(t) − n  j=1 a j (t)x(t − σ j (t)) − m  i=1 c i (t)x  (t − τ i (t)) (2) where c i (t) = b i (t)(1 − τ  i (t)), i = 1, 2, . . . , m. 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 that S =  i S i and diamS i ≤ δ} 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 E 1 and E 2 be Banach spaces, D ⊂ E 1 , A : D → E 2 be a continuous and bounded operator. If there exists a constant k ≥ 0 satisfying α E 2 (A(S)) ≤ kα E 1 (S) for 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 < 4 ∞ 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 l(L) = sup{δ > 0|δα X (B) ≤ α Y (L(B)) for any bounded set B ⊂ DomL}. 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 N : ¯ Ω → Y 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, N : ¯ Ω → Y 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 (R1) Lx = λNx + λr, ∀λ ∈ (0, 1), ∀x ∈ ∂Ω  DomL; (R2) [QN(x) + Qr, x][QN(−x) + Qr, x] < 0, ∀x ∈ ∂Ω  KerL. 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 exists a x ∈ ¯ Ω satisfying Lx = Nx+r. In the rest of this article, we set Y = C ω , X = C 1 ω Lx = dx dt (3) 5 and Nx = − n  j=1 a j (t)x(t − σ j (t)) − m  i=1 c i (t)x  (t − τ i (t)), (4) then Equation (2) is equivalent to the equation Lx = Nx + r (5) where r = r(t). Clearly, Equation (2) has an ω-periodic solution if and only if Equation (5) has a solution x ∈ C 1 ω . Lemma 2. [7] The differential operator L is a Fredholm operator with index 0, and satisfies l(L) ≥ 1. Lemma 3. [9] If k =  n i=1 |c i | 0 , then N : Ω → C ω is a k-set contractive operator. Lemma 4. [8] Suppose τ ∈ C 1 ω 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 ∗ ∈ (−∞, +∞) max t∈[t ∗ ,t ∗ +T ] |x(t)| ≤ |x(t ∗ )| + 1 2 T  0 |x  (s)|ds. 6 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 Γ(t ) =  n j=1 a j (µ j (t)) 1−σ  j (µ j (t)) −  m i=1 b  i (γ i (t)) 1−τ  i (γ i (t)) . Theorem 1. Assume the following conditions hold: (H1) If Γ(t) > 0, ∀t ∈ [0, ω] (or Γ(t) < 0, ∀t ∈ [0, ω]); (H2) 1 2  n j=1 |a j | 0 ω +  m i=1 |b i | 0 |1 − τ  i | 1/2 0 < 1 and  m i=1 |b i | 0 |1 − τ  i | 0 < 1. 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 Lx = λNx + λr (6) where L and N are defined by Equations (3) and (4), respectively. Then x(t) satisfies x  (t) = λ   r(t) − n  j=1 a j (t)x(t − σ j (t)) − m  i=1 c i (t)x  (t − τ i (t))   . (7) Integrating both sides of Equation (7) over [0, ω] gives ω  0   r(t) − n  j=1 a j (t)x(t − σ j (t)) + m  i=1 b  i (t)x(t − τ i (t))   dt = 0 (8) i.e., ω  0   n  j=1 a j (t)x(t − σ j (t)) − m  i=1 b  i (t)x(t − τ i (t))   dt = ¯rω. (9) 7 Let t − σ j (t) = s, i.e., t = µ j (s). Lemma 4 implies that a j (µ j (s)) 1 − σ  j (µ j (s)) ∈ C ω , a j (µ j (s)) 1 − σ  j (µ j (s)) x(s) ∈ C ω . 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 ω  0 a j (µ j (s)) 1 − σ  j (µ j (s)) ds = ω−σ j (ω)  −σ j (0) a j (µ j (s)) 1 − σ  j (µ j (s)) ds = ω  0 a j (t)dt = ω¯a j , j = 1, . . . , n, (10) ω  0 b  i (γ i (s)) 1 − τ  i (γ i (s)) ds = ω−τ i (ω)  −τ i (0) b  i (γ i (s)) 1 − τ  i (γ i (s)) ds = ω  0 b  i (t)dt = 0, i = 1, . . . , m. (11) Noting that Γ(t) > 0, we have ¯ Γ = 1 ω ω  0 Γ(t)dt = 1 ω ω  0   n  j=1 a j (µ j (t)) 1 − σ  j (µ j (t)) − m  i=1 b  i (γ i (t)) 1 − τ  i (γ i (t))   dt = n  j=1 ¯a j > 0. (12) Furthermore ω  0 a j (t)x(t − σ j (t))dt = ω−σ j (ω)  −σ j (0) a j (µ j (s)) 1 − σ  j (µ j (s)) x(s)ds = ω  0 a j (µ j (s)) 1 − σ  j (µ j (s)) x(s)ds, j = 1, . . . , n. (13) 8 Similarly ω  0 b  i (t)x(t − τ i (t))dt = ω−τ i (ω)  −τ i (0) b  i (γ i (s)) 1 − τ  i (γ i (s)) x(s)ds = ω  0 b  i (γ i (s)) 1 − τ  i (γ i (s)) x(s)ds, i = 1, . . . , m. (14) Combining (13) and (14) with (9) yields ω  0 Γ(t)x(t)dt = ¯rω. (15) Since Γ(t) > 0, it follows from the extended integral mean value theorem that there exists η ∈ [0, ω] satisfying x(η) ω  0 Γ(t)dt = ¯rω, (16) i.e., x(η) = ¯r ¯ Γ . (17) By Lemma 5, we obtain |x(t)| ≤ |x(η)| + 1 2 ω  0 |x  (t)|dt. So |x| 0 ≤ | ¯r ¯ Γ | + 1 2 ω  0 |x  (t)|dt. (18) 9 [...]