In this paper, under appropriate hypotheses, we have the existence of a solution semigroup of partial differential equations with delay operator. These equations are used to describe time–age-structured cell cycle model. We also prove that the solution semigroup is a frequently hypercyclic semigroup.

1. Introduction

Partial differential equations with delay have been studied for many years and arise in various applications, like biology, medicine, control theory, climate models, and many others. For example, the following mathematical problem involving a delayed nonlocal dynamical described by a particular partial differential equation as follows:

()

where u_t : σ ∈ [−τ, 0] → u(t + σ) ∈ X = L^p, τ > 0 and with an initial condition:

()

This equation is considered as a particular time–age-structured cell cycle model that was motivated by the biological process of hematological cells.

When c(x) = x and g(t, u, u_t) = λu, λ is the constant, then equation can be written as follows:

()

with the initial condition:

()

This equation is so called the Lasota equation (see, e.g., [1–5]) and the references cited therein.

This equation has been developed as a model of the dynamics of a self-reproducing cell population, such as the population of developing red blood cells (erythrocyte precursors). It also has been applicable to a conceptualization of abnormal blood cell production, such as leukemia. Although this equation is linear but the solution also has chaotic behavior and is studied by many authors

In this paper, we are interested in g(t, u, u_t) = Φu_t, Φ is a delay operator, and the first-order partial differential equation with delay is expressed as follows:

()

with an initial condition:

()

where Φ, w, and f see later hypotheses.

Here, we study the partial differential equations with delay in an abstract way.

Under the appropriate hypotheses that follow Bátkai and Piazzera’s [6] study, we can switch the linear delay differential equations to be an abstract Cauchy problem in an appropriate Banach space. That means, we can use a semigroup approach to deal with those equations.

We introduce the abstract delay equation and hypotheses first. List the standing hypotheses as follows:

(H1)
X is a Banach space.
(H2)
Usually B just needs to be a closed, densely defined, linear operator, but here we choose to let the equations have the frequently hypercyclic property.
(H3)
f ∈ L^p([−1, 0], X), 1 ≤ p < ∞.
(H4)
Φ ∈ W^1,p([−1, 0], X) → X is a bounded linear operator, called the delay operator.
(H5)
Θ : = X × L^p([−1, 0], X).

Under these hypotheses and for given elements x ∈ X and f ∈ L^p([−1, 0], X).

The following initial value problem will be called an abstract delay equation:

()

For a function u : [−τ, ∞) → X, t ≥ 0 and u_t : σ ∈ [−τ, 0] → u(t + σ) ∈ X.

Chaotic phenomena are interesting and abundant topics in different areas and attract many mathematicians (see, e.g., [1, 2, 7–10]). Alberto Conejero et al. [7] introduced different kinds of chaotic operators, such as Devaney chaos, frequent hypercyclicity, and so on. In [7] the authors provide a lot of examples. Here, we are interesting in a tape of C₀-semigroup so called frequently hypercyclic semigroup. Motivated by Birkhoff’s ergodic theorem, Bayart and Grivaux [11] introduced the notion of frequently hypercyclic operators trying to quantify the frequency with which an orbit meets the open set. This concept was extended to C₀-semigroup in [12].

When a semigroup {S(t)}_t≥0 is a frequently hypercyclic semigroup, then for every t₀ > 0 the operator S(t₀) is frequently hypercyclic, but the chaotic semigroup does not necessarily satisfy this condition. By the results of Bayart and Bermúdez [13], there are chaotic C₀-semigroup {S(t)}_t≥0 such that no single operator S(t) is chaotic and a C₀-semigroup {S(t)}_t≥0 containing a nonchaotic operator S(t₀), t₀ > 0 and a chaotic operator S(t₁) for some t₁ > 0. However, if a frequently hypercyclic semigroup {S(t)}_t≥0 satisfies frequently hypercyclic criterion, then S(t) is also chaotic for every t > 0 [12, Proposition 2.7]. This is one of the reason for us to study frequently hypercyclic semigroups.

