Well-posedness of a nonlinear model of proliferating cell populations with inherited cycle length

Acta Mathematica Scientia 2016,36B(5):1225–1244 http://actams.wipm.ac.cn

WELL-POSEDNESS OF A NONLINEAR MODEL OF PROLIFERATING CELL POPULATIONS WITH INHERITED CYCLE LENGTH∗ Abdul-Majeed AL-IZERI Khalid LATRACH† Universit´e Blaise Pascal (Clermont II) Laboratoire de Math´ematiques, CNRS UMR 6620 Campus des C´ezeaux - B.P. 80026, 63171 Aubi`ere Cedex France E-mail : Abdul− Majeed.Al− [email protected]; [email protected] Abstract This paper deals with a nonlinear initial boundary values problem derived from a modified version of the so called Lebowitz and Rubinow’s model [16] discussed in [8, 9] modeling a proliferating age structured cell population with inherited properties. We give existence and uniqueness results on appropriate weighted Lp -spaces with 1 ≤ p < ∞ in the case where the rate of cells mortality σ and the transition rate k are depending on the total density of population. General local and nonlocal reproduction rules are considered. Key words

evolution equation; local and nonlocal boundary conditions; quasi-accretive operators; mild solutions, strong solutions; local and global solutions 47H06; 34A12; 35F20

2010 MR Subject Classification

1

Introduction

In the works [8, 9], the author discussed the well-posedness and various mathematical aspects of solution to the Cauchy problem    ∂f (t, a, l) = − ∂f (t, a, l) − σ(a, l)f (t, a, l) + (P f )(t, a, l), ∂t ∂a (1.1)  f (0, a, l) = f (a, l), 0

where

(P f )(t, a, l) =

Z

l2 l1

Z

l′

k(a, l, a′ , l′ )f (t, a′ , l′ )da′ dl′ ,

0

t > 0, 0 < a, a′ < l, 0 < l1 < l, l′ < l2 and f0 stands for the initial data. The function f := f (t, a, l) denotes the cell density with respect to cell cycle length l, and age a at time t, and σ(a, l) denotes the rate of cell mortality. By cell cycle length we mean the time between cell birth and cell division. It is an inherent characteristic of cells determined at birth, i.e., the duration of the cycle from cell birth to cell division is determined at birth. The constant l1 (resp. l2 ) denotes the minimum cycle length (resp. the maximum cycle lenght). The function ∗ Received

May 13, 2015; revised September 2, 2015. author: Khalid LATRACH.

† Corresponding

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k(a, l, a′ , l′ ) denotes the transition rate at which cells change their cell cycle length from l′ to l and their age from a′ to a. The problem (1.1) is complemented with a general biological reproduction rule given by
f (t, 0, l) = Kf (t, ·, ·) (l), (1.2)

where K is a bounded linear operator on suitable function spaces.
Rl In the case where P = 0 and Kf (t, ·, ·) (l) = l12 r(l, l′ )f (t, l′ , l′ )dl′ , where r(·, ·) is a measurable function, we find a model originally introduced by Lebowitz and Rubinow [16] for modeling proliferating microbial populations. It was extensively studied by many authors and various mathematical aspects concerning the Cauchy problem: well-posedness (generation results), spectral analysis, time asymptotic behavior of solution (when it exists) were discussed in many papers (see, for example, the works [7, 15, 17, 21, 22] and the references therein).

As it was observed by Rotenberg [19], it seems that the linear model is not adequate. Indeed, the cells under consideration are in contact with a nutrient environment which is not part of the mathematical formulation. Fluctuations in nutrient concentration and other density-dependent effects such as contact inhibition of growth make the transition rates functions of the population density, thus creating a nonlinear problem. On the other hand, the biological boundaries at l1 and l2 are fixed and tightly coupled through out mitosis. The conditions present at the boundaries are left throughout the system and cannot be remote. This phenomena suggests that at mitosis the daughter cells and parent cell are related by a nonlinear reproduction rule. At mitosis, the daughter and mother cells are related by a nonlinear reproduction rule which describes the boundary conditions. We point out that the well-posedness of nonlinear initial boundary value problems derived from Rotenberg model were already discussed in [12, 20] and [1]. However, it seems that the wellposedness of nonlinear time dependent versions derived from (1.1) has not yet been investigated. The main goal of this work is to present and to discuss two nonlinear versions of the model (1.1) in a weighted Lp -spaces, 1 ≤ p < ∞, on the set Ω : = {(a, l); 0 < a < l, l1 < l < l2 } where the total cross section σ and the transition rate k are assumed to be nonlinear functions of the total density of population f and, at the mitosis, daughter and mother cells are related by a nonlinear reproduction rule which describes constraints at boundaries. The structure of this work is as follows. In the next section we introduce the functional setting of the problem and the main assumptions. In Section 3 we study the initial boundary value problem (3.1) in the case where σ(·, ·, ·) and k(·, ·, ·, ·, ·) are nonlinear functions of the density of population f , and K is a nonlinear operator on suitable trace spaces modeling the biological rule. Following the same strategy as in the works [12, 20] for Lebowitz-Rubinow’s model (also see [4] and [13]) we discuss existence and uniqueness of solutions to problem (3.1). The main result of this section is Theorem 3.5 which asserts that, under reasonable assumptions, problem (3.1) has a unique mild solution on appropriate weight Lp spaces where 1 ≤ p < ∞. If p > 1, then this solution is also a weak solution of the problem. Further, if the initial data ω ω belongs to the domain of TK (see Section 2 for the definition of TK ), then we obtain a strong solution. Further, we derive sufficient conditions guaranteeing that problem (3.1) possesses a unique global strong solution. Section 4 focuses on problem (4.1) where σ1 (·, ·, ·) is a nonlinear function of hf i(t) with

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Rl Rl hf i(t) := l12 0 f (t, a, l)dadl (the usual L1 norm of L1 (Ω)) and k1 (·, ·, ·, ·, ·) is a nonlinear function of the density of population f . Here the boundary conditions are modeled by Z l2 f (t, 0, l) = k2 (l, l′ , hf i(t), f (t, l′ , l′ ))dl′ + σ2 (l, hf i(t), f (t, l, l)), l1

where σ2 (·, ·, ·) and k2 (·, ·, ·, ·) are nonlinear functions of both f (t, ·, ·)(l) and hf i(t). After giving some preparatory results, we state the main result of this section (Theorem 4.6). We establish that under appropriate conditions, the existence of a local weak solution on weighted Lp -spaces ω with 1 < p < ∞. Moreover, if the initial data belongs to the domain of TK , then we have a local strong solution and, if further conditions (A7) is satisfied, then we get a unique global strong solution. Moreover, we have a unique global weak solution whenever the initial data ω does not belong to D(TK ). On the other hand, if p = 1, then problem (4.2) possesses a local mild solution on a weighted L1 -space. Our analysis uses the approach developed for semi-linear evolution equations in [18] (see also [13, 20]). For the sake of completeness, in Section 5, we recall and gather some facts from functional analysis required throughout of the paper.

2

Notations and Preliminaries

The goal of this section is to first fix the notations and to introduce the functional setting of the problem. To do so, let 1 ≤ p < ∞ and set
Xp = Lp Ω, dadl ,

where Ω : = {(a, l); 0 < a < l, l1 < l < l2 } and 0 < l1 < l2 < ∞. The space Xp endowed with its natural norm is a Banach space. We denote by Xp1 and Xp2 the boundary spaces Xp1 = Lp (Γ1 , dl), where

 Γ1 : = (0, l) : l ∈ (l1 , l2 )

Xp2 = Lp (Γ2 , dl), and

 Γ2 : = (l, l) : l ∈ (l1 , l2 ) .

