We study a particular first-order partial differential equation which arisen from a biologic model. We found that the solution semigroup of this partial differential equation is a frequently hypercyclic semigroup. Furthermore, we show that it satisfies the frequently hypercyclic criterion, and hence the solution semigroup is also a chaotic semigroup.

1. Introduction

The first-order partial differential equations appear in different branches of science and succeed to demonstrate events of nature. In this paper we focus on the particular form of partial differential equation with an initial condition where and are given continuous function defined on with The function in (1) is a given continuously differentiable function satisfying where ,  are nonnegative constants.

This equation is usually used to represent the models of age-structured populations. Populations of replicating and maturing cells are age structured in that the replenishment of new individuals into the population depends on the density of a cohort of older individuals. Many biological populations have similar models; the Lasota equation is the famous example which is an application of (1). It can be written as with the initial condition

Equation (6) has been developed as a model for the dynamics of a self-reproducing cell population, such as the population of developing red blood cell (erythrocyte precursors). It has also been applied to a conceptualization of abnormal blood cell production such as leukemia. Although this equation is linear, the solution has chaotic behavior. Recently, there has been many authors studying this problem (e.g., [120]).

A lot of researchers are interested in the chaotic behavior of differential equation and chaotic -semigroup. In this paper, we would like to study a type of semigroup, so-called frequently hypercyclic semigroup. The frequently hypercyclic semigroup has some restricted property to the chaotic -semigroup.

Motivated by Birkhoff's ergodic Theorem, Bayart and Grivaux introduced the notion of frequently hypercyclic operators in [5]. In that paper, they quantified the frequency with which an orbit meets the open sets, and several examples of frequently hypercyclic operator are given. Moreover, the authors also give an operator which is hypercyclic but not frequently hypercyclic. Mangino and Peris extended the concept of single operator to the continuous case and defined the frequently hypercyclic semigroup in [17]. When a semigroup is a frequently hypercyclic semigroup, then for every the operator is frequently hypercyclic, but the chaotic semigroup does not necessary satisfy this condition. By recent results of Bayart and Bermúdez [6], there are chaotic -semigroup such that no single operator is chaotic and a -semigroup containing a nonchaotic operator ,  and a chaotic operator for some .

However, if a frequently hypercyclic semigroup satisfies frequently hypercyclic criterion, then is also chaotic for every [17, Proposition 2.7]. That is one of the reasons for us to study frequently hypercyclic semigroup.

The arrangement of this paper as follows. we will find the solution semigroup of (1) and some prosperities of it in Section 2. We prove that the solution semigroup is a frequently hypercyclic semigroup and some useful propositions in Section 3. In Section 4, we find the set of period points of the solution semigroup of (1) and prove this solution semigroup is chaotic directly.

2. The Solution Semigroup of (1)

Using the method of characteristics to find the unique solution of problem (1)(3) is equivalent to solving the following two initial value problems: Under condition (3), there exists a unique solution of the initial value problem (8). Denote it by for and . For simplicity, we denote for and , where is the first point such that . We admit that .

For describing , we set From (3), it follows that is strictly decreasing and hence exist. Moreover, for , . In particular, From (11), is nonnegative and nondecreasing in . Furthermore, it is increasing and positive for .

Secondly, we are going to solve the initial value problem (9). According to conditions (4) and (5), the unique solution of (9) exists for all and .

The characteristic of (1) is given by . Thus, for each pair of (1) and (3), we have that where . Setting , we obtain , and hence Formula (14) shows the existence and uniqueness of solution of (1) and (3).

Let be the space of nonnegative and continuous functions defined on and the subspace of which contains all continuously differentiable functions. Because the functions and do not depend explicitly on , formula (14) can define a semigroup on of the form The space with the supremum norm is a Banach space. Since the transformations are continuous operators and is a dense subspace of , hence, is also a semigroup on .

Let with the metric where is the supremum norm on . Then, is a complete metric space. For convenience, we will denote it as .

Lemma 1. The space is invariant with respect to the semigroup given by (15).

Proof. We need only to show that for and . By (3), (11), and (15), we have and . From (5), is the unique solution of (9) for initial value . This implies that and the proof of this lemma is completed.

Since is complete separable metric space and for (Lemma 1), the semigroup can be considered on the space .

For proving Lemma 2, given any with , we will define a transformation on . From (4) and (5), it follows that is a bijection. Thus, there exists a unique function satisfying For every and , we can define a transformation by the formula for every .

Lemma 2. There exists a closed subset which is invariant with respect to .

Proof. Let , where will be determined later. By the differentiation of (18) with respect to and the fact that is a bijection, we obtain Using (9), we have and consequently for , , and . Since for and from (18), we get for .
For a given , by mean-value theorem, we have that
For estimate , according to (22) and (23), we need to estimate and .
From (5) there exist numbers and such that By the fact, , and , we may choose such that for , , and is chosen in (24). From (22) and (24), we have Using (19), (23), and (26), it is easy to see , and the proof of this lemma is complete.

