PERIODIC SOLUTIONS OF SYSTEMS OF AUTONOMOUS DIFFERENTIAL EQUATIONS WITH VARIABLE STRUCTURE AND IMPULSES

Abstract: Autonomous systems of differential equations with variable structure and impulsive effects are the main objects of our study. The structure changes and the impulsive effects realize at the switching moments, which are specific to each different solution of the system. In these moments, the trajectory of the corresponding initial value problem meets successively the switching sets. For this class of equations, sufficient conditions for the existence of periodic solutions are found. The results are based on the Brouwer’s fixed point theorem. We investigate the question of existence of periodic solutions of a generalized model of predator-prey community.


Introduction
The impulsive differential equations are convenient mathematical apparatus for modeling the dynamic processes, subjected to the discrete external influences over time.Frequently, the duration of the external effects are negligibly small compared with the period of investigation of the dynamic process.For this reason, it can be considered that these effects are in the form of impulses.The qualitative theory of impulsive differential equations is developing intensively due to the numerous applications: [2], [4], [5], [10], [11], [12], [20], [28], [29], [35], [37] and [38].In particular, the existence of periodic solutions of this type of equation is the object of study in a number of articles: [1], [3], [8], [9], [19], [22], [24], [27], [30], [31], [32], [40] and [42], and several monographs, we will point out: [6], [7], [14], [21] and [36].Should be noted that the above mentioned results refer only to the differential equations with permanent structure and fixed moments of impulsive effects.The authors' research in this paper aims partially fill this gap.Here, we study the periodic solutions of the equations with variable structure and variable impulsive effects.Recall that the periodic solutions of impulsive differential equations are used for describing the repetitive processes with any leaps of changes.This type of equations and their solutions are especially useful in modeling bioprocesses and eco-processes: [18], [23], [25], [26], [39] and [41].

Statement of the Problem and Preliminary Remarks
Denote the Euclidean norm and dot product in R n by .and ., ., respectively.For the points x(x 1 , x 2 , ..., x n ) and y(y 1 , y 2 , ..., y n ) in R n , we have x, y = x 1 y 1 + x 2 y 2 + ... + x n y n ; x = x, x The Euclidean distance between the nonempty sets X and Y , where An open ball with a center x 0 ∈ R n and radius δ = const > 0 is denoted by X and ∂X are notations for the closure and contour of the set X, respectively.The length of curve γ is denoted by l[γ].The closed segment with endpoints x and y is denoted by [x, y] = {z λ ∈ R n ; z λ = (1 − λ)x + λy, 0 ≤ λ ≤ 1} .Definition 1.The curve γ is said to be p-linear, if That is to say, p-linear curve is composed by p sequentially connected line segments.
Definition 2. [15] The domain G is said to be p-convex, where p is a natural number, if In other words, any two points of G can be connected by p-linear curve from G. It is clear that each 1-convex domain is convex.Definition 3. [15] The domain G is said to be bounded-connected, if Remark 1. From the definition above, it follows immediately that each bounded-connected domain is bounded.

Definition 4.
[33] Let for each i = 1, 2, ..., the following conditions be valid: Then: -The set Φ i is said to be positive reachable from the set X + i via system (6); -The set Φ i is said to be totally positive reachable via system (6), if [13] The solutions of system (6), i = 1, 2, ..., is said to be uniformly Lipschitz stable, if The validity of the assumptions below will be required during the basic consideration: Assumption A1.Each of the sets Φ i is totally positive reachable via system (6), i = 1, 2, .... Assumption A2.The moments of switching t 1 , t 2 , ... (0 < t 1 < t 2 < ...) do not possess a compression point, i.e. lim i→∞ t i = ∞.Definition 6.Let Assumption A1 be valid.Then the functions Θ + i : G i → R + , i = 1, 2, ..., are called a function of positive reachability of system (6), if Assumption A3.The functions

C11. The constants
As a result of [16] we will formulate the following theorem, in which the sufficient conditions are given under which Assumption A1 is valid.Theorem 1. [16] Let Conditions C1-C9 and C13 hold.
The next theorem is equivalent to Assumption A3.We obtain it as a result of article [15].

