STABILITY OF IMPULSIVE DIFFERENTIAL EQUATIONS WITH RANDOM IMPULSIVE MOMENTS

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STABILITY OF IMPULSIVE DIFFERENTIAL

EQUATIONS WITH RANDOM IMPULSIVE MOMENTS

Katya G. Dishlieva

¹

, Angel A. Dishliev

²

1Department of Differential Equations, Faculty of Applied Mathematics and Informatics, Technical University of Sofia, Bulgaria,

2Department of Mathematics, University of Chemical Technology and Metallurgy, Sofia, Bulgaria

ABSTRACT

The main objects of study in this paper are the impulsive autonomous nonlinear differential equations with random impulses. The sufficient conditions for several types of stability of non-zero solutions of such equations are found.

Keywords: Impulsive Equations, Random Impulsive Moments, Stability

I. INTRODUCTION

The impulsive differential equations can be classified according to several basic symptoms:

Order of differential equation: first order (see D. Bainov and all [1]), second order (see X. Li [2]), n-th order (see F.

Chen and X. Wen [3]), etc.;

Form of the right hand side of the equation: linear (see M. Akhmet and all [4]), nonlinear (see S. Mohamad and all [5]), autonomous (see M. Akhmet [6]), nonautonomous (see S. Tang and all [7]) etc.;

Type of the impulsive functions: linear (see D. Bainov and P. Simeonov [8], Chapter 1, Paragraph 2), nonlinear (see Z. He and W. Ge [9]), scalar functions (see D. Bainov and A. Dishliev [10]), non-scalar functions (see D.

Cheng and J. Yan [11]), random (see A. Samojlenko and O. Stanzhytskyi [12]), etc.;

Method of determining the impulsive moments: fixed (see X. Fu and X. Li [13]), variable (see D. Bainov and A.

Dishliev [14]), random (see K. Dishlieva [15]), etc.

In this paper, we study the impulsive systems of differential equations, which are: (i) first order, (ii) nonlinear, (iii) nonautonomous, (iv) with variable impulsive functions and (v) random moments of impulsive effects.

Conditionally, we can split the theory of stability of the solutions of impulsive equations in two main parts:

Classical theory of stability of these equations. There are investigated many types of stability in respect to the perturbations on: (i) the initial condition, (ii) the right hand side, (iii) the parameter, (iv) the integral manifolds, etc.

Here, we mention only a few monographs dedicated to the above types of stability: A. Samoilenko and N.

Perestyuk [16], I. Stamova [17], V. Lakshmikantham and all [18] and D. Bainov and P. Simeonov [19];

Specific theory of stability, where the perturbations are associated with the changes in elements which are specific to this type of equations. As an example we shall point the effects related to: (i) the impulsive moments, (ii) the

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impulsive hypersurfaces, (iii) the switching sets, (iv) the barrier curves etc. (see A. Dishliev and all [20], A.

Dishliev and D. Bainov [21]).

Here, we investigate classical stability on the initial value condition of the solutions of impulsive systems of differential equations with random moments of impulsive effects.

II. PRELIMINARY REMARKS

The object of study is the following initial problem:

 

^, ^, i

dx f t x t

dt   ; (1)



i ⁰

  

i i

  

i



x   x  I x  , i1, 2,...; (2)

 

0 0

x t x , (3)

where: the function f R: ^ D Rⁿ; the phase space D is a nonempty domain in Rⁿ; the functions I_i:DRⁿ;



IdI_i



:DD, where Id is an identity in Rⁿ; the impulsive moments  ₁, ₂,..., 0  t₀ ₁ ₂..., are random (occasional in their nature); the initial point



t x0, 0



R^D. For convenience, we introduce the notation



1, 2,...



    . The solution of the problem above will be denoted by x t t x



; ,0 0,



. Further, we shall use the designations: x_i x



_i; ,t x0 0,



, x_i^  x_i I_i

 

x_i , i1, 2,....