Let ω > 0 be an arbitrary real number and define the weighted space Xpω by
Xpω = Lp Ω, hω dadl ,

where the weight function hω (·, ·) is given by

hω (a, l) = e−ω(l−a) . The space Xpω endowed with the norm Z 1/p  Z p kf kp,ω = |f (a, l)| hω (a, l)dadl = Ω

l2

l1

Z

l

0

is a Banach space. Let Wpω be the partial Sobolev space defined by   ∂f Wpω = f ∈ Xpω such that ∈ Xpω . ∂a The space Wpω equipped with the norm kf kWpω =



p kf kp,ω

p 1/p

∂f

+

∂a p,ω

p

|f (a, l)| hω (a, l)dadl

1/p

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is a Banach space. Let us now recall the following lemma established in [1]. Lemma 2.1 Let f ∈ Wpω . If f|Γ1 ∈ Xp1 , then f|Γ2 ∈ Xp2 and vice versa. According to Lemme 2.1 in [1], if f|Γ1 ∈ Xp1 (resp. f|Γ2 ∈ Xp2 ), then f|Γ2 ∈ Xp2 (resp. f|Γ1 ∈ Xp1 ). So, we can define the space  f ω = f ∈ W ω such that f|Γ ∈ X 1 W p p p 1  = f ∈ Wpω such that f|Γ2 ∈ Xp2 .

fpω has traces f|Γ and f|Γ belonging to the boundary spaces Accordingly, any function f ∈ W 1 2 1 2 Xp and Xp , respectively. ω Let TK be the operator defined by  ω  T ω : D(TK ) ⊂ Xpω −→ Xpω ,   K   ∂f ω f −→ (TK f )(a, l) = (a, l),  ∂a     ω fpω : f|Γ = K(f|Γ )}. D(TK ) = {f ∈ W 1 2 Before going further we shall prove the following lemma which is required below.

Lemma 2.2 Let 1 ≤ p < +∞. If f ∈ Xp , then f ∈ Xpω and conversely. In particular we have ωl2 (i) kf kp,ω ≤ kf kp ≤ e p kf kp,ω , 1 l2 −l2
(ii) kf k1 ≤ eωl2 2 2 1 q kf kp,ω , where q denotes the conjugate exponent of p. Proof

Let f ∈ Xp . Since l ≥ a, we have Z Z l2 Z l p p e−ω(l−a) |f (a, l)| dadl ≤ kf kp,ω = l1

l2

l1

0

Z

0

l

p

p

|f (a, l)| dadl = kf kp

hence f ∈ Xpω . Conversely, let f ∈ Xpω . One can write Z l2 Z l eω(l−a) e−ω(l−a) |f (a, l)|p dadl kf kpp = l1

≤e

ω l2

0

Z

l2

l1

Z

0

l

e−ω(l−a) |f (a, l)|p dadl = eω l2 kf kpp,ω .

This proves the first assertion and (i). Rl Rl Rl Rl
(ii) It is clear that kf k1 = l12 0 |f (a, l)| dadl ≤ eωl2 l12 0 e−ω(l−a) |f (a, l)| dadl . Now applying H¨older’s inequality, we get  Z l2 Z l 1/p  2 2 1 p ωl2 l2 − l1 q −ω(l−a) kf k1 ≤ e e |f (a, l)| dadl 2 l1 0  l2 − l2  q1 1 ≤ eωl2 2 kf kp,ω . 2 This ends the proof.  For simplicity, we shall identify Xp1 and Xp2 with the space
Yp := Lp [l1 , l2 ], dl .

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Accordingly, the boundary operator K may be viewed as a map from Yp into itself. We close this section by recalling that the function sgn0 (·) is defined by   if r > 0,  1 sgn0 (r) =

  

0

if r = 0,

−1

if r < 0,

and the definition of the symbol [·, ·]s is given in Section 5 (see also [3, pp.102-103]).

3

Local Boundary Conditions

The goal of this section is to discuss existence and uniqueness results for the following initial boundary value problem  Z l2 Z l′
 ∂f ∂f   k(a, l, a′ , l′ , f (t, a′ , l′ ))da′ dl′ ,   ∂t (t, a, l) = − ∂a (t, a, l) − σ a, l, f (t, a, l) + l 0 1 (3.1) f (0, a, l) = f0 (a, l),     f (t, 0, l) = [Kf (t, ·, ·)](l),

where σ(·, ·, ·) and k(·, ·, ·, ·, ·) are nonlinear functions of the density of population f , f0 stands for the initial data and K denote a nonlinear operator from Yp into itself modeling the transition biological rule. We now introduce the following hypotheses required below. • (A1) There exists κ > 0 such that, for all f1 , f2 ∈ Yp , we have kK(f1 ) − K(f2 )k ≤ κ kf1 − f2 k . • (A2) The functions σ(·, ·, ·) and k(·, ·, ·, ·, ·) are measurables and there exist ζ ∈ L∞ (Ω) and ρ ∈ L∞ (Ω × Ω) such that |σ(a, l, z1 ) − σ(a, l, z2 )| ≤ |ζ(a, l)| |z1 − z2 | , and |k(a, l, a′ , l′ , f1 (a′ , l′ )) − k(a, l, a′ , l′ , f2 (a′ , l′ ))| ≤ |ρ(a, l, a′ , l′ )| |f1 (a′ , l′ ) − f2 (a′ , l′ )| , where z1 , z2 ∈ R, f1 , f2 ∈ Xpω . Further, σ(·, ·, ·) is assumed to be a Carath´eodory function. In the remainder of this section, the real ω will satisfy the condition  1  ω > max 0, ln(κ) , l1 where κ is the constant appearing in hypothesis (A1). The function σ(·, ·, ·) is assumed to be a Carath´eodory function (see Section 5), so its generates a Nemytskii operator Nσ given by  N : X ω → X ω , σ p p (Nσ f )(a, l) = σ(a, l, f (a, l)).

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Let B be the operator defined by  ω ω   B : Xp −→ Xp , Z l2 Z l′   f −→ k(a, l, a′ , l′ , f (t, a′ , l′ ))da′ dl′  l1

0

and put

(Fp f )(a, l) := (Bf )(a, l) − (Nσ f )(a, l). Now problem (3.1) may be written abstractly as  ∂f  ω  (t) + TK (f (t)) = Fp (f (t)),    ∂t

(3.2)

f (0, a, l) = f0 (a, l) ∈ Xpω ,     f (t, 0, l) = [Kf (t, ·, ·)](l).

For the sake of completeness, we shall recall the following result established in [1]. ω Lemma 3.1 If condition (A1) is satisfied, then TK is a ω-m-accretive operator on Xpω .

Remark 3.2 Note that the result of Lemma 3.1 holds also true for p = 1 (see Theorem 2 in [12]). Xpω

ω) Lemma 3.3 Let p ∈ [1, +∞) and assume that condition (A1) is satisfied. Then D(TK ω = Xp .
Xω ω ) p . The proof is similar to that of Proof It is enough to prove that C0∞ Ω ⊆ D(TK Theorem 3 in [12] and then it is omitted. 

Lemma 3.4 Let p ∈ [1, +∞). If hypothesis (A2) holds true and Nσ maps Xpω into itself, then there exists a constant C > 0 such that kFp (ϕ1 ) − Fp (ϕ2 )kp,ω ≤ C kϕ1 − ϕ2 kp,ω , Proof

∀ϕ1 , ϕ2 ∈ Xpω .

For all (ϕ1 , ϕ2 ) ∈ Xpω × Xpω , we have
Fp (ϕ1 ) − Fp (ϕ2 ) = B(ϕ1 ) − B(ϕ2 ) − Nσ (ϕ1 ) − Nσ (ϕ2 ) .