... (2004) [9] Luo, Y, Luo, ZG: Existence of positive periodic solutions for neutral multi-delay logarithmic population model Appl Math Comput 216, 1310–1315 (2010) [10] Petryshyn, WV, Yu, ZS: Existence theorems for higher order nonlinear periodic boundary value problems Nonlinear Anal 9, 943–969 (1982) 16 [11] Tang, ML, Liu, XG, Liu, XB: New results on periodic solution for a kind of Rayleigh equation Appl... Feldstein, A: Boundedness of a nonlinear nonautonomous neutral delay equation J Math Anal Appl 156, 293–304 (1991) [7] Liu, ZD, Mao, YP: Existence theorem for periodic solutions of higher order nonlinear differential equations J Math Anal Appl 216, 481–490 (1997) [8] Lu, SP, Ge, WG: Existence of positive periodic solutions for neutral logarithmic population model with multiple delays J Comput Appl Math... x]  ω n 2 = M3  r(t)dt − M3 0 ω  ω n aj (t)dt  r(t)dt + M3 j=1 0  2 = ω 2 M3 r − M3 ¯ n 0  n aj  r + M3 ¯ ¯ j=1  ω aj (t)dt (28) j=1 0  aj  ¯ j=1 < 0 Therefore, by Lemma 1, Equation (1) has at least an ω -positive periodic solution 1/2 Since |1−τi |0 < 1, then |1−τi |0 < |1−τi |0 So m i=1 |bi |0 |1−τi |0 < m i=1 1/2 |bi |0 |1−τi |0 From Theorem 1, we have Corollary 1 Assume that the... L: On a periodic neutral logistic equation Glasgow Math J 33, 281–286 (1991) [3] Gaines, RE, Mawhin, JL: Coincidence degree and nonlinear differential equation Lecture notes in Math., vol 568 Springer, Berlin (1997) [4] Kirlinger, G: Permanence in Lotka–Volterra equation, linked prey-predator system Math Biosci 82, 165–191 (1986) [5] Kuang, Y: Delay Differential Equations with Applications in Population. .. − b1 (γ1 (t)) = (cos2 t + 1) + sin t > 0, 8 64 4π + 1 1 1/2 |a1 |0 ω + |b1 |0 |1 − τi |0 = < 1 2 16 The conditions in Theorem 1 in this article are satisfied Hence Equation (29) has at least an 2π -positive periodic solution However, the condition (H2 ) in Theorem A(Theorem 3.1 in [9]) is not satisfied Since 1/2 |a1 |0 ω + |b1 |0 |1 − τi |0 = 8π + 1 > 1, 16 Theorem 3.1 in [9] can not be applied to this... conditions hold (H1 ) If Γ(t) > 0, ∀t ∈ [0, ω] (or Γ(t) < 0, (H2 ) 1 2 n j=1 |aj |0 ω + m i=1 1/2 |bi |0 |1 − τi |0 ∀t ∈ [0, ω]) < 1 and |1 − τi |0 < 1, 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... have no competing interests Authors’ contributions All authors contributed equally to the manuscript and read and approved the final manuscript Acknowledgments 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... Then k = ¯ |ci |0 < 1 ≤ l(L) 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 1 define a bounded bilinear form [·, ·] on Cω × Cω as follows: ω [y, x] = y(t)x(t)dt 0 Define Q : Y → CokerL by 1 Qy = ω ω y(t)dt 0 Obviously, x|x ∈ kerL ∂Ω = {x|x = M3 , x = −M3 } 12 (27) Without loss of generality, we may assume . 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. appeared upon acceptance. Fully formatted PDF and full text (HTML) versions will be made available soon. Positive periodic solutions for neutral multi-delay logarithmic population model Journal of. 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