Remark 1 [15]. We shall mention a few important properties of the solution x t t x



; ,0 0,



:

1. It is a piece wise continuous function with the points of discontinuity  ₁, ₂,... at which it is continuous on the left hand side, i.e. x



_i; ,t x0 0,



x



_i0; ,t x0 0,



, i1, 2,...;

2. Let the sets ^^* ^



^{ }¹^*^, ²^*^,...



^and ^^**^



^{ }¹^**^, ²^**^,...



be different. More precisely, let ₁^*₁^**,...,

* ** * **

1 1,

k k k k

 _  _   , where kN. Then the following inequalities are satisfied:



^{; ,}⁰ ⁰^, ^*

 

^{; ,}⁰ ⁰^, ^**



^, ⁰ ^k^min ^min



^*^k^, ^**^k



^;

x t t x  x t t x  t  t     ^{x t t x}



^{; ,}⁰ ⁰^,^^*

 

^^{x t t x}^{; ,}⁰ ⁰^,^^**



^,^^k^min ^{  }^t ^.

We consider the initial problem without impulsive effects



^,



dX f t X

dt  ; (4)

 

0 0

X t x . (5)

This problem will be named the degenerated of the basic problem. The reason for this name is that the problem (4), (5) is obtained from problem (1), (2), (3), assuming that the impulsive functions I_i

 

x 0 for xD and

1, 2,...

i . The solution of degenerated problem will be denoted by X t t x



; ,0 0



.

Remark 2 [15]. The following relationship is valid between the solution x t t x



; ,0 0,



of main problem with impulses (1), (2), (3) and the solutions of the corresponding degenerated system without impulses (4):

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 

0 0 0 1

1 1 1 2

0 0

2 2 2 3

; , , ;

; , ,

; , , ;

...

X t t x t t

X t x t

x t t x

X t x t



  

   



  

  

 

  



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We introduced the following conditions:

H1. The function f C R ^D R, ⁿ.

H2. For every point



t x₀, ₀



R^D, the solution of degenerated problem (4), (5) exists and is unique for tt₀. H3. The functions



IdIi



^:DD.

H4. It is fulfilled lim _i

i

  .

Lemma 1. Let the Conditions H1-H4 be valid.

Then for every initial point



t x₀, ₀



R^D and any choice of impulsive moments 



 1, 2,...



, the solution of problem (1), (2), (3) exists and is unique for tt₀.

Proof. Let the initial point



t x₀, ₀



R^D and the impulsive moments  satisfy the inequalities t₀ ₁ ₂,.... For t₀ t ₁, according to (6), we have x t t x



; ,0 0, 



X t t x



; ,0 0



. Then using Condition H2, we find that the solution of the main problem exists and is unique in the interval t₀ t ₁. By Condition H3, we derive

       

1 1 1 1 1; ,0 0, 1 1; ,0 0,

x^ x I x x  t x  I x  t x  



IdI1

  

x 1; ,t x0 0,

 





IdI1

  

X 1; ,t x0 0

 

D. For ₁ t ₂, using (6) again, we find that ^{x t t x}



^{; ,}⁰ ⁰^,^



^^{X t}



^{; ,}^¹ ^x¹^



, from where, having in mind Condition H2, we deduce that the solution of main problem exists and is unique in the interval ₁ t ₂. Using Condition H3, we find

           

2 2 2; ,0 0, 2 2; ,1 1

x^ IdI x  t x   IdI X   x^ D

etc. In this way we conclude that the solution of main problem exists and is unique in every one of the intervals



t0,1



,



 1, 2



,



 2, 3



,.... Finally, by means of Condition H4, we reach the conclusion that the solution of problem (1), (2), (3) exists and is unique in the set



0, 1

 

1, 2

 

2, 3



... 0, lim _i



0,



i

t      t  t



 

     

  .

The lemma is proved.

We shall formulate the following lemma summarizing the Gronwall’s inequality (see Theorem 1.9, [22]):

Lemma 2. The following conditions are valid:

1. Condition H4 is fulfilled;

2. The functions ,m p R: ^ R^ are continuous in any one of the intervals



t0,1



,



 1, 2



,



 2, 3



,...; 3. The constants m₀0, q_i0,i1, 2,...;

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4. The next inequality is satisfied

       

0

0 , 0

i

t

i i

t t t

m t m p s m s ds q m t t



 

 







 ^.