Using (A2) together with H¨ older’s inequality we get

|Fp (ϕ1 )(a, l) − Fp (ϕ2 )(a, l)| Z l2 Z l′ ≤ kρk∞ |ϕ1 (a′ , l′ ) − ϕ2 (a′ , l′ )| da′ dl′ + kζk∞ |ϕ1 (a, l) − ϕ2 (a, l)| l1

≤ kρk∞ e

ωl2

Z

0

l2 l1

Z

l′

′

′ 1

′ 1

′

e−ω(l −a ) q e−ω(l −a ) p |ϕ1 (a′ , l′ ) − ϕ2 (a′ , l′ )| da′ dl′

0

+ kζk∞ |ϕ1 (a, l) − ϕ2 (a, l)|  Z l2 Z l′ ′ ′ 1 ≤ kρk∞ eωl2 e−ω(l −a ) p |ϕ1 (a′ , l′ ) − ϕ2 (a′ , l′ )| da′ dl′ l1

0

+ kζk∞ |ϕ1 (a, l) − ϕ2 (a, l)|  Z l2 Z l′  1q  Z ωl2 ′ ′ ≤ kρk∞ e da dl l1

0

+ kζk∞ |ϕ1 (a, l) − ϕ2 (a, l)|

l2

l1

Z

0

l′

e

−ω(l′ −a′ )

′

′

′

′

p

′

|ϕ1 (a , l ) − ϕ2 (a , l )| da dl

′

 p1

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≤ kρk∞ eωl2

 l2  q1 2

2

1231

kϕ1 − ϕ2 kp,ω + kζk∞ |ϕ1 (a, l) − ϕ2 (a, l)| . p

p

Next, simple calculations using the estimate (|a + b|)p ≤ 2p (|a| + |b| ) lead to Z l2 Z l p e−ω(l−a) |Fp (ϕ1 )(a, l) − Fp (ϕ2 )(a, l)| dadl l1

0

p

≤ 2p kρk∞ eωl2 p p

+2p kζk∞ h

 l2  pq 2

2 Z l2 Z l l1

p

≤ 2p kρk∞ eωl2 p

p

kϕ1 − ϕ2 kp,ω

Z

l2

l1

Z

l

dadl

0 p

e−ω(l−a) |ϕ1 (a, l) − ϕ2 (a, l)| dadl

0

 l2  pq +1 2

2

i p p + kζk∞ kϕ1 − ϕ2 kp,ω .

Hence,  l 2 p i p1 h kFp (ϕ1 ) − Fp (ϕ2 )kp,ω ≤ 2 kρkp∞ eωl2 p 2 + kζkp∞ kϕ1 − ϕ2 kp,ω 2 ≤ C kϕ1 − ϕ2 kp,ω , h p where C = 2 kρk∞ eωl2 p

l22
p 2

p

+ kζk∞

i p1

. This proves Lemma 3.4.


Theorem 3.5 Let ω be an arbitrary real satisfying ω > max 0, l11 ln(κ) and let 1 ≤ p < ∞. Assume that conditions (A1) and (A2) hold true and Nσ maps Xpω into itself. Then problem (3.2) has a unique mild solution on Xpω for all f0 ∈ Xp . Further, if p ∈ (1, +∞), then this mild solution is in fact a weak solution and it is a strong solution whenever the initial data ω f0 belongs to D(TK ). ω Proof According to Lemma 3.1, Remark 3.2 and Lemma 3.3, TK is quasi-m-accretive ω ω in Xp with dense domain. It follows from Lemma 3.4 that Fp : Xp → Xpω is a Lipschitzian ω mapping on Xpω , so TK − Fp is also a quasi-m-accretive operator on Xpω . The first part of the theorem follows from Corollary 4.1 in [3] (here the initial data f0 is taken in Xp because ω ) = X and the fact that any function in X ω belongs to X (cf. Lemma 2.2)). We know D(TK p p p that, for 1 < p < ∞, the space Xpω possesses the Radon-Nikodym property, so by Theorem 5.3 ω we infer that this mild solution is a weak solution on Xpω . Further, if f0 ∈ D(TK ), then the use of Theorem 5.1 shows that this weak solution is, in fact, a strong solution on Xpω . 

Remark 3.6 (a) Recall that it is proved in [1, Lemma 1] that, if 0 ≤ κ < 1, then TK is a m-accretive on Xp . Accordingly, if (A2) holds true and Nσ maps Xp into itself, then the conculusion of Theorem 3.5 remains valid even if l1 = 0 (cf. [1, Theorem 4.1]). Evidently, in this case we have ω = 0. Note also that if κ > 1, then the condition l1 > 0 is necessary. (b) If P = 0, i.e., k(·, ·, ·, ·, ·) = 0, then Theorem 3.5 is nothing else but Theorem 3 in [12] for p = 1 and Theorem 4.2 in [1] for p ∈ (1, +∞). We have also the following proposition. Proposition 3.7 Let p ∈ [1, +∞) and let ω be as in Theorem 3.5. Let f1 , f2 ∈ C([0, T ], Xpω ) be two mild solutions to Problem (3.2) where T > 0. Given ǫ > 0, there exists β > 0 such that if kf1 (0) − f2 (0)kp,ω ≤ β, then kf1 − f2 k∞ ≤ ǫ.

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It is similar to the proof of Proposition 3.3 in [13], so it is omitted.



Nonlocal Boundary Conditions

In this section we will discuss existence and uniqueness of solutions to the following initial boundary value problem  Z l2 Z l′
∂f ∂f   (t, a, l) = − (t, a, l) − σ1 a, l, hf i f + k1 (a, l, a′ , l′ , f (t, a′ , l′ ))da′ dl′ ,    ∂t ∂a l 0  1  f (0, a, l) = f0 (a, l), (4.1)   Z  l  2   f (t, 0, l) = k2 (l, l′ , hf i, f (t, l′ , l′ ))dl′ + σ2 (l, hf i, f (t, l, l)), l1

where

hf i(t) :=

Z

l2 l1

Z

0

l

 f (t, a, l)da dl.

The function σ1 (·, ·, ·) is a nonlinear function of hf i while σ2 (·, ·, ·) and k2 (·, ·, ·, ·, ·) are nonlinear functions of both the density of population f and hf i. Here f0 denotes the initial data. Before going further, we introduce the following hypotheses required in the sequel. • (A3) The function σ1 (·, ·, ·) is measurable and for any r > 0, there exists Λr > 0 such that |σ1 (a, l, z1 ) − σ1 (a, l, z2 )| ≤ Λr |z1 − z2 | for all (a, l) ∈ Ω, z1 , z2 ∈ [−r, r]. • (A4) There exist a function σ1 : Ω × R −→ R and two constants σ 1 ∈ R and σ 1 > 0 such that σ 1 ≤ σ1 (a, l, z) ≤ σ 1 ,

∀(a, l, z) ∈ Ω × R.

• (A5) The function k1 (·, ·, ·, ·, ·) is measurable and satisfies |k1 (a, l, a′ , l′ , f1 (a′ , l′ )) − k1 (a, l, a′ , l′ , f2 (a′ , l′ ))| ≤ |ρ(a, l, a′ , l′ )| |f1 (a′ , l′ ) − f2 (a′ , l′ )| , where f1 , f2 ∈ Xpω and ρ ∈ L∞ (Ω2 ). • (A6) The functions σ2 (·, ·, ·) and k2 (·, ·, ·, ·) are measurable and there exist two constants C1 > 0 and C2 > 0 such that e ≤ C1 (|η − ηe| + |ζ − ζ|) e σ2 (l, η, ζ) − σ2 (l, ηe, ζ)

and

e ≤ C2 (|η − ηe| + |ζ − ζ|) e k2 (l, l′ , η, ζ) − k2 (l, , l′ , ηe, ζ)

for all (l, l′ ) ∈ [l1 , l2 ] × [l1 , l2 ] and η, ηe, ζ, ζe ∈ R. • (A7) There exist a function e k1 : Ω2 × R −→ R and two constants e k 1 ∈ R and e k1 > 0 such that

e1 a, l, a′ , l′ , z z, for all (a, l, a′ , l′ , z) ∈ Ω2 × R, k1 a, l, a′ , l′ , z = k and

e k1 ≤ e k1 (a, l, a′ , l′ , z) ≤ e k1 , for all (a, l, a′ , l′ , z) ∈ Ω2 × R.

