# Positive semigroups and perturbations of boundary conditions

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## Abstract

We present a generation theorem for positive semigroups on an \(L^1\) space. It provides sufficient conditions for the existence of positive and integrable solutions of initial-boundary value problems. An application to a two-phase cell cycle model is given.

## Keywords

Positive semigroup Perturbation of boundary conditions Steady state Cell cycle models## Mathematics Subject Classification

47B65 47H07 47D06 92C40## 1 Introduction

*A*is such that Eq. (1) with \(\varPsi =0\) generates a

*positive semigroup*on \(L^1\), i.e., a \(C_0\)-semigroup of positive operators on \(L^1\). We present sufficient conditions for the operators \(A, \varPsi _0,\) and \(\varPsi \) under which there is a unique positive semigroup on \(L^1\) providing solutions of the initial-boundary value problem (1). For a general theory of positive semigroups and their applications we refer the reader to [4, 7, 11, 14, 34]. An overview of different approaches used in studying initial-boundary value problems is presented in [13].

Our result is an extension of Greiner’s [19] by considering unbounded \(\varPsi \) and positive semigroups. Unbounded perturbations of the boundary conditions of a generator were studied recently in [1, 2] by using extrapolated spaces and various admissibility conditions. In the proof of our perturbation theorem we apply a result about positive perturbations of resolvent positive operators [3] with non-dense domain in *AL*-spaces in the form given in [37, Theorem 1.4]. It is an extension of the well known perturbation result due to Desch [15] and by Voigt [41]. For positive perturbations of positive semigroups in the case when the space is not an *AL*-space we refer to [5, 10]. We also present a result about stationary solutions of (1). We illustrate our general results with an age-size-dependent cell cycle model generalizing the discrete time model of [22, 25, 38]. This model can be described as a piecewise deterministic Markov process (see Sect. 5 and [34]). Our approach can also be used in transport equations [8, 23].

## 2 General results

- (i)
for each \(\lambda >0\), the operator \(\varPsi _0:\mathcal {D}\rightarrow L^1_{\partial }\) restricted to the nullspace \(\mathcal {N}(\lambda I-A)=\{f\in \mathcal {D}:\lambda f-Af=0\}\) of the operator \((\lambda I-A,\mathcal {D})\) has a positive right inverse, i.e., there exists a positive operator \(\varPsi (\lambda ):L^1_{\partial } \rightarrow \mathcal {N}(\lambda I-A)\) such that \(\varPsi _0\varPsi (\lambda )f_{\partial }=f_{\partial }\) for \(f_{\partial }\in L^1_{\partial }\);

- (ii)
the operator \(\varPsi :\mathcal {D} \rightarrow L^1_{\partial }\) is positive and there exists \(\omega \in {\mathbb {R}}\) such that the operator \(I_{\partial }-\varPsi \varPsi (\lambda ):L^1_{\partial }\rightarrow L^1_{\partial }\) is invertible with positive inverse for all \(\lambda >\omega \), where \(I_{\partial }\) is the identity operator on \(L^1_{\partial }\);

- (iii)
the operator \(A_0\subseteq A\) with \(\mathcal {D}(A_0)=\{f\in \mathcal {D}:\varPsi _0f=0\}\) is the generator of a positive semigroup on \(L^1\);

- (iv)for each nonnegative \(f\in \mathcal {D}\)$$\begin{aligned} \int _E Af(x)\,m(dx)-\int _{E_\partial }\varPsi _0f(x)\,m_{\partial }(dx)\le 0. \end{aligned}$$(2)

### Theorem 1

### Proof

*AL*-space with norm

### Remark 1

- (v)
\((A_0,\mathcal {D}(A_0))\) is densely defined and resolvent positive,

*substochastic semigroup*on \(L^1 \), i.e., a positive semigroup of contractions on \(L^1\). This is a consequence of the Hille–Yosida theorem, see e.g. [34, Theorem 4.4]. Thus it is enough to assume condition (v) instead of (iii). Observe also that (iii) and (iv) imply that \((0,\infty )\subseteq \rho (A_0)\).

### Remark 2

*stochastic semigroup*, i.e., a positive semigroup of operators preserving the \(L^1\) norm of nonnegative elements (see e.g. [7, Section 6.2] and [34, Corollary 4.1]).