Continuous Dependence
We will recall the following definition.
Definition 7. [14] We say that the solution x(t; x 0 ) of problem ( 1), ( 2), (3) depends continuously on the initial point, if Pay attention to the specific feature of the upper definition of continuous dependence.More precisely, the different solutions have different sets of switching (impulsive) moments.This means that in the general case, the inequalities t Consider the solutions x(t; x * 0 ) and x(t; x 0 ) in the arbitrary interval with the switching moments t * i and t i , i = 1, 2, ..., as its ends.Both solutions are subjected to the impulsive effects in different ends of this interval.Consequently (in the general case), they are not "close" inside in the interval.For example, it is possible that their difference to be greater than the impulsive effect at the left end of the interval considered.Furthermore, assuming that the corresponding switching moments vary "significantly", then a proximity between the solutions may exists only in a negligible part of the interval 0 ≤ t ≤ T .This assumption negates the practical value of concept of continuous dependence.Therefore, it is natural to suppose that Then in the surroundings of moments t 1 , t 2 , ... with a radius η, both solutions are not close.Exactly, this is recognized in Definition 7.
Proof.Let the point x 0 ∈ G 1 \D 1 .Let ε, η and T be the arbitrary positive constants.

It is satisfied (given the previous point and Condition C12
) 2.5.We find out lim By induction for i = 1, 2, ..., k, we have: By the equalities i.1, where i = 1, 2, ..., k, we obtain From the equalities i.2, where i = 1, 2, ..., k, we get whence, taking into consideration the inequalities (7), we reach the inclusion Finally, take into account the arbitrary choice of constants x 0 , ε, η, T, from ( 12) and ( 13), it follows that From Theorem 1 -Theorem 4 we get the following statements.
In the next corollary, we derive the sufficient conditions for continuous dependence of the solutions of original system (1), (2) without requiring for the solutions of initial values problems (6) to be uniformly Lipschitz stable (i.e.without Condition C4).

Assumptions A1-A3 are valid
Then Proof.Let x 0 be an arbitrary point from The solution x(t; x 0 ) of system (1), ( 2), (3) consistently meets the switching sets Φ 1 , Φ 2 , ..., Φ k 0 .The meetings take place in the moments t 1 , t 2 , ..., t k 0 (0 < t 1 < t 2 < ... < t k 0 ), respectively.Given Condition C15 and Assumption A1, we conclude that x(t k 0 ; x 0 ) = x k 0 ∈ Φ k 0 and We define the function Similar to Theorem 4 (see i.3), we have lim Therefore, the function In accordance with Condition C15, we have On the other hand, by Condition C13 (and in particular from Remark 1), it follows that G 1 is a bounded set.Thus, the conclusion is that the set J k 0 Φ k 0 is bounded.Again from Condition C15, we have that this set is convex.
We obtain that the set J k 0 Φ k 0 is bounded, closed and convex.The function Then from the Brouwer's fixed point theorem, it follows The theorem is proved.
From Theorem 1 -Theorem 3 and Theorem 5, we get next statement.
Then there exists an initial point x 0 ∈ J k 0 Φ k 0 such that the solution x(t; x 0 ) of problem ( 1), ( 2), ( 3) is periodic with a period t k 0 .
Then there exists an initial point x 0 ∈ J k 0 Φ k 0 such that the solution x(t; x 0 ) of problem ( 1), ( 2), ( 3) is periodic with a period t k 0 .