Then

     

0 0

0 1 exp , 0

i

t

i t

t t

m t m q p s ds t t



 

 





 



  ^. Further, we shall use the following definitions.

Definition 1. We shall say that:

(i) The solution of problem (1), (2), (3) is stable on the initial condition at the fixed impulsive moments, if



  const0

 

 



 1, 2,... ;



t012...

 

^{ }^{  }



^{, ,}^{t x}⁰ ⁰^,^



^^{0 :}

 

^{ }^x^*⁰ ^{D x}^, ⁰^*^^x⁰ ^^





^{; ,}⁰ ^*⁰^,

 ^

^{; ,}⁰ ⁰^,

^

^, ⁰

x t t x  x t t x   t t

    ;

(ii) The solutions of basic impulsive system (1), (2) are uniformly stable on the initial condition at the fixed impulsive moments, if (i) is true and  does not depend on the initial point, i.e. ^{   }^

 

^, ^;

(iii) The solution x t t x



; ,0 0,



is stable on the initial condition at the random impulsive moments, if (i) is true and  does not depend on the impulsive moments  ₁, ₂,..., i.e.   



, ,t x0 0



;

(iv) The solutions of system (1), (2) are uniformly stable on the initial condition at the random impulsive moments, if (i) is true and  does not depend on the initial point



t x0, 0



and the impulsive moments

1, 2,...

  , i.e. ^{  }^

 

.

We shall recall the following definition.

(i) The solution of problem without impulses (4), (5) is locally Lipschitz stable with Lipschitz constant L>0, if

 



^{ }^{ } ^{t x}⁰^, ⁰ ^^{0 :}

 

^{ }^x⁰^* ^D^, ^x⁰^*^^x⁰ ^^



^ ^{X t t x}



^{; ,}⁰ ^*⁰



^^{X t t x}

^

^{; ,}⁰ ⁰

^

^^{L x}⁰^*^^x⁰ ^, ^t^^t⁰^;

(ii) The solutions of system (4) are locally uniformly Lipschitz stable (with Lipschitz constant L>0), if



^{ }^ ^const^^{0 :}

 

^



^{t x}⁰^, ⁰



^^R^^^D

 

^{ }^x^*⁰ ^D^, ^x^*⁰^^x⁰ ^^





^{; ,}⁰ ⁰^*

 ^

^{; ,}⁰ ⁰

^

⁰^* ⁰ ^, ⁰

X t t x X t t x L x x t t

     ;

(iii) The solution of problem without impulses (4), (5) is a globally Lipschitz stable (with Lipschitz constant L>0), if



^{ }^x^*⁰ ^D



^ ^{X t t x}



^{; ,}⁰ ⁰^*



^^{X t t x}

^

^{; ,}⁰ ⁰

^

^^{L x}⁰^*^^x⁰ ^, ^t^^t⁰^;

(iv) The solutions of system (4) are globally uniformly Lipschitz stable (with Lipschitz constant L>0), if

 



^ ^{t x}⁰^, ⁰ ^^R^^^D



^{ }^x^*⁰ ^D



^ ^{X t t x}



^{; ,}⁰ ^*⁰



^^{X t t x}

^

^{; ,}⁰ ⁰

^

^^{L x}⁰^*^^x⁰ ^, ^t^^t⁰^.

Similarly, we introduce next definition.

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(i) The solution of problem (1), (2), (3) is locally Lipschitz stable at the fixed impulsive moments (with Lipschitz constant L>0), if

 



^{ }^ ^{ }¹^, ²^{,... ;} ^t⁰^^¹^^²^^...

 

^{ }^{ }

^

^{t x}⁰^, ⁰^,^

^

^^{0 :}

 

^{ }^x⁰^* ^D^, ^x⁰^*^^x⁰ ^^





^{; ,}⁰ ^*⁰^,

 ^

^{; ,}⁰ ⁰^,

^

⁰^* ⁰ ^, ⁰

x t t x  x t t x  L x x t t

     ;

(ii) The solutions of impulsive system (1), (2) are locally uniformly Lipschitz stable at the fixed impulsive moments (with constant L), if  does not depend on the initial point



t x0, 0



R^D, i.e. ^{  }^

 

^{. In this}

case, we have

 



   1, 2,... ; t012...