### Remark 3

- (a)
\((A,\mathcal {D})\) is closed,

- (b)
\(\varPsi _0\) is onto and continuous with respect to the graph norm \(\Vert f\Vert _A=\Vert f\Vert +\Vert Af\Vert \),

### Remark 4

### Remark 5

We now look at a simple example where Theorem 1 can be easily applied and it should be compared with [1, Corollary 25].

### Example 1

*A*to

It should be noted that in [34, Theorem 4.6] the assumption that the domain \(\mathcal {D}(A_{\varPsi })\) of the operator \(A_{\varPsi }\) is dense is missing. Making use of Theorem 1, we get the following result.

### Theorem 2

*B*is a bounded positive operator such that

### Theorem 3

Assume conditions (i)–(iv). Let \(\varPsi (0)\) be as in (11). If a nonnegative \(f_{\partial }\in L^1_{\partial }\) satisfies \(\varPsi (0)f_{\partial }\in L^1\) and \(f_{\partial }=\varPsi \varPsi (0)f_{\partial }\), then \(\varPsi (0)f_{\partial }\in \mathcal {D}(A_\varPsi )\) and \(A_\varPsi \varPsi (0)f_{\partial }=0\). Conversely, if \(A_\varPsi f=0\) for a nonnegative \(f\in \mathcal {D}(A_\varPsi )\) then \(f_{\partial }=\varPsi f\) satisfies \(\varPsi \varPsi (\lambda )f_{\partial }\le f_{\partial }\) for all \(\lambda >\max \{0,\omega \}\), where \(\omega \) is as in (ii).

### Proof

It follows from condition (i) that \(\varPsi (\lambda )f_{\partial }\in \mathcal {D}\), \(\varPsi _0\varPsi (\lambda )f_{\partial }=f_{\partial }\), and \(A\varPsi (\lambda )f_{\partial }=\lambda f_{\partial }\) for all \(\lambda >0\). We have \(\varPsi (\lambda )f_{\partial }\rightarrow \varPsi (0)f_{\partial }\) in \(L^1\), as \(\lambda \rightarrow 0\). Thus \(A\varPsi (\lambda )f_{\partial }\rightarrow 0\) in \(L^1\), as \(\lambda \rightarrow 0\). Recall from the proof of Theorem 1 that the operator \((\mathcal {A}+\mathcal {B})(f,0)=(Af,\varPsi f-\varPsi _0 f)\), \(f\in \mathcal {D}\), is a closed operator in the space \(L^1\times L^1_{\partial }\). The operators \(\varPsi \) and \(\varPsi _0\) are positive and we have \(\varPsi \varPsi (\lambda )f_{\partial }\rightarrow \varPsi \varPsi (0)f_{\partial }=f_{\partial }=\varPsi _0\varPsi (0)f_{\partial }\). Thus, \((\mathcal {A}+\mathcal {B})(\varPsi (\lambda )f_{\partial },0)\rightarrow (0,0)\) as \(\lambda \rightarrow 0\). This implies that \(\varPsi (0)f_{\partial }\in \mathcal {D}(A_{\varPsi })\) and \(A_{\varPsi }\varPsi (0)f_{\partial }=0\).

## 3 A model of a two phase cell cycle in a single cell line

*S*phase (DNA synthesis and replication), \(G_2\) phase (post DNA replication growth period), and

*M*(mitotic) phase (period of cell division). The Smith–Martin model [36] divides the cell cycle into two phases:

*A*and

*B*. The

*A*phase corresponds to all or part of \(G_1\) phase of the cell cycle and has a variable duration, while the

*B*phase covers the rest of the cell cycle. The cell enters the phase

*A*after birth and waits for some random time \(T_A\) until a critical event occurs that is necessary for cell division. Then the cell enters the phase

*B*which lasts for a finite fixed time \(T_B\). At the end of the

*B*-phase the cell splits into two daughter cells. We assume that individual states of the cell are characterized by age \(a\ge 0\) in each phase and by size \(x> 0\), which can be volume, mass, DNA content or any quantity conserved trough division. We assume that individual cells of size

*x*increase their size over time in the same way, with growth rate

*g*(

*x*) so that \(dx/dt=g(x)\), and all cells age over time with unitary velocity so that \(da/dt=1\). We assume that the probability that a cell is still being in the phase

*A*at age

*a*is equal to \(H(a)\), so the rate of exit from the phase

*A*at age

*a*is \(\rho (a)\) given by

*A*. We make the following assumptions:

- (I)
The function \(h\) in (12) is a probability density function so that \(h:[0,\infty )\rightarrow [0,\infty )\) is Borel measurable and the function \(H\) in (12) satisfies: \(H(0)=1\), \(H(\infty )=0\).

