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Positive periodic solutions for a class of delay differential equations q Zhijie Nan a, Weijun Chen b, Lin Li a,⇑ a b

Department of Mathematics, Physics and Information Engineering, Jiaxing University, Jiaxing, Zhejiang 314001, PR China Jiaxing Vocational and Technical College, Jiaxing, Zhejiang 314001, PR China

a r t i c l e

i n f o

a b s t r a c t In this paper, we employ ﬁxed point theorem and functional equation theory to study the existence of positive periodic solutions of the delay differential equation

Keywords: Delay differentiable equation Functional equation Positive periodic solution

x0 ðtÞ ¼ aðtÞxðtÞ bðtÞx2 ðtÞ þ cðtÞxðt sðtÞÞxðtÞ: Moreover, new explicit conditions for the related population growth models are also given. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction In this paper we consider the existence of positive periodic solutions of the delay differential equation

x0 ðtÞ ¼ aðtÞxðtÞ bðtÞx2 ðtÞ þ cðtÞxðt sðtÞÞxðtÞ;

ð1:1Þ

where a(t) > 0, b(t) > 0, c(t) P 0 and s(t) P 0 for t 2 R are continuously differentiable x-periodic functions. Delay differential equations (DDEs) are a large and important class of dynamical systems. The delays can represent gestation times, incubation periods, transport delays, or can simply lump complicated biological processes, which have been used in many branches of biological and medical modelling: population dynamics [12,14], drug therapy [18] and immune response [4]. In particular, our equation can be interpreted as the standard Malthus population model x0 = a(t)x that subjects to a perturbation with periodical delay. One question of this subject is whether these equations can support positive periodic solutions. Such problems have been studied extensively by a number of authors; see for example [6,10,13,19]. One of another model that is a prototype of Eq. (1.1), is the system of Volterra integro differential equation

" x0i ðtÞ

¼ xi ðtÞ ai ðtÞ

n X

bij ðtÞxj ðtÞ

j¼1

n Z X j¼1

#

t

C ij ðt; sÞg ij ðxj ðsÞÞds ; 1

which governs the population growth of interacting species xj(t), j = 1, 2, . . . , n. On the other hand, for basic theory of periodicity and the applications of Eq. (1.1) to a variety of dynamical models, see the work [1–3,5,9,15–17], and the references therein. It is our view to discuss the dynamical models and their applications of describing the nature with a large range of parameters involved. In [7,8], the authors mentioned that periodic variations in population numbers occurred when the temperature itself was varied periodically on a daily basis. This led us to investigate the existence of periodic solutions for single species delay models with periodic delay. For the study of periodic solutions of the approximation scheme, a general approach is to have a set of test equations which are as common as possible so that the explicit analytic conditions can be given,

q

Project supported by the Youth Teacher Training Plan of Zhejing Province (No. 2009164).

⇑ Corresponding author.

E-mail address: [email protected] (L. Li). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.10.013

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Z. Nan et al. / Applied Mathematics and Computation 218 (2011) 4647–4650

such as Halbach [8], and Freedman and Wu [7]. However, it is difﬁcult to determine its conditions. In this paper, based on the results in [7,8,11], we present the existence of positive periodic solutions of a functional equation, which will be applied directly to discuss the solutions of Eq. (1.1). 2. Mains results Let CP(x) be the set of all continuously differentiable x-periodic function, and

CPðL; xÞ ¼ f/ 2 CPðxÞ : j/ðxÞ /ðyÞj 6 Ljx yj;

L > 0;

8x; y 2 Rg:

We endow CP(x) with the norm k/k = max{j/(x)j : x 2 R}, then CP(x) and CP(L, x) are Banach space. Furthermore, setting

CPðM; L; xÞ ¼ f/ 2 CPðL; xÞ : j/ðxÞj 6 M; M > 0;

8x 2 Rg

and

CPþ ðM; m; L; xÞ ¼ f/ 2 CPðM; L; xÞ;

0 < m 6 /ðxÞ 6 M;

8x 2 Rg:

Concerning the linear functional equation

/ðf ðxÞÞ ¼ gðxÞ/ðxÞ þ FðxÞ:

ð2:2Þ

where g, F 2 CP(L;x) fulﬁlls g(x) > 0 and F(x) < 0 for every x 2 R, and f is Lipschitz continuously differentiable with the Lipschitz constant L and satisﬁes that jf(x) f(x + x)j = x for any x 2 R. Denote m(g) = minx2Rg(x), M(g) = maxx2Rg(x), m(F) = minx2RF(x) and M(F) = maxx2RF(x). We consider the following case

Mð1 mðgÞÞ 6 mðFÞ

ð2:3Þ

ðL þ 1ÞmðgÞ þ M mðFÞ 6 ðmðgÞÞ2 ;

ð2:4Þ

and

then the following results are obtained. Theorem 1. Assume that conditions (2.3) and (2.4) hold, then Eq. (2.2) has a continuously differentiable x-period solution / 2 CP(M, L; x).

