Differential functional von Foerster equations with renewal

The paper is dedicated to the memory of Professor A. Lasota. Von Foerster and Volterra-Lotka equations arise in biology, medicine and chemistry, [1,7,11,12,14]. The independent variables xj and the unknown function u stand for certain features and densities, respectively. It follows from this natural interpretation that xj > 0 and u > 0. Von Foerster model is essentially nonlocal, because it contains the total size of population ∫


Introduction
The paper is dedicated to the memory of Professor A. Lasota.Von Foerster and Volterra-Lotka equations arise in biology, medicine and chemistry, [1,7,11,12,14].The independent variables x j and the unknown function u stand for certain features and densities, respectively.It follows from this natural interpretation that x j 0 and u 0. Von Foerster model is essentially nonlocal, because it contains the total size of population u(t, x)dx.
Existence results for certain von Foerster type problems have been established by means of the Banach contraction principle, the Schauder fixed point theorem or iterative methods, see [2] and [3][4][5].Nonlocal terms always cause huge problems.Satisfactory conditions for convergence of iterative methods were provided in [9], where (for the sake of simplicity) some boundary data were prescribed.This forced additional restrictions on the (tangential!)flow of bicharacteristics near the lateral boundary.
In the present paper we generalize the L ∞ ∩ L 1 -convergence results of [9] to the case of renewal boundary conditions with natural assumptions on the flow of bicharacteristics.An associate result to [9] on fast convergent quasi-linearization methods has been published in [8].The renewal condition of the form u(t, 0) = ∞ 0 k(t, x, y) u(t, x)dx is interpreted as giving birth to young individuals at the age 0 by mature individuals (0 < x < +∞).In A. Lasota's investigations there were included some delay effects like u(t − r, x)dx.Such cases are also available using the method of the present paper.Investigations of finite difference schemes on large meshes with strongly nonlocal functionals were analyzed in [15].Since the renewal case is complex, we arrive at its reduction to the one with given data on the lateral boundary.In fact, the unknown renewal part is expressed by a Green function.This approach can be extended to many species models.There is also a difference between the renewal case and [9]: the renewal involves such coupling of the unknown function that excludes monotone iterations.

Formulation of the differential problem.
For each function w defined on [−τ 0 , a], we have the Hale functional w t (see [6]), which is the function defined on [−τ 0 , 0] by For each function u defined on E 0 ∪ E, we similarly write a Hale-type functional u (t,x) , defined on B by u (t,x) (s, y) = u(t + s, x + y) for (s, y) ∈ B. Let Consider the differential-functional equation where with the initial conditions and with the renewal condition where k : ∂E × R n + → R n + .We are looking for Caratheodory's solutions to (1)-(4), see [2] and [10].The functional dependence includes a possible delayed and integral dependence of the Volterra type.The Hale functional z[u] t takes into consideration the whole population within the time interval [t − τ 0 , t], whereas the Hale-type functional u (t,x) shows the dependence on the density u locally in a left neighbourhood of (t, x).For simplicity we assume in the paper that n = 1.Notice that it is possible to extend the result to the case n > 1 with quite technical multiple integrals on the lateral boundary.

Bicharacteristics
First, for a given function z ∈ C([−τ 0 , a], R + ), consider the bicharacteristic equations for problem (1), (3): Denote by ) the bicharacteristic curve passing through (t, x) ∈ E, i.e., the solution to problem (5).We consider its maximal (left) existence domain to be an interval [α(t, x), t], where α(t, x) = 0 or 0 < α(t, x) t.This alternative splits E into two parts E 0 [z] and E + [z].Next, we consider the following equation with the initial condition and (with the brief notation α = α(t, x)) In the latter equation ( 6) the existence of a suitable extension ṽ of v to the lateral boundary is complicated.We discuss this topic later on.
Assume that: (V0) v ∈ CB (E 0 , R + ) (non-negative, bounded and continuous function); (V2) the function v satisfies the Lipschitz condition (C0) c j : Ω 0 → R + are positive, continuous and A continuous function σ : [0, a] × R + → R + is said to be a Perron comparison function if σ(t, 0) ≡ 0 and the differential problem y = σ(t, y), y(0) = 0 has the only zero solution.We call it uniform if σ, multiplied by any positive constant, is also a Perron comparison function.We call it monotone if σ is non-decreasing in the second variable.
(W1) There exists a function + is bounded and continuous, and a • k ∞ • c < 1.The latter condition states that the length of the interval [0, a] is sufficiently small.This assumption is superfluous if k(t, x, y) = 0. Lemma 2.1 If the conditions (V0), (Λ1), (K0) are satisfied, then any solution u of equation ( 6) has the estimate and Proof.The first inequality is standard.The second one will be explained in the following subsection.