- (II)
The growth rate function \(g:(0,\infty )\rightarrow (0,\infty )\) is globally Lipschitz continuous and \(g(x)>0\) for \(x>0\).

*x*. It follows from assumption (II) that the function \(\mathfrak {Q}\) is strictly increasing and continuous. We denote by \(\mathfrak {Q}^{-1}\) the inverse of \(\mathfrak {Q}\). Define

*t*satisfying \(\mathfrak {Q}(x_0)+t>0\); otherwise we set \(\pi _{t}x_0=0\). Note that at time \(t=T\), the generation time, a “mother cell” of size \(\pi _{T}x_0\) divides into two daughter cells of equal size \(\frac{1}{2}\pi _{T}x_0\).

*f*be the probability density function of the size distribution at birth at time \(t_0\) of mother cells and let \(t_1>t_0\) be a random time of birth of daughter cells. Then the probability density function of the size distribution of daughter cells is given by [25, 38]

*P*defined by (15) is a positive contraction on \(L^1(0,\infty )\), the space of Borel measurable functions defined on \((0,\infty )\) and integrable with respect to the Lebesgue measure. Here we extend the probabilistic model to a continuous time situation by examining what happens at all times

*t*and not only at \(t_0,t_1,t_2,\ldots \).

*A*-phase and in the

*B*-phase at time

*t*, age

*a*, and size

*x*, respectively. Neglecting cell deaths the equations can be written as

*A*with intensity being dependent on maturity, not age. In the case of \(T_B=0\) there is only one phase present; a maturity structured model being a continuous time extension of [24] is studied in [27], while age and volume/maturity structured population models of growth and division were studied extensively since the seminal work of [12, 26, 33]. We refer the reader to [28] for historical remarks concerning modeling of age structured populations and to [35, 42] for recent reviews.

*m*is the product of the two-dimensional Lebesgue measure and the counting measure on \(\{1,2\}\), and \(\mathcal {E}\) is the \(\sigma \)-algebra of all Borel subsets of

*E*. We identify \(L^1=L^1(E,\mathcal {E},m)\) with the product of the spaces \(L^1(E_1)\) and \(L^1(E_2)\) of functions defined on the sets \(E_1\) and \(E_2\), respectively, and being integrable with respect to the two-dimensional Lebesgue measure. We say that the operator

*P*has a steady state in \(L^1(0,\infty )\) if there exists a probability density function

*f*such that \(Pf=f\). Similarly, a semigroup \(\{S(t)\}_{t\ge 0}\) has a steady state in \(L^1\) if there exists a nonnegative \(f\in L^1\) such that \(S(t)f=f\) for all \(t>0\) and \(\Vert f\Vert _1=1\) where \(\Vert \cdot \Vert _1\) is the norm in \(L^1\).

### Theorem 4

Assume conditions (I) and (II). There exists a unique positive semigroup \(\{S(t)\}_{t\ge 0}\) on \(L^1\) which provides solutions of (16)–(19) and \(\{S(t)\}_{t\ge 0}\) is stochastic. If \(H\in L^1(0,\infty )\) then the semigroup \(\{S(t)\}_{t\ge 0}\) has a steady state in \(L^1\) if and only if the operator *P* in (15) has a steady state in \(L^1(0,\infty )\).

We give the proof of Theorem 4 in the next section. Theorem 4 combined with [9] implies the following sufficient conditions for the existence of steady states of (16)–(19).

### Corollary 1

*a*. Conversely, if there is \(x_0\ge 0\) such that \(H(Q(\lambda (x_0)))>0\) and \(\mathbb {E}(T_A)>\sup _{x\ge x_0} (\mathfrak {Q}(\lambda (x))-\mathfrak {Q}(x)) \), then (16)–(19) has no steady states.

*k*is a positive constant, then it is known [22, 38, 39] that the operator

*P*has no steady state. We now consider a linear cell growth and assume that \(g(x)=k\) for all \(x>0\). We see that \(\mathfrak {Q}(x)=x/k\), the operator

*P*is of the form (see [39] or the last section)

### Corollary 2

Assume that \(g(x)=k\) for \(x>0\) and that \(h(a)>0\) for all sufficiently large \(a>0\). If \(\mathbb {E}(T_A)<\infty \) then the semigroup \(\{S(t)\}_{t\ge 0}\) has a unique steady state.