Corollary 1. Assume that conditions (2.3) and (2.4) hold, and

mð1 MðgÞÞ P MðFÞ;

ð2:5Þ +

then there exists a continuously differentiable x-period solution / 2 CP (M, m, L; x) for Eq. (2.2). Corollary 2. If conditions (2.3) and (2.4) are satisﬁed, and

mðgÞ > 1;

ð2:6Þ

then Eq. (2.2) has an unique continuously differentiable x-periodic solution / 2 CP(M, L; x). Furthermore, the unique solution is positive when (2.5) holds. Theorem 2. If Inequalities (2.3), (2.4) and (2.5) hold, then Eq. (1.1) has a positive x-periodic solution x(t), t 2 R. The following lemmas are useful to prove the main results above. Lemma 1. CP(M, L; x) and CP+(M, m, L; x) are sequentially compact convex subset of CP(x). Proof of Lemma 1. From the deﬁnition of CP(M, L; x) and CP+(M, m, L; x), we know that they are uniformly bounded closed subset of CP(x). Moreover, it is easy to prove k/1 + (1 k)/2 2 CP(M, L; x) for each /1,/2 2 CP(M, L; x) (same result for CP+(M, m, L; x)). Then CP(M, L; x) and CP+(M, m, L; x) are convex. Furthermore, from the Lipschits condition of /(x), it follows that CP(M, L; x) (CP+(M, m, L; x)) is equi-continuous. Therefore, we obtain that CP(M, L; x) and CP+(M, m, L; x) are sequentially subset of CP(x) by Ascoli–Arzela lemma. h Lemma 2 (Theorem 2.1 in [7]). Suppose that the equation

aðtÞ bðtÞ/ðtÞ þ cðtÞ/ðt sðtÞÞ ¼ 0;

ð2:7Þ

has a positive, x-periodic and continuously differentiable solution /(t). Then the model Eq. (1.1) has a positive x-periodic solution x(t).

Z. Nan et al. / Applied Mathematics and Computation 218 (2011) 4647–4650

4649

Proof of Theorem 1. Constructing the operator L/ðxÞ:

L/ðxÞ ¼

/ðf ðxÞÞ FðxÞ ; gðxÞ gðxÞ

ð2:8Þ

for any / 2 CP(M, L; x). Since jf(x) f(x + x)j = x and g, F 2 CP(L; x), it implies that L/ðxÞ is also x-periodic. In what follows, we will prove that L/ðxÞ is an operator of CP(M, L; x) ? CP(M, L; x). In fact, From (2.3), we have

/ðf ðxÞÞ FðxÞ þ 6 M mðFÞ 6 M: jL/ðxÞj 6 gðxÞ gðxÞ mðgÞ mðgÞ On the other hand, for each / 2 CP(M, L; x), x1, x2 2 R, from (2.4), we get

/ðf ðx1 ÞÞ Fðx1 Þ /ðf ðx2 ÞÞ Fðx2 Þ /ðf ðx1 ÞÞ /ðf ðx2 ÞÞ Fðx1 Þ Fðx2 Þ 6 þ jL/ðx1 Þ L/ðx2 Þj ¼ þ gðx1 Þ gðx1 Þ gðx2 Þ gðx2 Þ gðx1 Þ gðx2 Þ gðx1 Þ gðx2 Þ /ðf ðx1 ÞÞ /ðf ðx2 ÞÞ /ðf ðx2 ÞÞ /ðf ðx2 ÞÞ Fðx1 Þ Fðx2 Þ Fðx2 Þ Fðx2 Þ þ þ þ 6 gðx1 Þ gðx1 Þ gðx1 Þ gðx2 Þ gðx1 Þ gðx1 Þ gðx1 Þ gðx2 Þ 6

Ljf ðx1 Þ f ðx2 Þj Mjgðx1 Þ gðx2 Þj Ljx1 x2 j mðFÞLjx1 x2 j þ þ mðgÞ mðgÞ ðmðgÞÞ2 ðmðgÞÞ2