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We now define the operator F1p by (F1p f )(a, l) := (B1 f )(a, l) − (U f )(a, l) where

and

 U : X ω → X ω , p p (U f )(a, l) = σ1 (a, l, hf i)f,

 1 ω ω   B : Xp −→ Xp , Z l2 Z l′   k1 (a, l, a′ , l′ , f (t, a′ , l′ ))da′ dl′ . f −→ l1

0

Hence, problem (4.1) may be written abstractly as  ∂f  ω  (t) + TK (f (t)) = F1p (f (t)),    ∂t

(4.2)

f (0) = f0 ∈ Xpω ,     f (t, 0, l) = [Kf (t, ·, ·)](l),

where K denotes the following nonlocal boundary operator  ω   K : Yp × Xp −→ Yp , Z l2   k2 (l, l′ , hf i, v(l′ ))dl′ + σ2 (l, hf i, v(l)). (v, f )(l) −→ K(v, f )(l) = l1

In the remainder of this section, the space Yp × Xpω will be equipped with the norm k·k∗ given by h i1/p p p k(v, f )k∗ = kf kYp + kf kp,ω , ∀ (v, f ) ∈ Yp × Xpω . Lemma 4.1 Let r > 0 and p ∈ [1, +∞). If conditions (A3)–(A6) hold true. Then there exist two constants Cr > 0 and ϑ > 0 such that

(i) F1p (ϕ1 ) − F1p (ϕ2 ) p,ω ≤ Cr kϕ1 − ϕ2 kp,ω , ∀ϕi ∈ Brp,ω := {f ∈ Xpω : kf kp,ω ≤ r}, i = 1, 2. (ii) kK(v1 , ϕ1 ) − K(v2 , ϕ2 )kYp ≤ ϑ k(v1 , ϕ1 ) − (v2 , ϕ2 )k∗ , ∀(vi , ϕi ) ∈ Yp × Xpω , i = 1, 2. Proof

(i) For all (ϕ1 , ϕ2 ) ∈ Brp,ω × Brp,ω , we have
F1p (ϕ1 ) − F1p (ϕ2 ) = B1 (ϕ1 ) − B1 (ϕ2 ) − U (ϕ1 ) − U (ϕ2 ) .

Using (A5) together with H¨ older’s inequality we get Z l2 Z l′ 1 B (ϕ1 ) − B1 (ϕ2 ) ≤ kρk |ϕ1 (a′ , l′ ) − ϕ2 (a′ , l′ )| da′ dl′ ∞ l1

≤ kρk∞ eωl2 ≤ kρk∞ e

ωl2

≤ kρk∞ e

ωl2

Z

0

l2

l1

Z

l2

l1

l22

Z

′

′ 1

′

′ 1

′

′

1

e−ω(l −a ) q e−ω(l −a ) p |ϕ1 (a′ , l′ ) − ϕ2 (a′ , l′ )| da′ dl′

0

Z


1q

l′

l′

e−ω(l −a ) p |ϕ1 (a′ , l′ ) − ϕ2 (a′ , l′ )| da′ dl′

0

kϕ1 − ϕ2 kp,ω .

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ACTA MATHEMATICA SCIENTIA

Vol.36 Ser.B

1 l2 −l2
Applying Lemma 2.2, we get |hϕ1 i − hϕ2 i| ≤ kϕ1 − ϕ2 k1 ≤ eωl2 2 2 1 q kϕ1 − ϕ2 kp,ω . So, a simple calculation using the estimate (|a + b|)p ≤ 2p (|a|p + |b|p ) and (A3)–(A4) gives

|U (ϕ1 ) − U (ϕ2 )|p = |σ1 (a, l, hϕ1 i)(ϕ1 − ϕ2 ) + (σ1 (a, l, hϕ1 i) − σ1 (a, l, hϕ2 i))ϕ2 |p p

p

≤ 2p |σ1 (a, l, hϕ1 i)(ϕ1 − ϕ2 )| + 2p |(σ1 (a, l, hϕ1 i) − σ1 (a, l, hϕ2 i))ϕ2 | ≤ 2p σ p1 |ϕ1 − ϕ2 |p + 2p Λpr |hϕ1 i − hϕ2 i|p |ϕ2 |p  l2 − l2  pq p p p 1 eωl2 p kϕ1 − ϕ2 kp,ω |ϕ2 | . ≤ 2p σ p1 |ϕ1 − ϕ2 | + 2p Λpr 2 2 Hence, Z

l2

l1

≤2

p

Z

l

0

2p p kρk∞ eωl2 p l2 q

+2p Λpr ≤

p e−ω(l−a) F1p (ϕ1 )(a, l) − F1p (ϕ2 )(a, l) dadl kϕ1 −

 l2 − l2  pq 2

1

2

p ϕ2 kp,ω

Z

l2

l1

Z

l

dadl + 2

0

p

eωl2 p kϕ1 − ϕ2 kp,ω

Z

l2

l1

Z

l

p



2p σ p1

Z

l2

l1

Z

l

0

p

e−ω(l−a) |ϕ2 | dadl

0

p

e−ω(l−a) |ϕ1 − ϕ2 | dadl 

p p 2 kρk∞ eωl2 p l22p kϕ1 − ϕ2 kp,ω  l2 − l2  pq h 1 eωl2 p +2p 2p σ p1 + 2p Λpr 2 p

i kϕ2 kpp,ω kϕ1 − ϕ2 kpp,ω 2  l2 − l2  pq i h 1 eωl2 p kϕ1 − ϕ2 kpp,ω ≤ 2p kρkp∞ eωl2 p l22p kϕ1 − ϕ2 kpp,ω + 2p 2p σ p1 + 2p rp Λpr 2 2 and therefore,

1

Fp (ϕ1 ) − F1p (ϕ2 ) ≤ Cr kϕ1 − ϕ2 kp,ω , p,ω

i p1 h p l2 −l2
p where Cr := 2 kρk∞ eωl2 p l22p + 2p σ p1 + 2p rp Λpr 2 2 1 q eωl2 p . This proves (i). (ii) Now, we shall show the second estimate. For (v1 , ϕ1 ), (v2 , ϕ2 ) ∈ Yp × Xpω , we can write kK(v1 , ϕ1 ) − K(v2 , ϕ2 )kYp p  p1  Z l2  Z l2 ′ ′ ′ ′ ′ |k2 (l, l , hϕ1 i, v1 (l )) − k2 (l, l , hϕ2 i, v2 (l ))| dl dl ≤ l1

l1

+

Z

l2

p

|σ2 (l, hϕ1 i, v1 (l)) − σ2 (l, hϕ2 i, v2 (l))| dl

l1

 p1

= J1 + J2 . Using assumption (A6) together with Lemma 2.2 (ii), we get Z l2 |k2 (l, l′ , hϕ1 i, v1 (l′ )) − k2 (l, l′ , hϕ2 i, v2 (l′ ))| dl′ l1 l2

≤

Z

C2 (|hϕ1 i − hϕ2 i| + |v1 (l′ ) − v2 (l′ )|)dl′

l1

≤

 l2 − l2  1q 2

1

2

l2 C2 eωl2 kϕ1 − ϕ2 kω,p + C2

Applying H¨older’s inequality, we get Z Z l2 1 |v1 (l′ ) − v2 (l′ )| dl′ ≤ l2q l1

l2

l1

′

Z

l2

|v1 (l′ ) − v2 (l′ )| dl′ .

l1

′

p

|v1 (l ) − v2 (l )| dl

′

 p1

1

= l2q kv1 − v2 kYp .