Application
The classical Lotka-Volterra model describes fairly adequately the dynamics of an isolated community of predator-prey type without external influences.Assume that the community could modify its parameters of development.We suppose that these changes are performed instantaneously.The parameters change reflects on the sharp variation of the growth speed of the biomass of both species.Further, we will specify the moments in which these changes are made.The corresponding initial value problem has the form: where: -m = m(t) > 0 and M = M (t) > 0 are the prey and predator biomasses at the moment t ≥ 0; -r 1 i = const > 0 and r 2 i = const > 0, i = 1, 2, ..., are specific growth factors, relevant to the first species (prey) and the second (predator), respectively; -q 1 i = const > 0 and q 2 i = const > 0, i = 1, 2, ..., are the coefficients indicating interspecies competition.In the common case, they are different for the prey and predator; -m 0 > 0 and M 0 > 0 are the prey and predator biomasses at the initial moment t = 0.
It is known that for i = 1, 2, ..., the system ( 14), ( 15) possesses: 1. Unstable (saddle) stationary point (0, 0); 2. Stable stationary point (m 00 i , M 00 i ) = (r 2 i /q 2 i , r 1 i /q 1 i ); 3. First integral of the form 4. For each constant c ≥ 0, implicitly given curve is a trajectory of system ( 14), (15).This trajectory is closed.The initial point (m 0 , M 0 ) of this trajectory is suitably chosen.It is sufficient V i (m 0 , M 0 ) = c; 5.For each constant c > 0, the set Let c and C be arbitrary constants such that 0 < c < C. For i = 1, 2, ..., the phase space of system ( 14), ( 15) is defined as Assume that the biomass of the prey is useful.For this reason, predetermined quantity of this biomass is produced for a "relatively long period of time".For this purpose, in the form of impulsive effects, certain quantities of biomass (for example m i , i = 1, 2, ...) are withdrawn repeatedly from the prey.The moments t 1 , t 2 , ..., at which the impulsive effects are realized upon the modeled community, coincide with the moments when the trajectory of problem ( 14), (15), (16) and the domains We will specify that changes of parameters in the studied system are realized in moments t 1 , t 2 , .... Further, we denote by m c,max i and m C,max i , respectively the bigger solutions of the equations The constants m c,min i and m C,min i are smaller solutions of the above two equations.
Then the reachable sets have the form The moments of impulsive effects t 1 , t 2 , ... satisfy the equalities: the right hand side of ( 14) becomes zero.Therefore, we have (dm(t))/(dt) = 0.
It can be shown that just then the victim's biomass is maximum.Consequently, the withdrawal of biomass from the victim in these moments (t 1 , t 2 , ...) is justified.
Remark 5.It is natural to assume that the quantity of victim's biomass taken away at the moment t i depends on its volume at this moment.Usually, the quantity taken away is m i = p i .m(ti ), where the constant p i satisfies 0 ≤ p i ≤ 1, i = 1, 2, ....
Let the impulsive functions are defined by equalities In addition, we suppose that The inequalities (22) will be valid, if Note that, in case r 1 1 = r 1 2 = ..., r 2 1 = r 2 2 = ..., q 1 1 = q 1 2 = ... and q 2 1 = q 2 2 = ... the inequalities (23) are fulfilled.Remark 6.We consider that the inequalities (21) are natural and useful of the following three reasons: -Quantity of biomass m i (taken away from the prey at the moment t i ) is i+1 , we conclude that m(t i + 0) satisfies (21) or the next inequalities Obviously, the quantity of biomass m i is greater, if the inequalities (21) are true; -After the moment t i , the solution of problem starts from the initial point m(t i + 0), M 00 i+1 ).It can be shown that, if ( 21) is satisfied, then from this moment on (until to the next switching moment t i+1 ), the victim's biomass increases, which is favorable for the user.On the contrary, if the inequalities (24) are valid, immediately after the switching moment, the victim's biomass decreases; -The time interval between two adjacent moments of removal of victim's biomass ∆ i = t i+1 − t i is shorter in the case when m(t i + 0) satisfies ( 21) in comparison with the case when m(t i + 0) satisfies (24).In other words, the time for reproduction of withdrawn biomass in case ( 21) is shorter than in case (24).
Consider the following problem (model of predator-prey community with the impulsive extractions of biomass from the victim and impulsive change of the parameters): We will apply Corollary 4 for system (25)- (28).Therefore, we will check the validity of Conditions C1-C3, C5-C13, C15, C16 and Assumption A1.Sequentially for i = 1, 2, ..., we find: i .m)are the right hand sides of system (25), (26).Therefore, i .This means that the functions f i satisfy the Lipschitz conditions.
• Condition C2: The functions i , where c = V i (m 0 , M 0 ).Therefore, the solutions of systems ( 25)-( 28) exist and are unique in R.
• Condition C5: It is clear that the functions ϕ i (m, M ) = M − M 00 i are continuously differentiable in G c,C i .
The above inequalities indicate that J i (m, M ) ∈ G c,C i+1 \D c,C i+1 .
• Condition C13: The domains G c,C i are bounded-connected.
• Condition C15: It is enough to assume that (∃k 0 ∈ N ) such that the inequalities , M 00 k 0 < C are valid.
• Condition C16: We will assume that: r 1 i+k 0 = r 1 i , r 2 i+k 0 = r 2 i , q 1 i+k 0 = q 1 i , q 2 i+k 0 = q 2 i , q i+k 0 = q i , from where, it follows this condition.
• i .From the above two statements, it follows that the sets Φ c,C i are totally reachable.
Remark 7. Through Corollary 4, we find that system ( 25)- (28) with variable structure and impulses has a periodic solution.

Remark 4 .
Pay attention that for Condition C6: We obtain ∀(m, M ) ∈ Φ c,C i = (m, M ); m c,max i < m < m C,max i , M = M 00 i