 

^{ }^{  }

 

^^{0 :}

 

^



^{t x}⁰^, ⁰



^^R^^^D

 

^{ }^x⁰^* ^D^, ^x⁰^*^^x⁰ ^^





^{; ,}⁰ ^*⁰^,

 ^

^{; ,}⁰ ⁰^,

^

⁰^* ⁰ ^, ⁰

x t t x  x t t x  L x x t t

     ;

(iii) The solution x t t x



; ,0 0,



of problem (1), (2), (3) is locally Lipschitz stable at the random impulsive moments with constant L, if (i) is true and  does not depend on the impulsive moments  ₁, ₂,..., i.e.



t x0, 0



  . More precisely, we have

 



^{ }^{ } ^{t x}⁰^, ⁰ ^^{0 :}

 

^{ }^x^*⁰ ^D^, ^x⁰^*^^x⁰ ^^

 ^

^{ }^

^

^{ }¹^, ²^{,... ;}

^

^t⁰ ^{ }^¹ ^²^^...

^



^{; ,}⁰ ^*⁰^,

 ^

^{; ,}⁰ ⁰^,

^

⁰^* ⁰ ^, ⁰

x t t x  x t t x  L x x t t

     ;

(iv) The solutions of basic system (1), (2) are locally uniformly Lipschitz stable at the random impulsive moments with constant L, if



^{ }^ ^const^^{0 :}

 

^



^{t x}⁰^, ⁰



^^R^^^D

 

^{ }^x^*⁰ ^D^, ^x^*⁰^^x⁰ ^^

 ^

^{ }^

^

^{ }¹^, ²^{,... ;}

^

^t⁰^^¹^^²^^...

^



^{; ,}⁰ ^*⁰^,

 ^

^{; ,}⁰ ⁰^,

^

⁰^* ⁰ ^, ⁰

x t t x  x t t x  L x x t t

     ;

(v) The solution of problem (1), (2), (3) is a globally Lipschitz stable at the fixed impulsive moments with constant L, if



^{ }^x^*⁰ ^D



^ ^{x t t x}



^{; ,}⁰ ^*⁰^,^



^^{x t t x}

^

^{; ,}⁰ ⁰^,^

^

^^{L x}^*⁰^^x⁰ ^, ^t^^t⁰^;

(vi) The solution of problem (1), (2), (3) is a globally Lipschitz stable at the random impulsive moments with constant L, if



^{ }^x^*⁰ ^D

 

^{ }^

^

^{ }¹^, ²^{,... ;}

^

^t⁰^^¹^^²^^...



^ ^{x t t x}



^{; ,}⁰ ^*⁰^,^



^^{x t t x}

^

^{; ,}⁰ ⁰^,^

^

^^{L x}⁰^*^^x⁰ ^, ^t^^t⁰^;

(vii) The solutions of system (1), (2) are globally uniformly Lipschitz stable at the fixed impulsive moments with constant L, if

 



^ ^{t x}⁰^, ⁰ ^^R^^^D



^{ }^x^*⁰ ^D



^ ^{x t t x}



^{; ,}⁰ ^*⁰^,^



^^{x t t x}

^

^{; ,}⁰ ⁰^,^

^

^^{L x}⁰^*^^x⁰ ^, ^t^^t⁰^;

(viii) The solutions of system (1), (2) are globally uniformly Lipschitz stable at the random impulsive moments with constant L, if

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 



^ ^{t x}⁰^, ⁰ ^^R^^^D



^{ }^x^*⁰ ^D

 ^

^{ }^

^

^{ }¹^, ²^{,... ;}

^

^t⁰^{ }^¹ ^²^^...

^



^{; ,}⁰ ^*⁰^,

 ^

^{; ,}⁰ ⁰^,

^

⁰^* ⁰ ^, ⁰

x t t x  x t t x  L x x t t

     .