No.5

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A. Al-Izeri & K. Latrach: EXISTENCE AND UNIQUENESS RESULTS

A simple calculations using the estimate (|a + b|)p ≤ 2p (|a|p + |b|p ) gives Z l2 h ip  l2 − l2  q1 1 p 1 J1 ≤ l2 kϕ1 − ϕ2 kp,ω + C2 l2q kv1 − v2 kYp dl C2 eωl2 2 2 l1  l2 − l2  pq p +1 p p 1 ≤ 2p C2p eωl2 p 2 l2p kϕ1 − ϕ2 kp,ω l2 + 2p C2p l2q kv1 − v2 kYp 2  l2 − l2  pq i  p h p p 1 l2p , l2q kϕ1 − ϕ2 kp,ω + kv1 − v2 kYp , ≤ 2p C2p l2 max eωl2 p 2 2 and therefore J1 ≤ ϑ1 k(v1 , ϕ1 ) − (v2 , ϕ2 )k∗ , 1
1 1 l2 −l2
where ϑ1 = 2C2 l2p max eωl2 2 2 1 q l2 , l2q . On the other hand, using (A6) and the estimate (|a + b|)p ≤ 2p (|a|p + |b|p ), we get Z l2 p J2p = |σ2 (l, hϕ1 i, v1 (l)) − σ2 (l, hϕ2 i, v2 (l))| dl

l1 l2

≤

Z

l1 l2

≤

Z

l1

  p p 2p C1p |hϕ1 i − hϕ2 i| + |v1 (l) − v2 (l)| dl 2p C1p

≤ 2p C1p

 2 1 l − l2  q 2

1

2

 l2 − l2  pq 2

1

2

eωl2 kϕ1 − ϕ2 kp,ω

p

p

+ |v1 (l) − v2 (l)|

p



dl

p

eωl2 p l2 kϕ1 − ϕ2 kp,ω + 2p C1p kv1 − v2 kYp .

This yields the inequality J2 ≤ 2C1

 l2 − l2  pq 2

1

p

p

l2 eωl2 p kϕ1 − ϕ2 kp,ω + kv1 − v2 kYp

2 ≤ ϑ2 k(v1 , ϕ1 ) − (v2 , ϕ2 )k∗ , 1

where ϑ2 = 2C1 max(l2p



l22 −l21 2

 q1

 p1

eωl2 , 1). Putting ϑ = max(ϑ1 , ϑ2 ), we get

kK(v1 , ϕ1 ) − K(v2 , ϕ2 )kYp ≤ ϑ k(v1 , ϕ1 ) − (v2 , ϕ2 )k∗ . This proves (ii).



ω Remark 4.2 In the following lemma, we will show that TK is quasi-accretive on Xpω 1 whenever ω > max{0, l1 ln(ϑ)} where ϑ is the constant appearing in Lemma 4.1 (ii).

Lemma 4.3 Let p ∈ [1, +∞) and let ω be an arbitrary real number satisfying ω > max(0, l11 ln(ϑ)). If hypothesis (A6) holds true, then, there exist a constant δ := δ(p) > 0 such that (i) if p = 1, then, for all g1 , g2 ∈ X1ω , we have

1 ω −1 ω −1

(I − λTK ) (g1 ) − (I − λTK ) (g2 ) 1,ω ≤ kg1 − g2 k1,ω 1 − λδ for all λ ∈ (0, 1δ ); ω (ii) if p ∈ (1, +∞), then, for all ϕ1 , ϕ2 ∈ D(TK ), we have

ω ω (ϕ1 ) − TK (ϕ2 ), ψδ is ≥ −δ kϕ1 − ϕ2 kp,ω hTK

for some ψδ ∈ Jδ (ϕ1 − ϕ2 ) and λ ∈ (0, δ1 ).

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Proof

Vol.36 Ser.B

(i) Define the real δ(1) by δ(1) = ω + 1 and let g1 , g2 ∈ X1ω be such that ω −1 ϕ1 = (I + λTK ) (g1 ),

ω −1 ϕ2 = (I + λTK ) (g2 ).

Hence we have

ω −1 ω −1 kϕ1 − ϕ2 k1,ω = (I + λTK ) (g1 ) − (I + λTK ) (g2 ) 1,ω Z ∂(ϕ1 − ϕ2 ) ≤ −λ e−ω(l−a) sgn0 (ϕ1 − ϕ2 )dadl ∂a Ω Z + e−ω(l−a) (g1 − g2 )sgn0 (ϕ1 − ϕ2 )dadl Ω Z l2  Z l  ∂ −ω(l−a) ≤ −λ (ϕ1 − ϕ2 ) da dl e l1 0 ∂a Z l2  Z l  ∂e−ω(l−a)   +λ da dl |ϕ1 − ϕ2 | ∂a l1 0  Z l2  Z l + e−ω(l−a) |g1 − g2 | da dl l1

≤ −λ

Z

l2

l1



0

 ϕ1|Γ (l) − ϕ2|Γ (l) − e−ωl K(ϕ1|Γ , ϕ1 )(l) − K(ϕ2|Γ , ϕ2 )(l) dl 2 2 2 2

+λω kϕ1 − ϕ2 k1,ω + kg1 − g2 k1,ω .

ω Since ϕ1 , ϕ2 ∈ D(TK ), applying Lemma 4.1 (ii) we get n

o

kϕ1 − ϕ2 k1,ω ≤ λ e−ωl1 K(ϕ1|Γ2 , ϕ1 ) − K(ϕ2|Γ2 , ϕ2 ) Y − ϕ1|Γ2 − ϕ2|Γ2 Y 1

1

+λω kϕ1 − ϕ2 k1,ω + kg1 − g2 k1,ω

1
≤ λϑe−ωl1 kϕ1 − ϕ2 k1,ω + λ ϑe−ωl1 − 1 ϕ1|Γ2 − ϕ2|Γ2 Y1 +λω kϕ1 − ϕ2 k1,ω + kg1 − g2 k1,ω .

Since ϑe−ωl1 ≤ 1, we get
kϕ1 − ϕ2 k1,ω ≤ λ ω + 1 kϕ1 − ϕ2 k1,ω + kg1 − g2 k1,ω 1
kg1 − g2 k1,ω ≤ 1−λ ω+1 1 = kg1 − g2 k1,ω . 1 − λδ(1) ω (ii) Now, we consider the case p ∈ (1, +∞) with p > 1. Let ϕ1 , ϕ2 ∈ D(TK ) be such that 1 ϕ1 6= ϕ2 and consider the real δ(p) = ω + p . Putting ϕ = ϕ1 − ϕ2 , we can write ω ω hTK (ϕ1 ) − TK (ϕ2 ), ψδ is Z ∂ 1−p ≥ kϕkp,ω e−ω(l−a) |ϕ|p−1 (ϕ)sgn0 (ϕ)(a, l)dadl ∂a Ω    1−p Z l2  Z l kϕkp,ω ∂ p = e−ω(l−a) (|ϕ| ) (a, l)da dl p ∂a l1 0     Z l2 Z l ∂ e−ω(l−a) |ϕ|p Z l2 Z l 1−p  −ω(l−a)  kϕkp,ω p ∂e = (a, l)dadl − |ϕ| (a, l)dadl p ∂a ∂a l1 0 l1 0

No.5

A. Al-Izeri & K. Latrach: EXISTENCE AND UNIQUENESS RESULTS 1−p

=

kϕkp,ω

−ω

p Z

l2

l1

Z

Z

l2

l1 l

1237

  ϕ1|Γ (l) − ϕ2|Γ (l) p − e−ωl K(ϕ1|Γ , ϕ1 )(l) − K(ϕ2|Γ , ϕ2 )(l) p dl 2 2 2 2 p

e−ω(l−a) |ϕ| (a, l)dadl.

