On chaotic and stable behaviour of the von Foerster – Lasota equation in some Orlicz spaces

We study the chaotic and stable behaviour of the von Foerster–Lasota equation in Orlicz spaces with homogeneous φ-function of any positive degree. This work is, in particular, the generalization of the asymptotic properties of the von Foerster– Lasota equation in integrable spaces with exponent p greater than or equal to 1.


INTRODUCTION
In 1926 McKendrick [11] proposed the first age-dependent model of the dynamics of a population.He assumed that the state of a population in time t is described by a function u(.,t).The number of individuals in age from the interval [x 1 , x 2 ] equals x 2 x 1 u(x,t)dx.From this paper the equation follows, called in the literature as McKendrick equation or more often as von Foerster equation.
McKendrick's model was generalized in many ways, among others by Gurtin and MacCamy [5] or by the authors [3].This equation is part of the mathematical description of a particular population, as the population of red blood cells is (see [14]).It is the model with a feedback, because a circulatory system controls the global number of erythrocytes to some quantity optimum which can change.It happens, for example, during mountain trips or in a case of any disease of the respiratory system.
In their next paper [7] the authors of [14] introduced a new model of precursor cells.There the main assumption is the fact that cells mature with different intensity.The form of the equation is the following: where c : [0, 1] → R and f : [0, 1] × [0, ∞) → R are the given functions fulfilling suitable conditions.In this model x denotes the degree of cell differentiation (maturity) and x 1 x 0 u(t 0 , x)dx is the number of cells having at time t = t 0 the value x in the interval [x 0 , x 1 ].The coefficient c is the velocity of cell differentiation.Because of biological application the above equation is still the matter of interest for many mathematicians: Lasota and Pianigiani [8], Rudnicki [13], Łoskot [10], and Lasota and Szarek [9].In this paper we consider a simpler case of the equation, that is Study of periodic or chaotic solutions of the von Foerster-Lasota equation is interesting from a medical point of view.In this paper we consider the semidynamical system (T t ) t 0 that is connected with the presented equation and fix our attention on the existence of its periodic and chaotic solutions in some function spaces.Such behaviour is already well described in the space of continuously differentiable functions [6], in the L p space (p > 1) or in the space of Hölder continuous functions [2].This paper is the generalization of the results referring to the asymptotic properties of the above equation in the integrable spaces L p , but in our case with the exponent p < 1.In order to do that, we present the definitions of Orlicz spaces with Luxemburg and F-norm in Section 2. We mention also some basic properties of such spaces (see [12]).In Section 3 we define the dynamical system connected with the von Foerster-Lasota equation.We consider the conditions of chaos and stability of such a semidynamical system and use there Devaney's definition of chaos (see [4]).Let us remind here that according to Devaney, a dynamical system (F t ) t 0 defined in a metric space (V, d) is chaotic as • (F t ) t 0 has a property of sensitive dependence on initial conditions in the sense of Guckenhaimer, i.e. there is a positive real number M (a sensitivity constant) such that for every point v ∈ V and every ε > 0 there exist w ∈ B(v, ε) and t > 0, such that d(F t v, F t w) M; Section 4 contains the generalization of the von Foerster-Lasota equation and describes common asymptotic properties of two dynamical systems: basic and generalized.We will see that all properties of these two systems depend on one common value of the parameter γ.

