Research Article | Open Access
Ruyun Ma, Yanqiong Lu, "One-Signed Periodic Solutions of First-Order Functional Differential Equations with a Parameter", Abstract and Applied Analysis, vol. 2011, Article ID 843292, 11 pages, 2011. https://doi.org/10.1155/2011/843292
One-Signed Periodic Solutions of First-Order Functional Differential Equations with a Parameter
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 .
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  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  (iii)the model for the survival of red blood cells in an animal proposed by Wazewska-Czyzewska and Lasota in 
Recently, Cheng and Zhang  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  is not sharp, and the main results in  give no any information about the global structure of the set of positive periodic solutions. Moreover, satisfied (H1) in , 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 .
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 .
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.
(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 .
Proof of Theorem 2.3. Suppose on the contrary that there exists such that either
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 3.3. 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.
The paper is supported by the NSFC (no. 11061030), the Fundamental Research Funds for the Gansu Universities.
- W. S. Gurney, S. P. Blythe, and R. N. Nisbet, “Nicholsons blowflies revisited,” Nature, vol. 287, pp. 17–21, 1980.
- M. C. Mackey and L. Glass, “Oscillation and chaos in physiological control systems,” Science, vol. 197, no. 4300, pp. 287–289, 1977.
- M. Wazewska-Czyzewska and A. Lasota, “Mathematical problems of the dynamics of a system of red blood cells,” Matematyka Stosowana, vol. 6, pp. 23–40, 1976.
- S. N. Chow, “Existence of periodic solutions of autonomous functional differential equations,” Journal of Differential Equations, vol. 15, pp. 350–378, 1974.
- H. I. Freedman and J. Wu, “Periodic solutions of single-species models with periodic delay,” SIAM Journal on Mathematical Analysis, vol. 23, no. 3, pp. 689–701, 1992.
- J. Wu and Z. Wang, “Positive periodic solutions of second-order nonlinear differential systems with two parameters,” Computers & Mathematics with Applications, vol. 56, no. 1, pp. 43–54, 2008.
- S. Cheng and G. Zhang, “Existence of positive periodic solutions for non-autonomous functional differential equations,” Electronic Journal of Differential Equations, vol. 59, pp. 1–8, 2001.
- D. Jiang and J. Wei, “Existence of positive periodic solutions for nonautonomous delay differential equations,” Chinese Annals of Mathematics, vol. 20, no. 6, pp. 715–720, 1999 (Chinese).
- D. Ye, M. Fan, and H. Wang, “Periodic solutions for scalar functional differential equations,” Nonlinear Analysis, vol. 62, no. 7, pp. 1157–1181, 2005.
- Y. Li, X. Fan, and L. Zhao, “Positive periodic solutions of functional differential equations with impulses and a parameter,” Computers & Mathematics with Applications, vol. 56, no. 10, pp. 2556–2560, 2008.
- H. Wang, “Positive periodic solutions of functional differential equations,” Journal of Differential Equations, vol. 202, no. 2, pp. 354–366, 2004.
- Z. Jin and H. Wang, “A note on positive periodic solutions of delayed differential equations,” Applied Mathematics Letters, vol. 23, no. 5, pp. 581–584, 2010.
- K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
- P. H. Rabinowitz, “On bifurcation from infinity,” Journal of Differential Equations, vol. 14, pp. 462–475, 1973.
- P. H. Rabinowitz, “Some global results for nonlinear eigenvalue problems,” Journal of Functional Analysis, vol. 7, pp. 487–513, 1971.
- A. Ambrosetti and P. Hess, “Positive solutions of asymptotically linear elliptic eigenvalue problems,” Journal of Mathematical Analysis and Applications, vol. 73, no. 2, pp. 411–422, 1980.
- R. Ma, “Global behavior of the components of nodal solutions of asymptotically linear eigenvalue problems,” Applied Mathematics Letters, vol. 21, no. 7, pp. 754–760, 2008.
- Y. An and R. Ma, “Global behavior of the components for the second order m-point boundary value problems,” Boundary Value Problems, vol. 2008, Article ID 254593, 10 pages, 2008.
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.