0

ω Since ϕ1 , ϕ2 ∈ D(TK ), applying Lemma 4.1 (ii), we get ω ω hTK (ϕ1 ) − TK (ϕ2 ), ψδ is 1−p

p

p o kϕkp,ω n −ωl e 1 K(ϕ1|Γ2 , ϕ1 ) − K(ϕ2|Γ2 , ϕ2 ) Y − ϕ1|Γ2 − ϕ2|Γ2 Y ≥ −ω kϕkp,ω − p p p



1 1 p 1−p p 1−p ≥ −ω kϕkp,ω − ϑp e−ωl1 kϕkp,ω kϕkp,ω − ϑp e−ωl1 − 1 kϕkp,ω ϕ1|Γ2 − ϕ2|Γ2 Y . p p p

Since ϑp e−ωl1 ≤ 1, we get

 1 ω ω kϕkp,ω = −δ(p) kϕkp,ω . hTK (ϕ1 ) − TK (ϕ2 ), ψδ is ≥ − ω + p  Let g ∈ Xpω and consider the problem ω (I + λTK )f = g,

(4.3)

ω where f must be sought in ∈ D(TK ). For any λ > 0, the solution of problem (4.3) is given Z 1 a s−a a −λ e λ g(s, l)ds. f (a, l) = e K(f |Γ2 , f )(l) + λ 0

(4.4)

For a = l, we get f (l, l) := f|Γ2 = e

− λl

1 K(f |Γ2 , f )(l) + λ

Z

l

e

s−l λ

g(s, l)ds.

(4.5)

0

In order to write abstractly equations (4.4) and (4.5), we define the following linear operators

 A : Y −→ Y , λ p p l  u → (Aλ u)(l) := u(l)e− λ ;  B : Y −→ X ω , λ p p a  u → (Bλ u)(a, l) := u(l)e− λ ;   Cλ : Xpω −→ Yp , Z 1 l s−l   g → (Cλ g)(l) := e λ g(s, l)ds λ 0

and

   Dλ : Xpω −→ Xpω , For u ∈ Yp , we have p

kAλ ukYp =

1 g → (Dλ g)(a, l) := λ

  Z

l2

l1

Z

a

e

s−a λ

g(s, l)ds.

0

Z p l1 −p lλ1 u(l) dl ≤ e−p λ e

l2

l1

p

l1

p

|u(l)| dl = e−p λ kukYp ,

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ACTA MATHEMATICA SCIENTIA

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which implies l1

kAλ kL(Yp ) ≤ e− λ . Similarly, for u ∈ Yp , we can write Z Z l2 Z l p a kBλ ukpX ω = e−ω(l−a) e− λ u(l) dadl ≤ p

l1

l2

l1

0

This yields

kBλ kL(Yp ,X ω ) ≤

(4.6) Z

l

a

e−p λ |u(l)|p dadl ≤

0

λ kukpYp . p

 λ  p1

. (4.7) p Next, using H¨ older’s inequality and the fact that 1 ≤ eωl2 e−ω(l−s) , for g ∈ Xpω , we have Z l p Z l2 s−l 1 λ |g(s, l)| ds e dl kCλ gkpYp ≤ p 0 l1 λ p Z l2 Z l  Z l 1 ≤ p |g(s, l)| ds dl λ l1 0 0  p1 p Z l2  Z l  q1  Z l 1 p ds dl ≤ p |g(s, l)| ds λ l1 0 0 p Z Z l2q ωl2 l2 l −ω(l−s) p e |g(s, l)| dsdl ≤ pe λ l1 0 p

p

l q eωl2 p = 2 p kgkp,ω λ and therefore

1

kCλ kL(X ω ,Yp ) p

l q ωl2 ≤ 2e p . λ

(4.8)

In the same way we can write Z l p Z l2 Z l s−l p −ω(l−a) 1 λ e kDλ gkXpω ≤ e |g(s, l)| ds dadl λp 0 l1 0 p Z l Z l2 Z l 1 |g(s, l)| ds dadl ≤ p 0 λ l1 0  p1 p Z l2 Z l  Z l  q1  Z l 1 p ≤ p ds |g(s, l)| ds dadl λ l1 0 0 0 ≤

l2p ωl2 e kgkpp,ω λp

and consequently

l2 ωlp2 e . (4.9) λ Now using these operators and the fact that f must satisfy the boundary conditions, equations (4.4) and (4.5) may be written in the form  f = B K(f , f ) + D g, λ λ |Γ2 f |Γ = Aλ K(f |Γ , f ) + Cλ g. kDλ kL(Xpω ,Xpω ) ≤

2

2

Accordingly, (f |Γ2 , f ) is a solution to the fixed point problem Fλ (v, w) = (v, w) on the product space Yp × Xpω where   Fλ (v, w) = (F1λ , F2λ )(v, w) = Aλ K(v, w) + Cλ g, Bλ K(v, w) + Dλ g . (4.10)

No.5

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A. Al-Izeri & K. Latrach: EXISTENCE AND UNIQUENESS RESULTS

Lemma 4.4 If assumption (A6) holds true. There exists a constant λp > 0, such that ω for any λ ∈ (0, λp ) and g ∈ Xpω , there exists a function f ∈ D(TK ) such that ω f + λTK (f ) = g.

(4.11)

Proof As we have seen above, in order to solve problem (4.11), it suffices to prove that the fixed point problem Fλ (u, f ) = (u, f ) has a unique solution in Yp × Xpω where the operator Fλ is defined by (4.10). To do so, let λp be the real defined by n  o l1 λ  p1 λp := sup λ > 0 : e−p λ + ϑ<1 , p where ϑ is the constant given in Lemma 4.1 (ii). Let (u1 , f1 ), (u2 , f2 ) ∈ Yp × Xpω . Using Lemma 4.1 (ii) together with estimates (4.6)–(4.9), we get p

kFλ (u1 , f1 ) − Fλ (u2 , f2 )k∗

p

p

= kAλ K(u1 , f1 ) − Aλ K(u2 , f2 )k∗ + kBλ K(u1 , f1 ) − Bλ K(u2 , f2 )k∗ l1 λ p p ≤ e−p λ kK(u1 , f1 ) − K(u2 , f2 )k∗ + kK(u1 , f1 ) − K(u2 , f2 )k∗ p  l1 λ p p ≤ e−p λ + ϑ k(u1 , f1 ) − (u2 , f2 )k∗ . p This yields

 l1 λ  p1 ϑ k(u1 , f1 ) − (u2 , f2 )k∗ . kFλ (u1 , f1 ) − Fλ (u2 , f2 )k∗ ≤ e−p λ + p Taking into acount of the definition of the real λp , it is clear that, for each λ ∈ (0, λp ), the operator Fλ is then a strict contraction mapping. Applying the fixed point theorem of Banach ω we infer that, for such λ, problem (4.11) has a unique solution in D(TK ).  ω Remark 4.5 It follows from Lemmas 4.3 and 4.4 that TK is a quasi-m-accretive operator ω ω in Xp . Note also that, in this case, D(TK ) is a dense subset of Xpω (see, for example [12, 13]).

The main result of this section is the following.