From the properties of and , we have the following lemma.

Lemma 3. For every ,

Proof. Using formula (15), (19), and the definition of , we have for , , and .
Pluging into (28), we have Furthermore, since and , for . By (18) we have for every , and the assertion of this lemma is established now.

3. The Frequently Hypercycle Property of

At beginning of this section, we introduce some terminologies and propositions which will be used later. According to Devaney's definition, a semigroup defined in a metric space is chaotic if it has following properties:(1) has a property of sensitive dependence on initial conditions; that is, there is a positive real number such that for every point and every there is and , such that ;(2) is transitive; that is, for all nonempty open subsets , there is such that ;(3)the set of periodic points of is dense in .

We recall that the lower density of a measurable set is defined by where is the Lebesgue measure on . A -semigroup is called frequently hypercyclic if there exists such that The lower density of a set is defined by An operator is said to be frequently hypercyclic if there exists (called frequently hypercyclic vector) such that, for any non-empty open set , the set has positive lower density. In [6], Bayart and Bermúdez proved that if is a frequently hypercyclic vector for , then for the is a frequently hypercyclic vector for the operator .

Proposition 4 (see [17, Proposition 2.1]). Let be a -semigroup on a separable Banach space . Then, the following conditions are equivalent:(1) is frequently hypercyclic;(2)for every the operator is frequently hypercyclic;(3)there exist such that the operator is frequently hypercyclic.

According to this proposition one wants to show that a semigroup is frequently hypercyclic just needed to check the operator for some fixed is frequently hypercyclic. The following proposition described the sufficient condition for frequently hypercyclic operator. It is also called frequently hypercyclic criterion. Frequently hypercyclic criterion builds the relation between frequently hypercyclic semigroup and chaotic semigroup.

Proposition 5. Let be a continuous operator on a separable Banach space . If there exist a dense subset and a map satisfying(1); (2) is unconditionally convergent for all ;(3) is unconditionally convergent for all ;then is frequently hypercyclic.

The proof of this proposition can be found in [3].

Theorem 6. Suppose that is the closed set in Lemma 2; then the solution semigroup in Section 2 is frequently hypercyclic on .

Proof. To show the conclusion of this theorem to be true, we are planning to apply Proposition 5.
According to Proposition 4, to show that is frequently hypercyclic, we need only to prove that is frequently hypercyclic operator for some fixed .
For this purpose, we defined an operator on by It is obvious that the operator defined by (19) is a good candidate for checking condition (1) of Proposition 5. In fact, by Lemma 3 we have and condition (1) of Proposition 5 is satisfied.
For checking condition (2) of Proposition 5, we are going to find a dense subset of . The characteristic functions , , , are candidates. However, does not belong to . So we need to modify . For a suitable small positive constant and , , we define for , for , and smooth connecting the graph of for such that . We choose some sequences and , , , such that and as . Let we have and for .
Let where were defined as in (35)}. It is clear that is dense in , and hence is dense in also.
According to the definitions of and , for , we have that for and , provided is the solution semigroup. From (11) and the fact that is strictly decreasing, for any fixed , there exists such that for all , , we have that and for . This implies that From the previously mentioned, there have been only finite many such that , so is unconditionally convergent. This proves that condition (2) of Proposition 5 is satisfied.
From (19), (27), and the definitions of and , for , we have where for every . In fact, for is equal to for .
Using similar estimation of (23), we have that This implies that , for every , and hence . This shows that is unconditionally convergent. Thus, condition (3) of Proposition 5 is also satisfied. We complete the proof of this theorem now.

Although from frequently hypercyclic criterion we can get is chaotic for every , we can directly prove the conclusion without using frequently hypercyclic criterion and we state in next section.

4. The Chaotic Property of

From the definition of chaotic semigroup, we need to find the set of period points of for some which is dense in . For this purpose, we first find a special function such that We restrict on ; then by (22), (23), and (24), we have Since , this implies that is contraction on , and hence there exists a such that .

For any , we define as From (43), is also belonging to . Combining (41) and (43), we obtain Using (27), (44), and the fact that for , we have In other words, is a periodic point of .

Remark 7. It is not hard to prove that the set of periodic points of (45) is dense in and the solution semigroup defined by (15) is transitive in . As proved by Bayart and Matheron [4], the sensitive dependence of the -semigroup on initial conditions in the sense of Guckenheimer appears immediately from its transitivity and density of the set of its periodic points. This is expressed by the following corollary.

Corollary 8. The solution -semigroup defined by (15) is chaotic in .

Finally, we demonstrate two simple examples. The first one is

where are constants, and with the initial condition

It is easy to see that condition (3) is satisfied.

In fact, the solution semigroup of (46) is given by From the previous results, we know that is not only a frequently hypercyclic semigroup but also chaotic.

Another example is the Lasota equation (6) in Section 1. It is easy to see that condition (3) is satisfied. The solution semigroup of (6) is frequently hypercyclic and chaotic as well.


The author would like to thank the referee for useful suggestions for this research work.