Abstract and Applied AnalysisVolume 2011 (2011), Article ID 843292, 11 pagesdoi:10.1155/2011/843292

Research Article

One-Signed Periodic Solutions of First-Order Functional Differential Equations with a Parameter

Ruyun Ma and Yanqiong Lu

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 8 June 2011; Accepted 29 August 2011

Academic Editor: Ferhan M. Atici

Copyright © 2011 Ruyun Ma and Yanqiong Lu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study one-signed periodic solutions of the first-order functional differential equation by using global bifurcation techniques. Where are periodic functions with , , is a continuous -periodic function, and is a parameter. and there exist two constants such that , for and for .

1. Introduction

In recent years, there has been considerable interest in the existence of periodic solutions of the following equation: where are -periodic functions, and , is a continuous -periodic function, is a parameter. (1.1) has been proposed as a model for a variety of physiological processes and conditions including production of blood cells, respiration, and cardiac arrhythmias; see, for example, [1–12] and the references therein. Roughly speaking, represents the number of adult (sexually mature) members in a population at time is the per capita death rate, and is the rate at which new members are recruited into the population at time ( is the age at which members mature, and it is assumed that the birth rate at a given time depends only on the adult population size). The most famous models of this type are(i)the Nicholson's blowflies equation proposed in [1] to explain the oscillatory population fluctuations observed by A. J. Nicholson in 1957 in his studies of the sheep blowfly Lucilia cuprina: (ii)the model for blood cell populations proposed by Mackey and Glass in [2] (iii)the model for the survival of red blood cells in an animal proposed by Wazewska-Czyzewska and Lasota in [3]

Recently, Cheng and Zhang [7] studied the existence of positive -periodic solutions of the functional equation (1.1) under the assumptions:(H1), and for ;(H2) are periodic functions, , , is a -periodic function;(H3)there exist such that They proved the following.

Theorem A. Assume (H1)–(H3)hold. Then for each satisfying equation (1.1) has a positive periodic solution, where

However, the condition used in [7] is not sharp, and the main results in [7] give no any information about the global structure of the set of positive periodic solutions. Moreover, satisfied (H1) in [7], so a natural question is what would happen if is allowed to have some zeros in ? The purpose of this work is to study the global behavior of the components of one-signed solutions of (1.1) under the condition(H4); there exist two constants such that , for , and for .

The rest of this paper is organized as follows. In Section 2, we give some notations and the main results. Section 3 is devoted to proving the main results.

2. Statement of the Main Results

Let with the norm Then is a Banach space. Let be the Banach space with the norm .

It is well known that (1.1) is equivalent to where Notice that , we have where , and .

Define that is a cone in by It is not difficult to prove that and is completely continuous.

Let us consider the spectrum of the linear eigenvalue problem

Lemma 2.1. Assume that (H2) holds. Then the linear problem (2.7) has a unique eigenvalue , which is positive and simple, and the corresponding eigenfunction is of one sign.

Proof. It is a direct consequence of the Krein-Rutman Theorem [13, Theorem 19.3].

In the rest of the paper, we always assume that

Define by setting Then is completely continuous.

Let be such that Clearly, Let us consider as a bifurcation problem from the trivial solution and as a bifurcation problem from infinity. We note that (2.12) and (2.13) are the same and each of them is equivalent to (1.1).

Let under the product topology. We add the points to our space . Let denote the set of positive functions in and , and . They are disjoint and open in . Finally, let and .

Remark 2.2. It is worth remaking that if is a nontrivial solution of (1.1) and , and satisfy (H2)–(H4), then for some . To see this, define Thus (1.1) is equivalent to Obviously, satisfies (H2). From Lemma 2.1, the nontrivial solution for some .

The result of Rabinowitz [14] for (2.12) can be stated as follows: for each , there exists a continuum of solutions of (2.12) joining to infinity, and .

The result of Rabinowitz [15] for (2.13) can be stated as follows: for each , there exists a continuum of solutions of (2.13) meeting , and .

Our main result is the following.

Theorem 2.3. Assume (H2)–(H4) hold. Moreover, suppose that (H5) satisfies the Lipschitz condition in . Then (i)for , (ii)for , we have that either or

Corollary 2.4. Let (H2)–(H5) hold. Then (i)if , then (1.1) has at least two solutions and , such that is positive on and is negative on ;(ii)if , then (1.1) has at least four solutions , , , and , such that , are positive on and , are negative on .

Corollary 2.5. Let (H2)–(H5) hold. Then (i)if , then (1.1) has at least two solutions and , such that is positive on and is negative on ;(ii)if , then (1.1) has at least four solutions , , , and , such that , are positive on and , are negative on .

3. Proof of the Main Results

To prove Theorem 2.3, we give a Proposition.

Proposition 3.1. (i) The first-order boundary value problem has a unique solution for all if and only if . (ii) Assume that is a solution of (3.1). If and on any subinterval of , then on .

Proof. (i) The equation has a solution , where is a constant. If is a nontrivial solution, then by , we can get that .On the other hand, from , we can get that has a nontrivial solution , where .(ii) We claim that . Suppose on the contrary that there exists , such that ; it is not difficult to compute that has a solution Since , we have If , then there exists a neighborhood of , such that on . Thus, ; this contradicts with .If , then there exists a neighborhood of , such that on . By using a similar way, we can prove that , which also contradicts with .Hence on . Moreover, it follows that that is, Thus , that is .

Next, we prove Theorem 2.3 and Corollaries 2.4 and 2.5.

Proof of Theorem 2.3. Suppose on the contrary that there exists such that either or We divide the proof into two cases.Case 1 (). In this case, we know that Let us consider the functional differential equation By (H2), (H4) and (H5), there exists such that is strictly increasing on for . Then (3.10) can be rewritten to the form and since , Subtracting, we get That is, From Proposition 3.1, we deduce that , which contradicts with that . Hence, Case 2 (). In this case, we know that Let us consider (3.10); by (H2), (H4), and (H5), there exists such that is strictly increasing in for . Then and since , Subtracting, we get That is From Proposition 3.1, we deduce that , this contradicts with that . Therefore,

Proof of Corollaries 2.4 and 2.5. Since boundary value problem has a unique solution , we get Take as an interval such that and as a neighborhood of whose projection on lies in and whose projection on is bounded away from 0. Then by [15, Theorem 1.6, and Corollary 1.8], we have that for each , either(1) is bounded in in which case meets , or(2) is unbounded.Moreover, if (1) occurs and has a bounded projection on , then meets , where is another eigenvalue of (2.7).Obviously, Theorem 2.3 (ii) implies that (1) does not occur. So is unbounded.Remark 2.2 guarantees that is a component of solutions of (2.12) in which meets , and consequently is unbounded. Thus Similarly, we get By Theorem 2.3, for any ,  (3.26) and (2.12) imply that which means that the sets and are bounded for any fixed . This together with the fact that and join to infinity yields, respectively, that Combining (3.24), (3.25), and (3.28), we conclude the desired results.

Remark 3.2. The methods used in the proof of Theorem 2.3, Corollaries 2.4, and 2.5 have been used in the study of other kinds of boundary value problems; see [16–18] and the references therein.

Remark 3.3. The conditions in Corollaries 2.4 and 2.5 are sharp. Let us take Let and consider problem It is easy to see that . Since the conditions of Corollaries 2.4 and 2.5 are not valid. In this case, (3.31) has no nontrivial solution. In fact, if is a nontrivial solution of (3.31), then which is a contradiction.

Acknowledgment

The paper is supported by the NSFC (no. 11061030), the Fundamental Research Funds for the Gansu Universities.

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