On a mathematical model of age-cycle length structured cell population with non-compact boundary conditions

Boulanouar, Mohamed

doi:10.1002/mma.3206

Cited by 6 publications

(27 citation statements)

References 9 publications

(20 reference statements)

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“…The second one follows from . Indeed, we have estimated the type ω 0 ( U α , β ), of the full semigroup U α , β =( U α , β ( t )) t ≥0 , by proving

μ (a, l) < ω 0 U α, β

For more explanation, we refer to and the references therein.…”

Section: Introductionmentioning

confidence: 85%

“…The aim of this section can now be announced as follows: Theorem Suppose that

((), {\overset{boldA}{false}}_{boldk}^{bold-italic 1})

((), {boldA}_{boldk}^{bold-italic 2})

((), {boldA}_{bold-italicμ}^{bold-italic 1})

((), {boldA}_{bold-italicμ}^{bold-italic 2})

, and

((), {\overset{boldA}{false}}_{bold-italicη}^{bold-italic 1})

hold and let 0≤ α < 1 and β ≥0 be such that

((), {boldA}_{boldk}^{' bold-italic 1})

holds. Then

ω ess ((), U α, β) \leq - μ true1

where,

μ true1 : = (a, l) \in Ω μ (a, l)

Proof First, since together with , we infer that

ω ess ((), U α, β) = ω ess ((), V α, 0) \leq ω 0 ((), V α, β) .

Next, using , (3.4)](with β = 0) and , Corollary 4.7], we get that

0 \leq V α, 0 (t) \leq e^{- μ true1 t} T

…”

Section: Essential Type Of Full Semigroupmentioning

confidence: 93%

“…Let then

{\hat{A}}_{k}^{1}

be the following assumption:

\begin{matrix} ({\hat{A}}_{k}^{1}) : & k \infty (\cdot) : = l^{'} \geq 0 |k (\cdot, l^{'})| \in Y 1 . \end{matrix}

Remark

((), {\overset{bold-italicA}{false}}_{boldk}^{bold-italic 1})

is stronger than

((), {boldA}_{boldk}^{bold-italic 1})

because of

\overset{true¯}{κ} \infty = l^{'} \geq 0 \int_{0}^{\infty} |k (l, l^{'})| d l \leq \int_{0}^{\infty} k \infty (l) d l = (∥∥, k \infty) Y 1 < \infty .

Accordingly, all results of and ) related to

((), {boldA}_{boldk}^{bold-italic 1})

are still true whenever

((), {\overset{boldA}{false}}_{boldk}^{bold-italic 1})

holds true. Remark Suppose that

((), \overset{boldA}{false})

…”

Section: Essential Type Of Full Semigroupmentioning

confidence: 99%

“…If

((), {boldA}_{boldk}^{' bold-italic 1})

holds, then T α , β generates, on L 1 , a C 0 ‐semigroup T α , β =( T α , β ( t )) t ≥0 . Furthermore, If

((), {boldA}_{bold-italicμ}^{bold-italic 1})

holds, then V α , β generates a C 0 ‐semigroup V α , β =( V α , β ( t )) t ≥0 . If

((), {boldA}_{bold-italicμ}^{bold-italic 1})

and

((), {boldA}_{bold-italicη}^{bold-italic 1})

hold, then U α , β generates a C 0 ‐semigroup U α , β =( U α , β ( t )) t ≥0 . If

((), {boldA}_{bold-italicμ}^{bold-italic 1})

((), {boldA}_{bold-italicμ}^{bold-italic 2})

, and ( A μ − η )hold, then the C 0 ‐semigroup U α , β =( U α , β ( t )) t ≥0 is contractive provided that

α + β \overset{true¯}{κ} \infty < 1

. Proof T α , β is a generator because of , Theorem 4.4]. The points (1), (2), and (3), respectively, follow from , Theorem 5.1], , Theorem 6.2], and , Theorem 6.3].…”