These types of stability have been studied by a number of authors, we quote the following: F. Dannan and S.

Elaydi [23] (Lipschitz stability of the solutions of differential equations without impulsive effects) and A. Dishliev and D. Bainov [24] (Lipschitz stability of the solutions of impulsive equations).

The main objective of this study is to find the sufficient conditions ensuring different types of stability of the solution of problem (1), (2), (3) at the random impulsive moments.

III. MAIN RESULTS

The following theorems are valid.

Theorem 1. Assume that:

1. Conditions H1-H4 hold.

2. There exists a function C_Lip C R ^,R^ such that:

2.1.



 t t0



x x', ''D



 f t x

 

, ' f t x



, ''



C_Lip

 

t x'x'' ;

2.2.

 

0 CLip s ds

  



^.

3. There exist the constants C_I_i 0 such that:

3.1.



x x^{', ''}D



 Ii

 

x^' Ii

 

x^'' CI_i^. x^'x^{'' ,} i^{1, 2,...}; 3.2.

1,2,...

Ii

i

C





 ^.

Then the solutions of impulsive system (1), (2) are uniformly stable on the initial condition at the random impulsive moments.

Proof. Let: the initial point



t x0, 0



R^D; const0;  



 1, 2,...



be an arbitrary set of impulsive moments and x is an arbitrary point of D . For the solutions of impulsive system (1), (2) with the initial point ^*₀



t x0, 0



and

 

t x0, ^*0 , the next equalities are valid:

       ^ ^ 

0

0 0 0 0 0 0 0

; , , , ; , , ; , ,

i

t

i i

t t t

x t t x x f s x s t x ds I x t x



   

 

 







^;

         

0

* * * *

0 0 0 0 0 0 0

; , , , ; , , ; , ,

i

t

i i

t t t

x t t x x f s x s t x ds I x t x



   

 

 







^,

from where we find ^{x t t x}



^{; ,}⁰ ⁰^*^,^



^^{x t t x}

^

^{; ,}⁰ ⁰^,^

^     ^ ^ ^ ^

0

* *

0 0 , ; ,0 0, , ; ,0 0,

t

x x f s x s t x  f s x s t x  ds

  





    ^ ^ ^ ^

0

*

0 0 0 0

; , , ; , , ,

i

i i o i i

t t

I x t x I x t x t t



   

 





  ^.

Using the Conditions 2.1 and 3.1, by the inequality above, for tt₀, we establish

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

^{; ,}⁰ ^*⁰^,

 ^

^{; ,}⁰ ⁰^,

^

x t t x  x t t x 

    ^ ^

0

* *

0 0 . ; ,0 0, ; ,0 0,

t t Lip

x x C s x s t x  x s t x  ds

  





  ^ ^

0

*

0 0 0 0

; , , ; , ,

i i

I

t t

C x s t x x s t x



 

 





 ^.

We apply Lemma 2 for each tt₀ and obtain



^{; ,}⁰ ^*⁰^,

 ^

^{; ,}⁰ ⁰^,

^

x t t x  x t t x 

  ^{ }

0 0

*

0 0 . 1 _i .exp

i

t

I Lip

t t t

x x C C s ds



 

 

 



 





  ^{ }

*

0 0

1,2,... 0

. 1 _I_i .exp _Lip

i

x x C C s ds





 

 



 



 ^ ^x^*⁰^^x⁰ ^.^C^,

Where constant

 

₀

^{ }

1,2,...

1 .exp

Ii Lip

i

C C C s ds





 

   

 

 

^.

If we substitute ^{  }^

 

^^_C, we reach the conclusion that

 



^ ^{t x}⁰^, ⁰ ^^R^^^D

 

^{ }^x^*⁰ ^D^, ^x^*⁰^^x⁰ ^^



^ ^{x t t x}



^{; ,}⁰ ^*⁰^,^



^^{x t t x}

^

^{; ,}⁰ ⁰^,^

^

^^^, ^t^^t⁰^.

The theorem is proved.