Theorem 4.6 Let r > 0, ω > max 0, l11 ln(ϑ) and assume that conditions (A3)–(A6) are satisfied. Then the following hold true. (i) If p ∈ (1, +∞), then, for each f0 ∈ Xp , problem (4.2) has a local weak solution. If ω the initial data f0 belongs to D(TK ), then problem (4.2) has a local strong solution. Assume, further, that (A7) is satisfied, then problem (4.2) has a unique global strong solution for each ω ω f0 ∈ D(TK ) and it is a unique global weak solution if f0 ∈ / D(TK ). (ii) If p = 1, then, for each f0 ∈ X1 , problem (4.2) has a local mild solution. ω Proof (i) According to Remark 4.5, TK is quasi-m-accretive on Xpω . It follows from 1 Lemma 4.1 that Fp is locally Lipschitzian on Xpω . Hence, the use of the fact that Xpω is reflexive together with Theorem 5.2, implies the existence of a local strong solution f to (4.2) ω whenever the initial data f0 belongs to D(TK ). ω ω Next, let f0 ∈ / D(TK ). Since D(Tk ) is dense in Xpω , there exists a sequence (fn )n∈N ω contained in D(TK ) such that fn → f0 as n → ∞ and put M := sup{kfn kp,ω : n = 0, 1, · · · }. A similar proof to that of Theorem 5.2 allows us deduce the existence of real TM > 0 such that

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ACTA MATHEMATICA SCIENTIA

for each n ∈ N, the problem

Vol.36 Ser.B

  ω  ∂f (t) + TK (f (t)) = F1p (f (t)), ∂t  f (0) = f ∈ X ω n

p

possesses a unique strong solution, say gn , on the interval (0, TM ). To prove the existence of a local weak solution, we have only to check that the sequence (gn )n∈N is a Cauchy sequence in the Banach space C(0, TM ; Xpω ). This follows immediately from (5.3). To discuss the existence of a unique global strong solution to problem (4.2), we assume that condition (A7) is satisfied. If g ∈ J(f ), then one can write (−F1p (f ), g)s = (σ1 (a, l, hf i)f − B1 (f ), g)s  Z l2  Z l  −ω(l−a) p−1 = kgk1−p e sgn0 (g)|g| σ1 (·, ·, hf i)f da dl p,ω −

Z

l2

l1

Z ·

l2

l1

Z Z

l1 l

0

e−ω(l−a)

0

l′

e

ω(l′ −a′ ) −ω(l′ −a′ )

e

0

2

2

sgn0 (g) |g|

≥ σ 1 kf kp,ω − e k 1 e2ω(l1 −l2 ) kf kp,ω 2

This yields to

≥ σ 1 kf kp,ω − e k 1 e2ω(l1 −l2 )

p−1

Z

l2

l1

Z

l

   ′ ′ e k1 (·, ·, ·, ·, f )f da dl da dl

dadl

0

l22 − l12 2 kf kp,ω . 2

k 1 e2ω(l1 −l (−F1p (f ), g)s ≥ σ 1 kf k2p,ω − e

2

2 ) l2

− l12 kf k2p,ω ≥ −α kf k2p,ω , 2

l2 −l2

where α : = e k 1 e2ω(l1 −l2 ) 2 2 1 − σ 1 . Applying Theorem 5.2, we conclude that f is a global strong solution. (ii) By hypothesis f0 ∈ X1ω . Let r > 1 be a real such that kf0 k1,ω ≤ r − 1. Let R be the function defined on X1ω by   if kgk1,ω ≤ r,  g R(g) = r  g if kgk1,ω ≥ r.  kgk1,ω

ω We know from Lemmas 4.3 and 4.4 (see also Remark 4.5) that TK is ω-m-accretive. The use 1 Lemma 4.1 (i) together with [11] shows that the function F1 (R(·)) is 2Cr -Lipschitz on X1ω where Cr is the Lipschitz constant of F11 on Br1,ω (cf. Lemma 4.1 (i)). Consequently, the operator ω ω TK − F11 (R(·)) is (ω + 2Cr )-m-accretive on X1ω or, equivalently, TK − F11 (R(·)) is quasi-accretive on X1ω . Applying Corollary 4.1 in [3] we infer that the problem  f ′ (t) + T ω (f (t)) = F1 R(f (t))
, 1 K f (0) = f0

has a unique mild solution f on X1ω . ω ω Next, using the fact that D(TK ) is dense in X1ω , we can choose h0 ∈ D(TK ) such that 1,ω ω 1 ω kh0 − f0 k1,ω ≤ 1/2. It is clear that h0 ∈ Br . Set η0 = TK (h0 )−F1 (R(h0 )) = TK (h0 )−F11(h0 ).

No.5

A. Al-Izeri & K. Latrach: EXISTENCE AND UNIQUENESS RESULTS

1241

The use of (5.2) yields ku(t) − h0 k1,ω ≤ e

ωt

kh0 − f0 k1,ω +

Z

t

et−s [−η0 , u(s) − h0 ]s ds

0

  1 ≤ eω t kh0 − f0 k1,ω + kη0 k1,ω (1 − eω t ) . ω Accordingly ku(t)k1,ω ≤ kh0 k1,ω + 1 ≤ r for 0 ≤ t ≤ Tr where Tr > 0 is suitably chosen. This shows that u(·) is a mild solution to problem (4.2) in the interval [0, Tr ] which completes the proof.  Remark 4.7 It should be noticed that in the case where the perturbation is trivial (P = 0), problem (4.1) coincide with problem (DE)–(DC) in [20]. So, Theorem 4.6 extends Theorems 5.5 and 6.5 in [20] to a more general framework.

5

Annex

Let (X, k · k) be a real Banach space. An operator A : D(A) ⊆ X → 2X is said to be accretive if the inequality kx − y + λ(u − v)k ≥ kx − yk holds for all λ ≥ 0, x, y ∈ D(A) and u ∈ Ax, v ∈ Ay. If, in addition, R(I + λA) (i.e., the range of the operator I + λA), is for one, hence for all, λ > 0, precisely X, then A is called m-accretive. Accretive operators were introduced by Browder [10] and Kato [14] independently. Finally, A is said to be quasi-accretive (quasi-m-accretive), if there exists w ∈ R such that A + wI is accretive (respectively m-accretive), in this case we say also that A is w-accretive (w-m-accretive respectively). Notice that A is accretive if and only if A is quasi-accretive with w = 0. The operators which are quasi-m-accretive play an important role in the study of nonlinear partial differential equations. Consider the Cauchy problem   u′ (t) + A(u(t)) ∋ f (t), t ∈ (0, T ), (5.1)  u(0) = x0 ∈ D(A), where A is quasi-m-accretive on X and f ∈ L1 (0, T, X). Given ǫ > 0. An ǫ-discretization on [0, T ] of the equation u′ (t) + A(u(t)) ∋ f (t) consists of a partition 0 = t0 ≤ t1 ≤ t2 ≤ · · · ≤ tN of the interval [0, tN ] and a finite sequence (fi )N i=1 ⊆ X such that ti − ti−1 < ǫ for i = 1, · · · , N, T − ǫ < tN ≤ T, N Z X i=1

ti

kf (s) − fi kds < ǫ.

ti−1

ǫ A DA (0 = t0 ≤ t1 ≤ t2 ≤ · · · ≤ tN ; f1 , · · · , fN ) solution to (5.1) is a piecewise constant function z : [0, tN ] → X whose values zi on (ti−1 , ti ] satisfy the finite difference equation

zi − zi−1 + A(zi ) ∋ fi , ti − ti−1

i = 1, 2, · · · , N.

