Positive periodic solutions for neutral multi-delay logarithmic population model

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.[PUBLICATION ABSTRACT]

Tang and Tang Journal of Inequalities and Applications 2012, 2012:10 http://www.journalofinequalitiesandapplications.com/content/2012/1/10

RESEARCH Open Access

Positive periodic solutions for neutral multi-delay logarithmic population model

Mei-Lan Tang* and Xian-Hua Tang

* Correspondence: mailto:[email protected]

Web End =csutmlang@163. mailto:[email protected]

Web End =com School of Mathematical Science and Computing Technology, Central South University, Changsha, 410083, China

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.

Keywords: positive periodic solution, existence, k-set contractive operator, logarithmic population model

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 >0 be a constant, C = {x : x C(R, R), x(t + ) = x(t)}, with the norm defined by |x|0 = maxt[0,] x(t)

, and C1 = {x : x C1(R, R), x(t + ) = x(t)}, with the norm defined by ||x||0 = max{|x|0, |x|0}, then C, C1 are both Banach spaces. Let

h = 1

0 h(t)dt, h C.

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:

dNdt = N(t)

r(t)

n

j=1aj(t) ln N(t j(t))

m

(1)

where r(t), aj(t), bi(t), sj(t), i(t) are all in C with r > 0, sj(t) > 0 and i(t) > 0, t [0,

], j {1, 2, ..., n}, i {1, 2, ..., m}. Furthermore, bi(t) C1(R, R), sj(t) C1(R, R), i(t)

i=1bi(t) ddt ln N(t i(t))

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C2 (R, R) and j(t) < 1, i(t) < 1, j {1, 2, ..., n}, i {1, 2, ..., m}.

Since j(t) < 1, t [0, ], t j(t) has a unique inverse. Let j(t) be the inverse of t - sj(t). Similarly, t - i(t) has a unique inverse, denoted by gi(t).

For convenience, denote (t) =

n

i=1b i(i(t))1 i(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 >0 such that |(t)| > , t [0, ].

(H2)

n j=1

j=1aj(j(t))1 j(j(t)) m

|aj|0 + mi=1 |bi|0 |1 i|1/20 < 1 and mi=1 |bi|0|1 i|0 < 1.

Then Equation (1) has at least an -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): |(t)| > in Theorem A by different assumption (t) >0, t [0, ], (or (t) < 0, t [0, ]). Obviously, it is more easy to check (t) > 0, t [0, ], than to find a constant >0 such that |(t)| > , t [0, ]. At the same time, the assumption (H2) in Theorem

A will be greatly weakened.

n j=1

|aj|0 + mi=1 |bi|0|1 i|1/20 < 1 in Theorem A is

replaced by 12

n j=1

|aj|0 + mi=1 |bi|0 |1 i|1/20 < 1 in this article.

2 Main lemmas

Under the transformation N(t) = ex(t), then Equation (1) can be rewritten in the following form:

x (t) = r(t)

n

j=1

aj(t)x(t j(t))

m

i=1

ci(t)x (t i(t)) (2)

where ci(t) = bi(t)(1 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 aE(S) =

inf { > 0| there is a finite number of subsets Si S such that S =

i Siand

diamSi }then aE is called non-compactness measure of S or Kuratowski distance (see [1]), where diamSi denotes the diameter of set Si.

Definition 2. Let E1 and E2 be Banach spaces, D E1, A : D E2 be a continuous and bounded operator. If there exists a constant k 0 satisfying E2(A(S)) kE1(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 < and ImL is closed in Y [3].

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Assume that L : DomL X Y is a Fredholm operator with index 0, from [3], we know that sup{ >0|aX(B) aY (L(B))} exists for any bounded set B DomL, so we can define

l(L) = sup{ > 0|X(B) 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, X be an open and bounded set, and let N :

Y be a k-set contractive operator

with k < l(L). By using the homotopy invariance of k-set contractive operators topological degree D[(L, N), ], 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 :

Yis a k-set contractive operator with k < l(L), where X is

bounded, open, and symmetric about 0 . Furthermore, we also assume that(R1) Lx = Nx + r, (0, 1), x DomL;

(R2) [QN(x) + Qr, x][QN(x) + Qr, x] < 0, x 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

satisfying Lx = Nx + r.

dt (3)

m

i=1

|ci|0, then N : C is a k-set contractive operator. Lemma 4. [8]Suppose C1and (t) < 1, t [0, ]. Then the function t - (t) has

a inverse (t) satisfying C(R, R) with (a + ) = (a) + .