Theorem 2. Assume that:

1. Conditions H1-H4 are valid.

2. The solutions of system without impulses (4) are locally uniformly Lipschitz stable with constant L . 3. There exist the constants L_i 0 such that:

3.1.



x x^{', ''}D







IdIi

  

x^'  IdIi

 

x^'' L xi^. ^'x^{'' ,} i^{1, 2,..}^.; 3.2. There exists a constant  0 such that

 

1,...,

0,1, 2,... ⁱ _j

j i

i L L



  



 ^.

Then the solutions of impulsive system (1), (2) are locally uniformly Lipschitz stable with constant L at the random impulsive moments.

Proof. Let the initial point



t x0, 0



R^D and  



 1, 2,...



, t₀ ₁ ₂..., be an arbitrary set of impulsive moments. We suppose that x is a point of D , which satisfies the inequality ₀^* x^*0x0   , in where constant  satisfies Definition 3, part (ii). We introduce the notations:

   

* * * * *

0 0

; , , , , 1, 2,...

i i i i i i

x x  t x  x^ x I x i . Consistently, we make the following considerations.

For t₀  t ₁, it is fulfilled



; ,0 0,

 

; ,0 0



x t t x  X t t x and ^{x t t x}



^{; ,}⁰ ^*⁰^,^{ }

 

^{X t t x}^{; ,}⁰ ⁰^*



^.

Since the solution X t t x



; ,0 0



is uniformly Lipschitz stable with constant L , we obtain

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

^{; ,}⁰ ^*⁰^,

 ^

^{; ,}⁰ ⁰^,

^ 

^{; ,}⁰ ⁰^*

 ^

^{; ,}⁰ ⁰

^

x t t x  x t t x   X t t x X t t x (7)

* *

0 0 0 0 , 0 1

L x x L x x t t 

       .

In particular, for t₁, from the inequality above it follows

  ^ ^

* * *

1 1 1; ,0 0, 1; ,0 0, 0 0

x x  x  t x  x  t x  L x x . (8)

We apply Condition 3.1 for i1

    ^ ^{ }

* * * *

1 1 1 1 1 1 1 1 1 1 0 0

x^x^  IdI x  IdI x L x x LL x x   

 . (9)

For ₁ t ₂, it is satisfied



^{; ,}⁰ ⁰^,

 

^{; ,}¹ ¹



x t t x  X t x^ and ^{x t t x}



^{; ,}⁰ ^*⁰^,^

 

^^{X t}^{; ,}^¹ ^x¹^*^



^.

Analogously to (7), given (8) we find



^{; ,}⁰ ^*⁰^,

 ^

^{; ,}⁰ ⁰^,

^ 

^{; ,}¹ ¹

 

^{; ,}¹ ¹^*



x t t x  x t t x   X t x^ X t x^ (10)

* 2 * *

1 1 1 0 0 0 0 , 1 2

L x^ x^ L L x x L x x  t 

         .

For t₂, by (10) it follows x₂^*x₂ L L x² ₁ ₀^*x₀ , from where, analogously to (9) we derive the estimate

    ^ ^{ }

* * * 2 *

2 2 2 2 2 2 2 2 2 1 2 0 0

x^x^  IdI x  IdI x L x x L L L x x   

 . By induction for each i1, 2,... , we have



^{; ,}⁰ ^*⁰^,

 ^

^{; ,}⁰ ⁰^,

^ 

^{; ,}ⁱ ⁱ

 

^{; ,}ⁱ ^*ⁱ



x t t x  x t t x   X t x^ X t x^ (11)

* 1 * *

1 2... 0 0 0 0 , 1

i

i i i i i

L x^ x^ L L L^ L x x L x x  t _

         ;

* 1 *

1 1 ⁱ 1 2... 0 0

i i i

x_ x_ L L L^ L x x ;

    ^ ^ ^

* *

1 1 1 1 1 1

i i i i i i

x_^ x^_  IdI_ x_  IdI_ x_ L_i_₁ x^*₂x₂ L L Lⁱ^¹ _{1 2}...L_i_₁ x^*₀x₀   

 . By (7) and (11), we conclude that, if x₀^*D and x₀^*x₀   , then



^{; ,}⁰ ^*⁰^,

 ^

^{; ,}⁰ ⁰^,

^

^*⁰ ⁰ ^,

^

⁰^, ¹

^ ^

¹^, ²

^ ^

²^, ³

^

^...

x t t x  x t t x   L x x t t         .