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Such a function z = (zi )N i=1 is called an ǫ-approximate solution to the Cauchy problem (5.1) if it further satisfies kz(0) − x0 k ≤ ǫ. It is well known (see [3, Corollary 4.1]) that (5.1) has a unique mild solution in the sense that there exists a unique continuous function u : [0, T ] → D(A) such that u(0) = x0 , and moreover, for each ǫ > 0 there is an ǫ-approximate solution z of u′ + A(u) ∋ f on [0, T ] such that ku(t) − z(t)k ≤ ǫ for all t ∈ [0, T ] with u(0) = x0 . If u is the mild solution of problem (5.1), then for each (x, y) ∈ A and 0 ≤ s ≤ t ≤ T, we have Z t

ku(t) − xk ≤ ew(t−s) ku(s) − xk +

ew(t−τ ) [f (τ ) − y, u(τ ) − x]s dτ,

(5.2)

s

here the function [·, ·]s : X × X → R is defined by [y, x]s = sup{x∗ (y) : x∗ ∈ J1 (x)}, where ∗ J1 : X → 2X is the duality mapping on X, i.e., J1 (x) = {x∗ ∈ X ∗ : x∗ (x) = kxk, kx∗ k = 1}. This means that u is an integral solution to equation (5.1) in the sense of B´enilan [5] and moreover both concepts of solution coincide under our context. If u, v are integral solutions of u′ (t) + A(u(t)) ∋ f (t) and v ′ (t) + A(v(t)) ∋ g(t), respectively, with f, g ∈ L1 (0, T, X), then Z t wt ku(t) − v(t)k ≤ e ku(0) − v(0)k + ew(t−s) kf (s) − g(s)kds. (5.3) 0

A strong solution of problem (5.1) is a function u ∈ W 1,∞ (0, T ; X), i.e., u is locally absolutely continuous and differentiable almost everywhere, and u′ (t) + A(u(t)) ∋ f (t) for almost all t ∈ [0, T ]. Concerning the existence of strong solutions, the following theorem is known (see [3, Theorem 4.5] or [6, p. 108]). Theorem 5.1 If X is a Banach space with the Radon-Nikodym property, A : D(A) ⊆ X → 2X is a quasi-m-accretive operator, and f ∈ BV(0, T ; X), i.e., f is a function of bounded variation on [0, T ], then problem (5.1) has a unique strong solution whenever x0 ∈ D(A). Our results rely on the following theorem. Theorem 5.2 [3, p.150] Let X be a reflexive Banach space and let A be a quasi-maccretive operator in X. Let F : X → X be locally Lipschitz. Then, for each y0 ∈ D(A), there is a local strong solution to the problem   u′ (t) + Au(t) ∋ F (u(t)),  u(0) = y0 . 2

Moreover, if h−F u, wi ≥ −α kuk + β,

∀(u, w) ∈ J where

J(u) = {w ∈ X ∗ : hu, wi = kuk2X , and kwkX ∗ = kukX }. Then, the solution is global. On the other hand, we say that u ∈ C(0, T ; X) is a weak solution of problem (5.1) if there are sequences (un ) ⊆ W 1,∞ (0, T ; X) and (fn ) ⊆ L1 (0, T ; X) satisfying the following four conditions: 1. u′n (t) + Aun (t) ∋ fn (t) for almost all t ∈ [0, T ], n = 1, 2, · · · ;

No.5

A. Al-Izeri & K. Latrach: EXISTENCE AND UNIQUENESS RESULTS

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2. lim kun − uk∞ = 0; n→∞

3. u(0) = x0 ; 4. lim kfn − f k1 = 0. n→∞ With respect to the existence of weak solutions, the following result, which is an easy consequence of Theorem 5.1, is important. Theorem 5.3 Let X be a Banach space with the Radon-Nikodym property. Then problem (5.1) admits a unique weak solution which is the unique integral solution of this problem. We now recall some important facts regarding accretive operators which will be used in our paper (see, for example [3]). Proposition 5.4 Let A : D(A) → 2X be an operator on X. The following conditions are equivalent: • A is an ω-accretive operator, • the inequality [u−v, x−y]s ≥ −ωkx−yk, holds for every x, y ∈ D(A) and u ∈ Ax, v ∈ Ay, • For each 0 < λ < ω1 the resolvent Jλ := (I +λA)−1 : R(I +λA) → D(A) is a single-valued 1 1−ωλ -lipschitzian mapping. Let Σ be a subset of RN . A function g : Σ × C −→ C is said to satisfy the Carath´eodory conditions on Σ × C if (1) the function t −→ g(t, u) is measurable on Σ for all u ∈ C, (2) the function u −→ g(t, u) is continuous on u for almost all t ∈ Σ. If g satisfies the Carath´eodory conditions, then we can define an operator Ng on the set of functions ψ : Σ −→ C by (Ng ψ)(z) := g(z, ψ(z)) for every z ∈ Σ. The operator Ng is called the Nemytskii operator generated by g. In Lp -spaces the Nemytskii operator has been extensively investigated (see [2] and the references therein). However, we recall the following result due to Krasnoselskii which states a basic fact for the theory of these operators on Lp -spaces. Proposition 5.5 Assume that g is a Carath´eodory function. If the operator Ng acts from Lp1 into Lp2 , then Ng is continuous and takes bounded sets into bounded sets. For the proof of this proposition we refer, for example, to [2]. References [1] Al-Izeri A, Latrach K. A nonlineat age-structured of populations dynamics with inherited properties. Mediterr J Math, DOI: 10.1007/S00009-015-0575-6 [2] Appell J, Zabrejko P P. Nonlinear Superposition Operators. Cambridge: Cambridge University Press, 1990 [3] Barbu V. Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer, 2010 [4] Barbu V, Iannelli M. The semigroup approach to non-linear age-structured equations. Rend Istit Univ Trieste, 1997, 28: 59–71 ´ [5] B´ enilan Ph. Equations D’´ evolution dans un espace de Banach Quelconque et Applications. Orsay: Th` ese ´ de doctorat d’Etat, 1972 [6] B´ enilan Ph, Crandall M G, Pazy A. Evolution equations governed by accretive operators. Forthcoming [7] Boulanouar M. A mathematical study in the theory of dynamic population. J Math Anal Appl, 2001 255: 230–259 [8] Boulanouar M. A mathematical analysis of a model of structured population (I). Differential Integral Equations, 2012, 25: 821–852

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[9] Boulanouar M. A mathematical analysis of a model of structured population (II). J Dyn Control Syst, 2012, 18: 499–527 [10] Browder Felix E. Nonlinear mappings of nonexpansive and accretive type in Banach spaces. Bull Amer Math Soc, 1967, 7(3): 875–882 [11] De Figueiredo D, Karlovitz L. On the radial projection in normed spaces. Bull Amer Math Soc, 1967, 73: 364–368 [12] Garcia-Falset J. Well-posedness of a nonlinear evolution equation arising in growing cell population. Math Meth Appl Sci, 2011, 34: 1658–1666 [13] Garcia-Falset J, Latrach K, Zeghal A. Existence and uniqueness results for a nonlinear evolution equation arising in growing cell populations. Nonlinear Anal, 2014, 97: 210–227 [14] Kato T. Nonlinear semigroups and evolution equations. J Math Soc Japan, 1967, 19: 508–520 [15] Latrach K, Mokhtar-Kharroubi M. On an unbounded linear operator arising in the theory of growing cell population. J Math Anal Appl, 1997, 211: 273–294 [16] Lebowitz J L, Rubinow S I. A theory for the age and generation time distribution of a microbial population. J Math Biol, 1974, 7(3): 17–36 [17] Lods B, Mokhtar-Kharroubi M. On the theory of a growing cell population with zero minimum cycle length. J Math Anal Appl, 2002, 266: 70–99 [18] Matsumoto T, Oharu S. Semilinear evolution equation with nonlinear constraints and applications. J Evol Equ, 2002, 2: 197–222 [19] Rotenberg R. Transport theory for growing cell populations. J Theor Biol, 1983, 103: 181–199 [20] Shanthidevi C N, Matsumoto T, Oharu S. Nonlinear semigroup approach to age structured proliferating cell population with inherited cycle length. Nonlinear Anal Real World Appl, 2008, 9: 1905–1917 [21] Webb, G F. A model of proliferating cell populations with inherited cycle lenght. T Math Biol, 1986, 23: 269–282 [22] Webb G F. Dynamic of structured populations with inherited properties. Compt Math Appl, 1987, 13: 749–757