ORLICZ SPACES AND LUXEMBURG NORM
Definition 2.1.Let X be a real vector space.A functional ρ : X → [0, ∞] is called a modular if there holds for arbitrary x, y ∈ X Let (Ω, Σ, µ) be a measure space, where Ω is a nonempty set, Σ is a σ -algebra of a subset of Ω, and µ is a nonnegative, complete measure not vanishing identically.A real function ϕ : R + → R + , where R + = [0, ∞), is called ϕ-function if it is nondecreasing and continuous and such that ϕ(0 Let X be the set of all real-valued, Σ-measurable, and finite µ-almost everywhere functions on Ω, with equality µ-almost everywhere.Then for every Definition 2.3.The modular space X ρ will be called Orlicz space and denoted by L ϕ (Ω, Σ, µ) (or briefly L ϕ ): and is strongly stable for γ − 1 p .For 0 < p < 1 the modular ρ(x) = b a |x(t)| p dt is p-convex modular, and such an Orlicz space is only the Fréchet space with the F-norm |x| F = b a |x(t)| p dt.This paper is an attempt at the generalization of the earlier results concerning the asymptotic properties of the von Foerster-Lasota equation in the L p (0, 1) (p 1) space.Therefore, our intention is to consider the Orlicz space X ρ with the ϕ-function homogeneous of any degree, i.e. for all real k > 0, ϕ(kx) = k α ϕ(x), where α is a real number (a degree).
In this section we introduce also some definitions and notations which will appear in the subsequent chapters.
Definition 2.5.A function v 0 ∈ V is a periodic point of the semigroup (T t ) t 0 with a period t 0 0 if and only if T t 0 v 0 = v 0 .A number t 0 > 0 is called a principal period of a periodic point v 0 if and only if the set of all periods of v 0 is equal to Nt 0 .
Definition 2.6.The semigroup (T t ) t 0 is strongly stable in V if and only if for every v ∈ V , lim t→∞ T t v = 0 in V.

CHAOTIC AND STABLE SOLUTIONS OF THE VON FOERSTER-LASOTA EQUATION
We consider the partial differential equation with the initial condition where v belongs to some normed vector space V of functions defined on [0, 1].Define the function T t by the formula where u is the unique solution of (3.1) and (3.2), see [6].If for every v ∈ V and t 0 the function T t belongs to V , then the family (T t ) t 0 is a semigroup on the space V .Now we will formulate some theorems describing the chaotic and stable behaviour of the above dynamical system.We will consider these properties in some Orlicz spaces with the ϕ-function fulfilling the following Assumption.The ϕ-function ϕ : R + → R + is homogeneous of the degree 0 < p < 1.
So, under this assumption we consider the ϕ-function ϕ(x) = Cx p , where C > 0 and 0 < p < 1.It is the only possible form of the ϕ-function fulfilling the above assumption.
p , then for any t 0 > 0 there exists a periodic point v 0 ∈ L ϕ of the dynamical system (T t ) t 0 .
for x e −t at the suitable choice of t.We should show that the above function w belongs to the space L ϕ (0, 1).
From the assumption we have e −t(γ p+1) < 1. Therefore ρ [0,1] (β w) → 0, as β → 0 + .It turns out from this fact that v 1 , v 2 ∈ L ϕ (0, 1).So w ∈ L ϕ (0, 1).Besides, from the above equality we can draw the following conclusion: From the estimation it turns out that for t large enough we obtain We learn from the above that the intersection of two sets B(v 2 , ε 2 ) and T t (B(v 1 , ε 1 )) is not empty.So we conclude that the dynamical system (T t ) t 0 is transitive in the space L ϕ (0, 1).
As proved in the paper of Banks et al. [1] the sensitive dependence of the dynamical system on initial conditions in the sense of Guckenhaimer appears immediately from its transitivity and density of the set of its periodic points.This is expressed by the following corollary: p , then the dynamical system (T t ) t 0 is chaotic in the sense of Devaney in the L ϕ (0, 1) space.Theorem 3.5.If γ − 1 p , then the semigroup (T t ) t 0 is strongly stable in the L ϕ (0, 1) space.
Proof.Let v ∈ L ϕ (0, 1) be an arbitrary function.For s > 0 we obtain From the above we get Due to e 1+γ p 1 one has |v| F [0,e −t ] → 0 as t → ∞.This proves the strong stability of the system (T t ) t 0 in the L ϕ (0, 1) space.