The structure of this paper is following. In Section 2, we will introduce some useful terminologies and proposition. In Section 3, first, we prove that the semigroup generated by A₀ (see later) is frequently hypercyclic in Theorem 1. Then, we describe the solution semigroup {U(t)}_t≥0 using and purtubation theorem. Then by constructing the new set having frequently hypercyclic characteristics with positive lower density, we prove the purtubation of a frequently hypercyclic semigroup is also a frequently hypercyclic semigroup in Theorem 2. In Section 4, we give some examples.

2. Terminologies

First, we introduce some useful terminologies and propositions. We recall that the lower density of a measurable set M ⊂ R₊ is defined by the following equation:

()

where μ is the Lebesgue measure on R₊. A C₀-semigroup {S(t)}_t≥0 is called frequently hypercyclic on the sunspace M⊆X if there exists x ∈ M such that

for any nonempty open set U ⊂ M.

The lower density of a set

is defined by the following equation:

()

An operator S ∈ L(X) is said to be frequently hypercyclic on the sunspace M⊆X if there exists x ∈ M (called frequently hypercyclic vector) such that for any nonempty open set U ⊂ M, the set has positive lower density. If x ∈ X is a frequently hypercyclic vector for {S(t)}_t≥0, then, for every t > 0, the x is also a frequently hypercyclic vector for the operator S(t), for detail see Mangino and Peris’s [12] study.

Proposition 1. See [14, Proposition 2.1].

Let {S(t)}_t≥0 be a C₀-semigroup on a separable Banach space X. Then, the following conditions are equivalent:

(1)
{S(t)}_t≥0 is frequently hypercyclic,
(2)
for every t > 0, the operator S(t) is frequently hypercyclic, and
(3)
there exist t₀ > 0 such that the operator S(t₀) is frequently hypercyclic.

By Proposition 1, to prove S(t) is frequently hypercyclic just need to prove S(t₀) is frequently hypercyclic operator for some fixed t₀.

3. The Frequently Hypercyclic Semigroup {S(t)}_t≥0

In order to use the semigroup approach to deal with (DE)_p, we switch (DE)_p to an abstract Chachy problem and hope the solution of (DE)_p is equal to the solution of the abstract Chachy problem.

If u : [−τ, ∞) → X is a classical solution of (DE)_p, then the function:

()

is a classical solution of the abstract Chachy problem:

()

with

, where

denotes the distributional derivative domain:

()

Reversely, for every classical solution U of (ACP)_p, the function u is defined as follows:

()

Then, u is a classical solution of (DE)_p and (Π₂∘U)(t) = u_t for all t ≥ 0, where Π₁ is the canonical project from Θ onto X and similarly Π₂ is the canonical project from Θ to L^p([−1, 0], X).

As we want, we can transform the problem of solving the delay equation (DE)_p to solving (ACP)_p.

We write:

()

as the sum A = A₀ + C, where

and

From Bátkai and Piazzera [6], under appropriate assumption, we can see that A₀ is the generator of semigroup

and the solution semigroup

is given by the following equation:

()

where the semigroup {S(t)}_t≥0 is generated by operator

is the nilpotent left semigroup on L^p([−1, 0], X) and S_t : X → L^p([−1, 0], X) is defined by the following equation:

()

Remark 1. We consider the set Ω = {v ∈ X : v(0) = 0} is the subset X (as H1). The semigroup {S(t)}_t≥0 is generated by operator B is described by S(t)x(λ) = x(λe^−t) and is frequently hypercyclic on Ω, for detail see Hung and Chang’s [15] study.

Theorem 1. If the solution semigroup {S(t)}_t≥0 is frequently hypercyclic, then the solution semigroup is also frequently hypercyclic on .