Section: Full Model (11)–(12)mentioning

confidence: 99%

“…Before we start the second part of the aim of this section, we put

\begin{matrix} ((), {\overset{boldA}{false}}_{bold-italicη}^{bold-italic 1}) : & \overset{η}{false} : = \int Ω (a^{'}, l^{'}) \in Ω (||, η (a, l, a^{'}, l^{'})) normald a normald l < \infty . \end{matrix}

Remark We notice that

((), {\overset{boldA}{false}}_{bold-italicη}^{bold-italic 1})

is stronger than

((), {boldA}_{bold-italicη}^{bold-italic 1})

. Indeed,

\overset{true¯}{η} = (a^{'}, l^{'}) \in normalΩ \int normalΩ |η (a, l, a^{'}, l^{'})| d a d l \leq \int_{0}^{\infty} (a^{'}, l^{'}) \in normalΩ |η (a, l, a^{'}, l^{'})| d a d l = \hat{η} < \infty .

Therefore, all results of and related to

((), {boldA}_{bold-italicη}^{bold-italic 1})

are still true whenever

((), {\overset{boldA}{false}}_{bold-italicη}^{bold-italic 1})

…”

Section: Essential Type Of Full Semigroupmentioning

confidence: 99%

See 4 more Smart Citations

On a mathematical model of age‐cycle length structured cell population with non‐compact boundary conditions (III)

Boulanouar¹

2015

Math Methods in App Sciences

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This work is a natural continuation of two other works in which a mathematical model has been studied. This model is based on age‐cycle length structured cell population. The cellular mitosis is mathematically described by a non‐compact boundary condition. We investigate the asymptotic behavior of the generated semigroup, and we prove that the cell population possesses Asynchronous Growth Property. Copyright © 2015 John Wiley & Sons, Ltd.

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“…The second one follows from . Indeed, we have estimated the type ω 0 ( U α , β ), of the full semigroup U α , β =( U α , β ( t )) t ≥0 , by proving

μ (a, l) < ω 0 U α, β

For more explanation, we refer to and the references therein.…”

Section: Introductionmentioning

confidence: 85%

“…The aim of this section can now be announced as follows: Theorem Suppose that

((), {\overset{boldA}{false}}_{boldk}^{bold-italic 1})

((), {boldA}_{boldk}^{bold-italic 2})

((), {boldA}_{bold-italicμ}^{bold-italic 1})

((), {boldA}_{bold-italicμ}^{bold-italic 2})

, and

((), {\overset{boldA}{false}}_{bold-italicη}^{bold-italic 1})

hold and let 0≤ α < 1 and β ≥0 be such that

((), {boldA}_{boldk}^{' bold-italic 1})

holds. Then

ω ess ((), U α, β) \leq - μ true1

where,

μ true1 : = (a, l) \in Ω μ (a, l)

Proof First, since together with , we infer that

ω ess ((), U α, β) = ω ess ((), V α, 0) \leq ω 0 ((), V α, β) .

Next, using , (3.4)](with β = 0) and , Corollary 4.7], we get that

0 \leq V α, 0 (t) \leq e^{- μ true1 t} T

…”

Section: Essential Type Of Full Semigroupmentioning

confidence: 93%

“…Let then

{\hat{A}}_{k}^{1}

be the following assumption:

\begin{matrix} ({\hat{A}}_{k}^{1}) : & k \infty (\cdot) : = l^{'} \geq 0 |k (\cdot, l^{'})| \in Y 1 . \end{matrix}

Remark

((), {\overset{bold-italicA}{false}}_{boldk}^{bold-italic 1})

is stronger than

((), {boldA}_{boldk}^{bold-italic 1})

because of

\overset{true¯}{κ} \infty = l^{'} \geq 0 \int_{0}^{\infty} |k (l, l^{'})| d l \leq \int_{0}^{\infty} k \infty (l) d l = (∥∥, k \infty) Y 1 < \infty .