From Condition H4, we have



t0,1







 1, 2







 2, 3



 ...



t0,



. Then the last estimate acquires the form



^{; ,}⁰ ^*⁰^,

 ^

^{; ,}⁰ ⁰^,

^

^*⁰ ⁰ ^, ⁰

x t t x  x t t x   L x x tt . The theorem is proved.

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IV. APLICATION

As an illustrative example we shall look at the development of isolated population subjected to external pulse effects. Usually, these impacts are expressed in the withdrawal or (more rarely) in the addition of certain quantities of biomass to the population. The duration of such external interventions is negligible compared with the total duration of the research process, therefore it can be considered that they are carried out instantaneously in the form of impulses. The next initial problem appears as an adequate mathematical model of such processes

 

, _i

dN r

N K N t t

dt K   ; (12)



_i 0

  

_i _i

 

_i

N t  N t  p N t , i1, 2,...; (13)

 

0 0

N N , (14)

where:

- ^N^^{N t}

 

is the biomass amount at the moment t0;

- t t₁, ₂,... are the moments at which the impulsive effects are carried out. There is 0  t₁ t₂ ... and lim _i

i t

 ; - K const 0 is the capacity of the environment;

- r const 0 is the reproductive potential of the population;

- p p₁, ₂,... are the coefficients reflecting the volume of the relevant withdrawals. It is fulfilled 0p_i 1, i1, 2,...;

- N is the biomass amount at the initial moment ₀ t0, where KN₀K and 0  const1.

We shall show that the described impulsive logistic model of an isolated population satisfies the conditions of Theorem 2. The validity of Condition H1 is obvious. For each value of the initial point N0



0,K



D, the solution N t



;0,N₀



of degenerated problem without impulses (12), (14) is defined and unique for every t0. Furthermore, we have



⁰



0



0⁰

  

0 ;0, , 0

exp

N t N KN K t

N K N rt

     

   .

This means that Condition H2 is satisfied by the model. From (13), we can see that



IdIi

 

N  



¹ p Ni



. Since N



0,K



D and taking into account the inequalities 0 1 p_i1, i1, 2,..., it follows that



1p N_i



D or stated another way



IdI_i



:DD. Therefore, Condition H3 is valid for the model considered. We assumed to above that lim _i

i

t ,which means that Condition H4 is fulfilled.

For each two initial points



^{t N}⁰^, ⁰



^,



^{t N}⁰^, ⁰^*



^

^

^0,^{ }

^{ }

^0,^K

^

^^R^^^D^{, we have}



^{; ,}⁰ ⁰^*

 ^

^{; ,}⁰ ⁰

^

N t t N N t t N

   ^ ^  ^ ^  ^ ^ 

*

0 0

* *

0 0 0

0 0 exp 0 exp

KN KN

N K N r t t

 

   

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   

0

 

⁰^* ⁰

max ;

exp

d KN

K N K N N

dN N K N r t t 

   

   

           

*

0 0

2

1 N N

  .

The inequality above shows that the solutions of the degenerate problem (12), (14) are locally uniformly Lipschitz stable (with constant L 1₂ ).

For every two points N N', ''D, we have



IdIi

  

N^'  IdIi

  

N^''  ¹ pi



N^'N^'' L Ni ^'N^{'' ,} i^{1, 2,...}.

This means that Condition 3.1 of Theorem 2 is satisfied. Finally, we shall impose the restriction

     

2 1,...,

0 : 0,1, 2,... 1 ⁱ

j i

const i p

 

     



  ^,

which coincides with Condition 3.2 of Theorem 2. Thus, we find out that all the solutions of the impulsive logistic model (12), (13) with the initial points



0, N0



, N0



K K,



, where 0  1 are locally uniformly Lipschitz stable with constant L for random (occasional) impulsive moments.

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