The fixed point equation.
Let where we put L W (s) = 0 for s ∈ [−τ 0 , 0], and Consider the operator T : Z → Z given by the formula where u = u[z] ∈ C 1 (B, R + ) is the solution of ( 6)- (7) with the initial condition u[z](t, x) = v(t, x) on E 0 .The function u = u[z] has the following integral representation where η(s) = η[z](s; t, x) and α = α(t, x) (α depends on z).By Lemma 2.1, we write (11) in the following way for t 0. The bicharacteristics admit the group property: Hence ( 13) can be written in the form where η(s) = η[z](s; 0, y), α = α(t, x) and S t is the set of x ∈ R n + such that α(t, x) > 0.
Proof.The renewal condition (4) for (t, x) ∈ ∂E can be rewritten as follows The first term is a bounded operator of v, see (12).From (12), the second term is equal to St k(t, x, y) v(α, η(α)) exp t α λ s, η(s), u (s,η(s)) , z s ds dy, where η(s) = η[z](s; t, y) and α = α(t, y).By (K0) the second term is small (has the norm less than 1).The above lemma explains the estimate of Lemma 2.1.The next statement is crucial in our paper.
Lemma 2.5 Under all previous assumptions, any solution z of the fixed point equation for (14) has the representation where the Green function G has the same estimate as the first kernel in (14), multiplied by some constant.
Proof.The assertion follows from a Neumann series expansion for u aided by the renewal condition.
Proof.(of Theorem 3.1) Denote Then we have the estimates where P (k) (s; t, x) = ∆η (k) (s; t, x) + ∆u (k) s + ∆z (k) s .In the above estimate the constant C is generic (depending on the data).
Denote Lλ = a 0 L λ (s)ds and Consider the comparison equations with ψ (0) (t) = Z(t) and ψ(0 The remaining part of the proof is split into several auxiliary lemmas.
The constants Ĉk have an upper estimate of the form AQ k , thus ψ(t) ≡ 0 in a neighbourhood of 0 (because ψ(t) AQ k t kθ/2 ).Lemma 3.4 Under the assumptions of Theorem 3.1 the sequences {z Proof.We intend to find the following estimates where the series k C k t l k is convergent in a neighbourhood of 0. The assertion is seen if we replace the comparison equations ( 17 If we put l 0 = 0, p 0 = 2/θ, l k = kθ/2, p k = 4 k for k = 1, 2, . . .and exploit Lemma 3.2, then C k , Ck can be defined as making the series k C k t l k convergent, hence the series ψ (0) + ψ (2) + . . .uniformly converges, and z (k) has a limit, which is continuous.
Using the same notation as in the proof of Theorem 3.1, we have the estimates of the form where Q is a generic constant.Hence the sequences {ψ (k) } and { ψ(k) } tend uniformly to 0 as k → +∞.
that is: any bicharacteristic curve passing through the points (0, y) and (t, x) has the same value at s ∈ [0, a].If we change the variables y = η[z](0; t, x), then, by the Liouville theorem, the Jacobian J = det ∂c ∂x is given for α = α(t, x) by the formula J(α; t, x) = exp − t α tr ∂ x c(s, η i [z](s; 0, y), z s )ds .

0 MC k+1 t l k+1 C t 0 M
)-(18) by the inequalitiesCk t l k C t 0 L * c e Lca+ Lλ C k s l k ds + Ce Lλ t λ σ s, (C k + Ck )s l k + L * c e Lca C k t l k +1 /(l k + 1) ds, W σ s, (C k + Ck )s l k + L * c e Lca C k t l k +1 /(l k + 1) ds with suitable C 0 t l0 Z(a)and with some C0 a C L * c e Lca+ Lλ Z(a) + Ce Lλ M λ a 0 σ s, C0 + Z(a) + a L * c e Lca Z(a) ds.
y) exp