Proof. First, we note that W is nonempty. For example, g(0)(s) = y(s) = s ∈ Ω, g(ω)(s) = se^−ω, and .

According to Proposition 1, to prove is frequently hypercyclic is equal to proving that is frequently hypercyclic operator for some fixed t₀ > 0.

For every nonempty open subset U = U₁ × U₂ ⊂ W, without losing the generality we can suppose U₁ = {z ∈ Ω : ‖z − y‖_X < ε₁}, U₂ = {h ∈ W^1,p([−1, 0], X) : ‖h − g‖_p < ε₂} for some ε₁, ε₂ > 0 and . We need to check for U the existence of frequently hypercyclic vector such that the set has positive lower density.

Since the semigroup {S(t)}_t≥0 generated by operator B is frequently hypercyclic, there exists a frequently hypercyclic vector x ∈ Ω such that for any neighborhood with radius r > 0 and center y = g(0), () then the set has positive lower density.

Instead of proving in U directly, we prove belongs to a subset of U. Now, we construct such subset of U. Choose r₁ > 0 such that and let U^′ = {z : ‖z − y‖_X < r₁}, it is clear that U^′ ⊂ U₁. Thus, corresponding U^′, there exists a set and E₁ has positive lower density.

Let U^″ = {h ∈ W^1,p([−1, 0], X) : h(0) = y^′ ∈ U^′}, then:

()

From above inequalties, we have U^″ ⊂ U₂ and U^′ × U^″ ⊂ U₁ × U₂.

Next, we need to find the set E, E₁ is natural suggestion and find a frequently hypercyclic vector, is natural suggestion for .

From Equation (15), we have the following form:

()

Then, we consider the second component of Equation (18), that is . From the perform of T₀, we have T₀(nt₀)f(−τ) = f(nt₀ − τ) = 0, where n > n₀ for some n₀ and we extent f to be zero out of [−τ, 0].

Then, Equation (18) becomes:

()

and we have the following equation:

()

Since S(nt₀)x ∈ U^′ in the first component of Equation (20) that is ‖S(nt₀)x − y‖ < r₁. Then, we move on the second component of Equation (20).

If g is the second component of

and n ∈ E₁, n > n₀, we get the following equation:

()

where we define an operator

and y^″ = S(nt₀)x ∈ U^′. In particular, when ω = y, we have

From the above, we have the following equation:

()

Since y^″ in U^′.

From the above conclusion, we get S(nt₀)x − y in

in U^″, and:

()

. We get

is a frequently hypercyclic vector and

has postive lower density. Therefore,

is the set that we wanted. This implies that

is a frequently hypercyclic operator on U and the theorem is proved.

Since we have the structure of , we can move on to study the existence of the solution semigroup of Equation (5), we add the condition (H6) and state as follows:

(H6) There exist some t₀ > 0 and 0 ≤ q < 1 such that:

()

Under the condition (H6), we consider the perturbation case. The existence of solution semigroup { U(t) }_t≥0 generated by A follows from Theorem 3.2 Miyadera–Voigt in [6].

The semigroup {U(t)}_t≥0 is given by the Dyson–Phillips series and denoted as

for all x ∈ X, t ≥ 0 and converging uniformly on compact subset of R⁺. Here, V ∈ L(C(R⁺, L(X))) is the abstract Volterra operator defined by the following equation:

()

and

()

Here, F ∈ C(R⁺, L(X)), z_n ∈ D(A₀) is a sequence such that .

Let F (t) to be , from Equation (25), we have and we also have is norm continuous for t ≥ 0 for detail see Bátkai and Piazzera’s [6] study. For any 0 < β < 1, we can choose some t₀ > 0 such that for all 0 ≤ t ≤ t₀.

We will use the frequently hypercyclic property of to prove U(t) is frequently hypercyclic and we write the result as Theorem 2.

Theorem 2. When (H1)–(H6) are satisfied the solution semigroup {U(t)}_t≥0 generated by A is frequently hypercyclic.