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)| +

3 Main results

Let j(t) be the inverse of t - sj(t), gj(t) be the inverse of t - i(t) and

(t) =

Page 3 of 9

In the rest of this article, we set Y = C, X = C1

Lx = dx

and

ci(t)x (t i(t)), (4)

then Equation (2) is equivalent to the equation

Lx = Nx + r (5)

where r = r(t). Clearly, Equation (2) has an -periodic solution if and only if Equa

tion (5) has a solution x C1.

Lemma 2. [7]The differential operator L is a Fredholm operator with index 0, and satisfies l(L) 1.

Lemma 3. [9]If k =

Nx =

n

j=1

aj(t)x(t j(t))

n i=1

1

2

T

0

|x (s)|ds.

nj=1aj(j(t))1 j(j(t)) mi=1b i(i(t))1 i(i(t)) .

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Theorem 1. Assume the following conditions hold: (H1) If (t) > 0, t [0, ] (or (t) < 0, t [0, ]);

(H2) 12

n j=1

a

j

0 +

|bi|0|1 i|1/20 < 1 and mi=1 |bi|0|1 i|0 < 1.

Then Equation (1) has at least an -positive periodic solution.

Proof. Suppose that x(t) is an arbitrary -periodic solution of the following operator equation

Lx = Nx + r (6)

where L and N are defined by Equations (3) and (4), respectively. Then x(t) satisfies

x (t) =

r(t)

n

j=1aj(t)x(t j(t))

m

m i=1

i=1ci(t)x (t i(t))

. (7)

Integrating both sides of Equation (7) over [0, ] gives

0

r(t)

n

j=1aj(t)x(t j(t)) +

m

i=1b i(t)x(t i(t))

dt = 0 (8)

i.e.,

0

n

j=1aj(t)x(t j(t))

m

i=1b i(t)x(t i(t))

dt = r. (9)

Let t - sj(t) = s, i.e., t = j(s). Lemma 4 implies that

aj(j(s))1 j(j(s))

C, aj(j(s)) 1 j(j(s))

x(s) C.

Lemma 4 implies j(0 + ) = j(0) + , gi(0 + ) = gi(0) + , j {1, ..., n}, i {1, ..., m}. Noting that sj(0) = sj(), i(0) = i(), then

0

aj(j(s))1 j(j(s))

ds =

j()

j(0)

aj(j(s))1 j(j(s))

ds =

0

aj(t)dt = j, j = 1, . . . , n, (10)

b i(i(s))1 i(i(s))

ds =

i()

i(0)

b i(i(s))1 i(i(s))

ds =

0

b i(t)dt = 0, i = 1, . . . , m. (11)

Noting that (t) > 0, we have

= 1

(t)dt = 1

0

n

j=1aj(j(t))1 j(j(t))

m

i=1b i(i(t))1 i(i(t))

dt =

n

j=1

aj > 0. (12)

Furthermore

0

aj(t)x(t j(t))dt =

j()

j(0)

aj(j(s))1 j(j(s))

x(s)ds

=

aj(j(s))1 j(j(s))

x(s)ds, j = 1, . . . , n.

(13)

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Similarly

0

b i(t)x(t i(t))dt =

i()

i(0)

b i(i(s))1 i(i(s))

x(s)ds

=

b i(i(s))1 i(i(s))

x(s)ds, i = 1, . . . , m.

(14)

Combining (13) and (14) with (9) yields

0

(t)x(t)dt = r. (15)

Since (t) > 0, it follows from the extended integral mean value theorem that there exists h [0, ] satisfying

x()

0

(t)dt = r, (16)

i.e.,

x() = r

. (17)

By Lemma 5, we obtain

|x(t)| |x()| +

1

2

0

|x (t)|dt.

So

|x|0 | r

| +

1

2

0

dt. (18)

Multiplying both sides of Equation (7) by x(t) and integrating them over [0, ], we have

0

x (t)

|x (t)|2dt

=

0

x (t)2dt

=

0

x (t)2dt

0

r(t)x (t)dt

m

=

0

n

j=1aj(t)x(t j(t))x (t)dt

0

i=1ci(t)x (t i(t))x (t)dt

|r|0

0

|x (t)|dt+

n

j=1

|aj|0|x|0

0

x (t) dt

+

m

i=1

0

|ci(t)x (t i(t))x (t)|dt.

(19)

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Cauchy-Schwarz inequality implies

0

|x (t)|2dt =

0

n

r(t)x (t)dt

0

j=1aj(t)x(t j(t))x (t)dt

0

m

i=1ci(t)x (t i(t))x (t)dt

|r|0 +

(20)

n

j=1

|aj|0|x|0

|x (t)|2dt

1/2

1/2+

m

i=1

|ci(t)x (t i(t))|2dt

1/2 |x (t)|2dt

1/2

.