## 4 Proof of Theorem 4

*A*on \(\mathcal {D}\) by setting \(Af=(A_1f_1,A_2f_2)\) for \(f=(f_1,f_2)\in \mathcal {D}\). We take operators \(\varPsi _0,\varPsi :\mathcal {D}\rightarrow L^1_{\partial }\) of the form

*A*restricted to \(\mathcal {D}(A_0)=\{(f_1,f_2)\in \mathcal {D}_1\times \mathcal {D}_2: \text {B}^{-}f_1=0, \text {B}^{-}f_2=0\}\) is the generator of the semigroup \(\{S_0(t)\}_{t\ge 0}\) given by

*P*is as in (15). Consequently, equation \(\varPsi \varPsi (0)f_{\partial }=f_{\partial }\) has a solution in \(L^1_{\partial }\) if and only if the equation \(Pf_{\partial ,1}=f_{\partial ,1}\) has a solution in \(L^1(0,\infty )\). Observe also that the operator \(\varPsi \varPsi (0)\) preserves the \(L^1_{\partial }\) norm on nonnegative elements. Hence, if \(f_{\partial }\in L^1_{\partial }\) is such that \(\varPsi \varPsi (0)f_{\partial }\le f_{\partial }\) then \(\varPsi \varPsi (0)f_{\partial }= f_{\partial }\). Thus the assertion follows from Theorem 3.

## 5 Final remarks

*a*,

*x*,

*i*), where \(i=1\) if a cell is in the phase

*A*, \(i=2\) if it is in the phase

*B*, the variable

*x*describes the cell size, and

*a*describes the time which elapsed since the cell entered the

*i*th phase. Let \(t_0=0\). If we observe consecutive descendants of a given cell and the

*n*th generation time is denoted by \(t_n\), then \(t_{n+1}=s_{n}+T_{B}\) where \(s_n\) is the time when the cell from the

*n*th generation enters the phase

*B*, \(n\ge 0\). A newborn cell at time \(t_n\) is with age \(a(t_n)=0\) and with initial size equal to \(x(t_n^-)/2\), where \(x(t_n^-)\) is the size of its mother cell. The cell ages with velocity 1 and its size grows according to the equations \(x'(t)=g(x(t))\) for \(t\in (t_n,s_n)\). If the cell enters the phase

*B*then its age is reset to 0 and its size still grows according to \(x'(t)=g(x(t))\) for \(t\in (s_n,s_n+T_B)\). We have

*jump times*. At jump times the process is given by (30) and (31). If the distribution of

*X*(0) has a density

*f*then

*X*(

*t*) has a density

*S*(

*t*)

*f*, i.e.,

*x*(

*a*) of the cell at time

*a*, if

*x*(0) has a density \(f_{\partial ,1}\). Thus the density of the mass \(x(t_{1})\) is given by

*P*is as in (15). Now, if the operator

*P*has a steady state \(f_{\partial ,1}\in L^1(0,\infty )\) so that \(f_{\partial ,1}\) satisfies (29) and if \(f_{\partial ,2}\) is as in (32), then \(f^*=(f_1^*,f_2^*)\) given by

*P*has a unique steady state.

### Remark 6

It should be noted that in the two-phase cell cycle model in [31] the rate of exit from the phase *A* depends on *x*, not on *a*, and that there is no such equivalence between the existence of steady states as presented in Theorem 4. Our results remain true if we assume as in [31] that division into unequal parts takes place. Methods as in [31, 34] can also be used in our model to study asymptotic behaviour of the semigroup \(\{S(t)\}_{t\ge 0}\). For a different approach to study positivity and asymptotic behaviour of solutions of population equations in \(L^1\) we refer to [32].

*a*and

*x*at time

*t*in the phase

*A*for \(i=1\) or

*B*for \(i=2\), i.e., \(\int _{a_1}^{a_2}\int _{x_1}^{x_2}p_i(t,a,x)dadx\) is the number of cells with age between \(a_1\) and \(a_2\) and size between \(x_1\) and \(x_2\) at time

*t*in the given phase. Then \(p_1\) and \(p_2\) satisfy Eqs. (16), (18), (19) while the boundary condition (17) takes the form

*x*and gives birth to two daughters of size

*x*entering the phase

*A*at age 0.

### Theorem 5

Assume conditions (I) and (II). Then there exists a unique positive semigroup on \(L^1\) which provides solutions of (16), (34), (18), (19).

## Notes

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