Proof. Without losing generality, we can suppose any nonempty open set as a neighborhood Z^′ with radius r and center at origin. To prove the frequently hypercyclic property of U(t₀), we need to find a frequently hypercyclic vector x and a set such that M has positive lower density for some particular operator U(t₀).

Since is frequently hypercyclic operator, there exists a frequently hypercyclic vector such that for a neighborhood with radius and center at origin, the set has positive lower density.

is a good candidate for the frequently hypercyclic vector of U(t₀). Then, we figure out the set which has frequently hypercyclic property.

When m = n + 1 for and let we have .

To estimate , we need to estimate and for all k ≥ 1.

From Equation (15), we have the following equation

()

We calculate the first component of

, as follows:

()

The second component of is .

We calculate their norms separately:

()

and

()

Combine the above estimates, we get the following equation:

()

Then, we consider for all n ≥ 1.

When n = 1, , we already know.

When n = 2, we get the following equation:

()

Moreover, for 0 ≤ t₁ ≤ t₀.

From Equation (25) and by induction, we have the estimate:

()

Consider

, we have the following equation:

()

So for all m ∈ M^′. This implies is a frequently hypercyclic vector and M^′ as we want. We finish the proof.

4. Examples

Example 1. We consider the following delay equation:

()

where w ∈ X, f ∈ L^p([−1, 0], X), 1 ≤ p < ∞.

Let Φ : C([−1, 0], X) → X be the bound linear operator given by Riemann–Stieltjes integral:

()

Since W^1,p([−1, 0], X) is continuously embedded in C([−1, 0], X), Φ defines a bounded operator from W^1,p([−1, 0], X) to X. So, we can translate Equation (35) to (DE)_p form.

To ensure the existence of the solution semigroup U(t) of (DE)_p, we need to check (H6) is satisfied.

For 0 < t < 1, we obtain the following equation:

()

where

. Finally, we conclude that

Now choose t₀ small enough such that , then condition (H6) is satisfied with . Apply Theorem 2, we have {U(t)}_t≥0 is frequently hypercyclic.

Example 2. The important special case of Example 1 is the following form:

()

where f ∈ W^1,p([−1, 0], X), B_k ∈ L(X) and h_k ∈ [−1, 0] for k = 0, 1, …, n.

The operator Φ is defined by the following equation:

()

The result of Example 1 is coming from we can choose q small enough when Φ is defined as Equation (36). Similar results for Example 2 were proved by Kunisch and Schappacher [16].

Remark 2. can extend to , where c(x) is a continuous function and satisfied the following equation:

()

The solution semigroup { S(t) }_t≥0 generated by is also frequently hypercyclic, for details see Hung and Chang’s [15] study.

5. Conclusion

By finding a frequently hypercyclic vector and the set has frequently hypercyclic characteristic with positive lower density, we have the results in Theorems 1 and 2. Finally, we give two examples satisfying Theorem 2, then they have the frequently hypercyclic solution semigroup.

Conflicts of Interest

The author declares that there are no conflicts of interest.

Open Research

Data Availability

The data used to support the findings of this study are included within the article.

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Frequently Hypercyclic Semigroup Generated by Some Partial Differential Equations with Delay Operator

Abstract

1. Introduction

2. Terminologies

3. The Frequently Hypercyclic Semigroup {S(t)}_t≥0

4. Examples

5. Conclusion

Conflicts of Interest

Open Research

Data Availability

References

References

Information

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Frequently Hypercyclic Semigroup Generated by Some Partial Differential Equations with Delay Operator

Abstract

1. Introduction

2. Terminologies

3. The Frequently Hypercyclic Semigroup {S(t)}t≥0

4. Examples

5. Conclusion

Conflicts of Interest

Open Research

Data Availability

References

References

Related

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3. The Frequently Hypercyclic Semigroup {S(t)}_t≥0