GENERALIZATION OF THE VON FOERSTER-LASOTA EQUATION
Let us consider a more general form of the equation with the initial condition where v belongs to some normed vector space V of functions defined on [0, 1] and λ : [0, 1] → R is a given continuous function.Let a semidynamical system T t be given by the formula where u(t, x) is the unique solution of (4.1), (4.2) and is given by the formula We are interested in finding a connection between two equations: (3.1), presented in Section 3, and (4.We assume that the above integral is convergent.
It means that h Remark 4.2.Many dynamical properties transfer by topological equivalence, for example, stability, the density of the set of periodic points, the existence of fixed points of the equation or its periodic orbits and many others.
The connection between the dynamical systems (3.1) and (4.1) described above can be illustrated by the following diagram: We can show that this diagram is commutative.
holds.Then we have the following equivalence: the function u belongs to the space L ϕ (0, 1) if and only if u ∈ L ϕ (0, 1).
Due to the commutativity of the diagram we have the topological equivalence of the systems (T t ) t 0 and ( T t ) t 0 .Therefore the properties of the system (T t ) t 0 described in Section 3 (the existence of the periodic points for any time, the density of the set of periodic points, transitivity and also stability) transfer on the system ( T t ) t 0 .All these properties depend on the value γ = λ (0).Moreover, Theorem 4.3 shows that the solutions of equations (3.1) and (4.1) stay in the same space L ϕ (0, 1) under some assumptions concerning the function λ .It provides the property of sensitive dependence on initial conditions for the system ( T t ) t 0 , which is a metric, not topological property.So if, by appropriate assumptions, the system (T t ) t 0 is chaotic or stable, or has got a dense subset of its periodic points, then the system ( T t ) t 0 has got exactly the same properties.Thus all properties of the system ( T t ) t 0 depend on the value γ = λ (0).If λ (0) > − 1 p , then for any t 0 the periodic solution exists and the set of periodic points (4.1) is dense in the L ϕ (0, 1) space.If λ (0) − 1 p , then the system is strongly stable.

CONCLUSIONS
This work is the generalization of the results included in paper [2], which treat the existence of periodic solutions, problem of chaos, and stability of the von Foerster-Lasota equation in integrable spaces with the exponent p, where 1 p < ∞.The value of the coefficient γ is decisive.If γ > − 1 p , then for any t 0 the periodic solution exists for which t 0 is its principal period.Under the same inequality the set of periodic points (3.1) is dense in the L p (0, 1) space.If γ − 1 p , then the system is strongly stable.So all properties of the system (T t ) t 0 , in this Banach space, are similar as in the Fréchet space presented above.Thus the results of this paper allow us to judge about asymptotic properties of the von Foerster-Lasota equation in any L p space irrespective of the value of the exponent p.

Example 2 . 4 .
called the Luxemburg norm.It is known that the space L ϕ with the norm x L is a Banach space.The L p spaces over the interval [a, b] are examples of the Orlicz spaces with the modular ρ(x) = b a |x(t)| p dt which is convex for p 1. The paper [2] considers the properties of the von Foerster-Lasota equation in such a space.All of them depend on the critical value of the coefficient γ which equals − 1 p .It means that the solution of the equation displays chaotic behaviour in the sense of Devaney for γ > − 1 p 1).It is easy to check that if u and u are the solutions of equations (3.1) and (4.1), respectively, we have the equality u(t, x) = κ(x) u(t, x), (4.4)where κ(x) = e x 0 λ (0)−λ (s) s ds and γ = λ (0).(4.5) 1-Convex modulars are called convex.|x + y| |x| + |y|, 3. for each scalar a, |ax| = |x| when |a| = 1, 4. for each scalar a k and a if a k → a and |x k Definition 4.1.Two dynamical systems (F t ) t 0 and (G t ) t 0 , where F t : X → X and G t : Y → Y, are topologically equivalent if there exists homeomorphism h : X → Y , such that the following diagram is commutative: