Hereditary Evolution Processes Under Impulsive Effects

In this note, we deal with a model of population dynamics with memory effects subject to instantaneous external actions. We interpret the model as an impulsive Cauchy problem driven by a semilinear differential equation with functional delay. The existence of delayed impulsive solutions to the Cauchy problem leads to the presence of hereditary impulsive dynamics for the model. Furthermore, using the same procedure we study a nonlinear reaction–diffusion model.


Introduction
We investigate on the presence of hereditary dynamics for a model driven by the parametric differential equation with memory effects. This equation is usually adopted for describing the dynamics of a population: u(t, x) represents the density of the population at time t and place x; −b(t, x) is the removal coefficient including the death rate and the displacement of the population; the nonlinearity g is the population development law involving a memory term expressed by its integral argument. Delayed and hereditary models are more appropriate than the classic ones, such as in the study of some pest populations where the individual's maturation time is not negligible or his current life is influenced by his own past. We will treat the above equation as a special case of the following semilinear differential equation with functional delay: where {A(t)} 0≤t≤T is a family of linear operators generating an evolution system and f is a nonlinear function. For every t ∈ [0, T ], the functional argument y t of f is defined as usual by and belongs to the space C 0 of all piecewise continuous functions defined on a given interval [−τ, 0], τ > 0. The map y t represents the history of the process starting from the present time t and going back up to t−τ ; the past from time t = 0 is provided by a given function ϕ ∈ C 0 . In many fields of applied sciences the past of the system influences its evolution, consequently it is necessary to implement models in which the dependence of the studied system on the past is formalized. In this direction, several articles on the subject have appeared in recent years (see, e.g. [14,18], or the interesting book [5]). In our model, possible external actions aimed at regulating the evolution of the phenomena taking place at pre-established times are allowed. In the general setting, these actions are represented by functions I i : C 0 → E, i = 1, . . . , p, causing sudden abrupt changes on the state of the system driven by (2). We consider impulse functions I i depending on the whole dynamic of the problem up to the time when the impulse has to act, so that the delay structure of the system is preserved also in this aspect of the problem, formally described by the equations y(t + i ) = y(t i ) + I i (y ti ), i = 1, . . . , p. The models involving impulsive functions are an excellent tool whenever a real world phenomenon exhibits instantaneous changes in state variables, as for example in real-time software verification, chemical process plants, mobile robotics, automotive control, nerve impulse transmission. In biology they are called "regulation functions" and are used for instance in the study of population dynamics to keep the population in a prescribed range.
To obtain the existence of solutions to our Cauchy impulsive problem, we break it down into an ordered sequence of non-impulsive Cauchy problems. This method works not only when the impulses are given at fixed times (see, e.g. [6,8]), but also in the case with impulses at variable times (see, e.g. [2,12]). It is worth emphasizing that the advantage of this approach in the study of the existence of solutions to impulsive problems of the first type, compared to others in the literature, is that it allows not to request hypotheses of any kind on the impulse functions. In the setting of delay problems, this technique leads to the construction of the solutions by means of an "extension-withmemory" process (see, e.g. [10] for the case of distributed delay, or e.g. [3] for the functional delay). We wish to underline that here we obtain the existence of delayed impulsive mild solutions considering problems whose initial data belong to the same space C 0 , unlike what is in [3] where the initial data are fixed in turn in the different spaces Furthermore, with the same technique we also study a model driven by the parabolic partial differential equation with memory effects. Reaction-diffusion equations with time delays like (4) appear in problems with delayed connections in which the processing of certain information is required, in relation not only to the dynamics of populations but also to chemical reactions and other physical phenomena. The paper is divided into two parts: Sect. 2 is devoted to the study the existence of solutions to the impulsive Cauchy problem with functional delay driven by Eq. (2); using the theory there developed, in Sect. 3, we provide the existence of evolutionary dynamics to the models driven, respectively, by (1) and (4).
At the end of the paper, we state an Appendix containing some background material, to make the article self-contained.

The Impulsive Cauchy Problem with Functional Delay
Let us consider the impulsive Cauchy problem with functional delay where {A(t)} 0≤t≤T is a family of linear operators; f : c has at most a finite number of jump discontinuities} , endowed with the norm is a normed (not Banach) space.
Remark 2.1. Our problem includes the possibility that an impulse I 0 : C 0 → E can be given in t 0 = 0. In this case, we shall rewrite problem (5) taking the function Notice that ϕ 0 belongs to C 0 as well.
On the linear part we assume that (see Sect. 3.2): is a dense subset of the Banach space E not depending on t, generating an evolution system {T (t, s)} 0≤s≤t≤T . We recall (see, e.g. [9]) that there exists a positive number D such that where L(E) is the space of all bounded linear operators from E to E furnished with the strong operator topology. Let us define the sets of piecewise continuous functions, for i = 1, . . . , p + 1, Notice that if y is a delayed impulsive mild solution, then y ∈ S([−τ, T ], E). We assume that the nonlinearity f : [0, T ] × E × C 0 → E satisfies the properties: (f1) f is such that f (t, ·, ·) is continuous, for every t ∈ [0, T ], and f ·, y(·), y (·) is measurable, for every y ∈ S([−τ, T ], E); We denote by 0 2 the zero-element of R 2 and by the symbol R 2 0,+ the standard positive cone R + 0 × R + 0 endowed with the partial ordering , where x y if and only if y − x ∈ R 2 0,+ , for x, y ∈ R 2 ; clearly, x ≺ y means that x y and x = y.
Let us fix i ∈ {0, · · · p} and L > 0. We consider the well-defined vectorial measure of noncompactness ν L i (cf. [13, Example 2.1.4]), acting on the bounded subsets of C([t i , t i+1 ], E) and taking values on R 2 0,+ , defined by where . It is known that ν L i is monotone and invariant with respect to the union with compact sets (hence also nonsingular).
Finally, for every

Existence of Solutions
In this Section we show the existence of delayed impulsive mild solutions to (5). Preliminarily eventually passing to a subsequence, is such that So the Lebesgue convergence theorem implies that lim n→∞ T 0 ψ n (s) ds = 0. Since b n ≥ 0, by (10) we obtain lim n→∞ b n = 0. Hence, (9) is satisfied.
Therefore, there exists H ∈ N such that b H < 1.
Now, we are in condition to provide our main result.
Proof. We proceed by steps.
Step 1. Let us consider the real interval [0, t 1 ] and the related Cauchy problem with functional delay arising from (5) A mild solution to (11) will be a function y : [−τ, t 1 ] → E such that Step 1.1. Let us define the integral operator where the map y if Step 1.2. We show that there exists a set in C([0, t 1 ], E) which is invariant under the action of Γ 1 .
Step 1.3. Consider the restriction of Γ 1 to IB R1 , i.e. the map The ball IB R1 is closed and convex in the space Step 1.4. The integral multioperator Γ 1,R1 has closed graph.
Of course, (30) and (7) provide First of all, from the above relation, we have the inequality Let us estimate (cf. (7)) Now, fixed t ∈ [0, t 1 ], by (f2) and (6) and by means of analogous arguments as above, we have that the set Moreover, by (14) and the definition of γ 1 , we have if s ≤ τ then and hence (35) provides We can conclude that, by (34), (38), (29) and (33), the following chain of inequalities holds: Since p L < 1 (cf. (29)) we deduce that the above relation is true only if Now, we show that (cf. (7)) As a consequence of (39) and (40) we also get By (37) and (42), we get Moreover, by (26) and (27), we know the set {f (·, y n (·), y n [ϕ] (·) } n to be semicompact. Put G 1 : The function y 1 [ϕ] satisfies (12) and hence is a mild solution for (11).
We associate to y 1 [ϕ] the map ϕ 1 ∈ C 0 defined by and the corresponding Cauchy problem with functional delay A mild solution to (45) will be a function y : [t 1 − τ, t 2 ] → E such that Step 2.1. Let us define the integral operator where the function ϕ 1 is defined in (44) and the map y 1 [ϕ] is the mild solution of (11) obtained in Step 1.6.
Step 3.b. In the case p > 1, by iteration a mild solution to (5)

Hereditary Evolutionary Processes Under External Instantaneous Actions
Using the theory developed in Sect. 2, we can provide the existence of hereditary dynamics of two models driven respectively by the parametric differential equation (1) and a parabolic partial differential equation (4) with memory effects, regulated by external instantaneous actions.

The Population Dynamics Model
We consider the parametric differential equation which represents the dynamics of a population where: u(t, x) is the population density at time t and place x; −b(t, x) is the removal coefficient given by the death rate and the displacement of the population; g is the population development law involving a memory term expressed by the integral In a modeling process it is necessary to introduce memory terms whenever the past state of the system influences its future evolution. An example can be found in the study of some pests in which the maturation time of the individual is not negligible. The initial past is known and given by a function times are formalized. In details, we set where: 0 = t 0 < t 1 < · · · < t p < t p+1 = T ; for every i = 1, . . . , p, ·). We treat the model as a particular case of the abstract problem (5), We assume that the map b : On the map g : [0, T ] × R × R → R we suppose that the following properties hold: for a.e. t ∈ [0, T ] and every p 1 , p 2 ∈ R, w 1 , w 2 ∈ C 0 L 2 ; (g3) for every y ∈ S([−τ, T ], L 2 ([0, 1])), the map t → g t, y(t)(·), 0 −τ y(t + θ) (·)dθ is measurable; (g4) g(·, 0, 0) ∈ L 1 ([0, T ]); (g5) there exists h ∈ L 1 + ([0, 1]) such that Now, we define the following functions: To state and prove the existence theorem for the model, we establish the next proposition, which combines the results described in [ Further, we need the following result on function f .