Meanwhile

|ci(t)x (t i(t))|2dt

1/2

=

0

11 i(i(t))|

ci(i(t))x (t)|2dt

1/2

= (1 i(i(t)))|bi(i(t))x (t)|2dt

1/2

(21)

|1 i|1/20|bi|0

|x (t)|2dt

1/2

.

Substituting Equations (18) and (21) into (20) gives

|x (t)|2dt

|r|0 +

n

j=1

|aj|0|x|0

|x (t)|2dt

1/2

1/2

m

i=1

+ |1 i|1/20|bi|0

|x (t)|2dt

|r|0 +

(22)

n

j=1

|aj|0 r

|x (t)|2dt

1/2

1/2+

1

2

n

j=1

|aj|0 +

m

i=1

|1 i|1/20|bi|0

|x (t)|2dt

.

|aj|0 + mi=1 |1 i|1/20|bi|0 < 1, it follows from Equa

tion (22) that there exists constant M >0 such that

0

From the assumption 12

n j=1

x (t) 2dt

1/2

< M. (23)

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Then

|x|0

r

+ 1

2

0

x (t)

dt

r

+ 1

2M1/2 := M1. (24)

Again from (7), we get

|x |0 |r|0 +

n

j=1

|aj|0 |x|0 +

m

i=1

|ci|0|x |0. (25)

Condition

m i=1

|ci|0 mi=1 |bi|0|1 i|0 < 1 implies that

|x |0 |

r|0 + n

j=1

|aj|0M1 1 m

i=1

:= M2. (26)

Let M3 > max{M1, M2, |r/ nj=1j|}

and = {x|x C1, x < M3}. Then

k =

|ci|0

m i=1

|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 define a bounded bilinear form [, ] on C C1 as follows:

[y, x] =

0

y(t)x(t)dt. (27)

Define Q : Y CokerL by

Qy = 1

0

y(t)dt.

Obviously,

x|x kerL = {x|x = M3, x = M3}.

Without loss of generality, we may assume that x = M3. Thus

[QN(x) + Qr, x][QN(x) + Qr, x]

= M23

0

r(t)dt M3

n

j=1

0

aj(t)dt

0

r(t)dt + M3

n

j=1

0

aj(t)dt

(28)

= 2M23

r M3

n

j=1

aj

r + M3

n

j=1

aj

< 0.

Therefore, by Lemma 1, Equation (1) has at least an -positive periodic solution.

Since |1 i|0 < 1, then |1 i|0 < |1 i|1/20. So

m i=1 |bi|0|1 i|0 < mi=1 |bi|0|1 i|1/20. From Theorem 1, we have

Corollary 1. Assume that the following conditions hold (H1) If (t) > 0, t [0, ] (or (t) < 0, t [0, ]).

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|aj|0 + mi=1 |bi|0|1 i|1/20 < 1and |1 i|0 < 1, i = 1, ..., m.

Then Equation (1) has at least an -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:

dNdt = N(t) r(t)

18(cos2 t + 1) ln N(t )

(H2) 12

n j=1

164(3 cos t)

ddt ln N(t ) (29)

1

64 (3 cos t) cos t.

Let = 2. Corresponding to Equation (1), we have n = m = 1, a1(t) = 18(cos2 t + 1), b1(t) = 1

64 (3 cos t), s1(t) = 1(t) = So r =

1

128 > 0,

where r(t) = cos t

1

32 (cos2 t + 1) sin t

1(t) = 1(t) = 0, 1(t) = 1(t) = + t. Thus

(t) = a1(1(t)) b 1(1(t)) =

18(cos2 t + 1) +

164 sin t > 0,

1

2|a1|0 + |b1|0|1 i|1/20 =

4 + 1

16 < 1.

The conditions in Theorem 1 in this article are satisfied. Hence Equation (29) has at least an 2-positive periodic solution. However, the condition (H 2)in Theorem A(Theorem 3.1 in [9]) is not satisfied. Since

|a1|0 + |b1|0|1 i|1/20 =

8 + 1

16 > 1,

Theorem 3.1 in [9]can not be applied to this example. Let = 7

64. Although the condition (H 1)in Theorem A (Theorem 3.1 in [9]) is satisfied, it is more complex to check the condition |(t)| > , t [0, ] in Theorem A than to test (t) >0, t [0, ]. This example illustrates the advantages of the proposed results in this paper over the existing ones.

AcknowledgementsThe 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.

Authors contributionsAll authors contributed equally to the manuscript and read and approved the final manuscript.

Competing interestsThe authors declare that they have no competing interests.

Received: 7 March 2011 Accepted: 16 January 2012 Published: 16 January 2012

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doi:10.1186/1029-242X-2012-10Cite this article as: Tang and Tang: Positive periodic solutions for neutral multi-delay logarithmic population model. Journal of Inequalities and Applications 2012 2012:10.

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