Accordingly, all results of and ) related to

((), {boldA}_{boldk}^{bold-italic 1})

are still true whenever

((), {\overset{boldA}{false}}_{boldk}^{bold-italic 1})

holds true. Remark Suppose that

((), \overset{boldA}{false})

…”

Section: Essential Type Of Full Semigroupmentioning

confidence: 99%

“…If

((), {boldA}_{boldk}^{' bold-italic 1})

holds, then T α , β generates, on L 1 , a C 0 ‐semigroup T α , β =( T α , β ( t )) t ≥0 . Furthermore, If

((), {boldA}_{bold-italicμ}^{bold-italic 1})

holds, then V α , β generates a C 0 ‐semigroup V α , β =( V α , β ( t )) t ≥0 . If

((), {boldA}_{bold-italicμ}^{bold-italic 1})

and

((), {boldA}_{bold-italicη}^{bold-italic 1})

hold, then U α , β generates a C 0 ‐semigroup U α , β =( U α , β ( t )) t ≥0 . If

((), {boldA}_{bold-italicμ}^{bold-italic 1})

((), {boldA}_{bold-italicμ}^{bold-italic 2})

, and ( A μ − η )hold, then the C 0 ‐semigroup U α , β =( U α , β ( t )) t ≥0 is contractive provided that

α + β \overset{true¯}{κ} \infty < 1

. Proof T α , β is a generator because of , Theorem 4.4]. The points (1), (2), and (3), respectively, follow from , Theorem 5.1], , Theorem 6.2], and , Theorem 6.3].…”

Section: Full Model (11)–(12)mentioning

confidence: 99%

“…Before we start the second part of the aim of this section, we put

\begin{matrix} ((), {\overset{boldA}{false}}_{bold-italicη}^{bold-italic 1}) : & \overset{η}{false} : = \int Ω (a^{'}, l^{'}) \in Ω (||, η (a, l, a^{'}, l^{'})) normald a normald l < \infty . \end{matrix}

Remark We notice that

((), {\overset{boldA}{false}}_{bold-italicη}^{bold-italic 1})

is stronger than

((), {boldA}_{bold-italicη}^{bold-italic 1})

. Indeed,

\overset{true¯}{η} = (a^{'}, l^{'}) \in normalΩ \int normalΩ |η (a, l, a^{'}, l^{'})| d a d l \leq \int_{0}^{\infty} (a^{'}, l^{'}) \in normalΩ |η (a, l, a^{'}, l^{'})| d a d l = \hat{η} < \infty .

Therefore, all results of and related to

((), {boldA}_{bold-italicη}^{bold-italic 1})

are still true whenever

((), {\overset{boldA}{false}}_{bold-italicη}^{bold-italic 1})

…”

Section: Essential Type Of Full Semigroupmentioning

confidence: 99%

See 3 more Smart Citations

On a mathematical model of age‐cycle length structured cell population with non‐compact boundary conditions (III)

Boulanouar¹

2015

Math Methods in App Sciences

Self Cite

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show abstract

On a mathematical model of age–cycle length structured cell population with non‐compact boundary conditions (II)

Boulanouar¹

2015

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

This work is a natural continuation of an earlier one in which a mathematical model has been studied. This model is based on an age–cycle length structured cell population. The cellular mitosis is mathematically described by a non‐compact boundary condition. We investigate the spectral properties of the generated semigroup, and we give an explicit estimation of its type. Copyright © 2015 John Wiley & Sons, Ltd.

show abstract

An $$L^{p}$$-Approach to the Well-Posedness of Transport Equations Associated to a Regular Field: Part II

Arlotti

Lods

2019

Mediterr. J. Math.

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On a mathematical model of age-cycle length structured cell population with non-compact boundary conditions

Cited by 6 publications

References 9 publications

On a mathematical model of age‐cycle length structured cell population with non‐compact boundary conditions (III)

On a mathematical model of age‐cycle length structured cell population with non‐compact boundary conditions (III)

On a mathematical model of age–cycle length structured cell population with non‐compact boundary conditions (II)

An $$L^{p}$$-Approach to the Well-Posedness of Transport Equations Associated to a Regular Field: Part II

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