6

L2 jx1 x2 j MLjx1 x2 j Ljx1 x2 j mðFÞLjx1 x2 j mðgÞL2 þ ML þ mðgÞL mðFÞL þ þ ¼ jx1 x2 j mðgÞ mðgÞ ðmðgÞÞ2 ðmðgÞÞ2 ðmðgÞÞ2

6 Ljx1 x2 j: Therefore, L/ 2 CPðM; L; xÞ. By Schauder’s ﬁxed point theorem, there exists / 2 CP(M, L; x) such that /ðxÞ ¼ L/ðxÞ, i.e., /(f(x)) = g(x)/(x) + F(x). h Proof of Corollary 1. Consider the operator (2.8), for each / 2 CP+(M, m, L; x), from Inequality (2.5), we obtain

L/ðxÞ ¼

/ðf ðxÞÞ FðxÞ m MðFÞ P P m: gðxÞ MðgÞ

Hence, it follows from the proof of Theorem 1 that L/ðxÞ 2 CP þ ðM; m; L; xÞ. Therefore, by Schauder’s ﬁxed point theorem, there exists / 2 CP+(M, m, L; x) such that /ðxÞ ¼ L/ðxÞ, i.e., /(f(x)) = g(x)/(x) + F(x). h Proof of Corollary 2. If linear functional Eq. (2.2) satisﬁes Inequalities (2.3)-(2.4), from (2.6) and (2.8), we have

/1 ðf ðxÞÞ FðxÞ /2 ðf ðxÞÞ FðxÞ /1 ðf ðxÞÞ /2 ðf ðxÞÞ 6 1 k/1 /2 k < k/1 /2 k ¼ þ kL/1 L/2 k ¼ mðgÞ gðxÞ gðxÞ gðxÞ gðxÞ gðxÞ for each /1,/2 2 CP(M, L; x). Therefore, L is a contraction mapping deﬁned on CP(M, L; x). Then by Banach’s ﬁxed point principle, it follows that L has only one ﬁxed point in CP(M, L; x). Furthermore, if (2.5) is also satisﬁed, we obtain that L is a contraction mapping deﬁned on CP+(M, m, L; x), which implies that L has unique ﬁxed point in CP+(M, m, L; x). h Proof of Theorem 2. Let

FðtÞ :¼

aðtÞ bðtÞ ; gðtÞ :¼ and f ðtÞ :¼ t sðtÞ cðtÞ cðtÞ

for every t 2 R. The result follows from Corollary 1 and Lemma 2.

ð2:9Þ h

3. Example Consider Eq. (1.1)

x0 ðtÞ ¼ aðtÞxðtÞ bðtÞx2 ðtÞ þ cðtÞxðt sðtÞÞxðtÞ in the case that

aðtÞ ¼ 4 cos

2p

x

t 2 sin

2p

x

t

2p cos tþ2 ;

x

and

cðtÞ ¼ cos where x > 0.

2p

x

t þ 2;

sðtÞ ¼ cos

2p

x

t þ 1;

2p 2p bðtÞ ¼ sin t þ 10 cos tþ2 ;

x

x

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Z. Nan et al. / Applied Mathematics and Computation 218 (2011) 4647–4650

Obviously, a(t) > 0, b(t) > 0,c(t) P 0 and s(t) P 0 are continuously differentiable x-periodic functions for t 2 R. By (2.9), we get that FðtÞ ¼ cos 2xp t þ 2 sin 2xp t 4; gðtÞ ¼ sin 2xp t þ 10 and f ðtÞ ¼ t cos 2xp t 1. Hence,

g; F 2 CP

6p

x

;x ;

jf ðtÞ f ðt þ xÞj ¼ x;

6p and thus f(t) is Lipschitz continuous with x when x 6 4p. pﬃﬃﬃ Lipschitz constant pﬃﬃﬃ On the other hand, since mðFÞ ¼ 5 4; MðFÞ ¼ 5 4; mðgÞ ¼ 9 and M(g) = 11, the conditions (2.3)–(2.5) are equivapﬃﬃ pﬃﬃ pﬃﬃﬃ lent to 58þ4 6 M 6 68 54xp 5 and m 6 410 5 respectively, which follows that Eq. (1.1) has a positive, continuously differ1 entiable x-periodic (x 6 4p) solution x(t), t 2 R by taking M = 1, x = 3 and m ¼ 10 and the solution x(t) is unique since m(g) > 1.

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