By Ivanka Stamova, Gani Stamov

Using the idea of impulsive differential equations, this publication specializes in mathematical types which mirror present examine in biology, inhabitants dynamics, neural networks and economics. The authors give you the uncomplicated heritage from the elemental idea and provides a scientific exposition of modern effects relating to the qualitative research of impulsive mathematical types. including six chapters, the booklet offers many appropriate suggestions, making them on hand in one resource simply available to researchers attracted to mathematical versions and their functions. Serving as a precious reference, this article is addressed to a large viewers of execs, together with mathematicians, utilized researchers and practitioners.

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**Example text**

1. 8), we obtain that S' 2 AP. Let '; 2 AP. 1 it follows that S is a contracting operator in AP. 1). t0 /. t/je˛t . 1 is complete. 1. tk /; where t 2 R; ˛; ˇ; ˙1; ˙2; : : :. 1. Let the following conditions hold. 1. t/ are almost periodic. 2. 3 are met. 10). Proof. 1. 1. The results in this section show that by means of appropriate impulsive perturbations we can control the almost periodic dynamics of these equations. 2 An Impulsive Model of Hematopoiesis In this section the existence and asymptotic stability of a positive almost periodic solution for a nonlinear impulsive delay model of hematopoiesis is investigated.

23) holds. 23) is called the Razumikhin condition, and the corresponding technique is known as the Razumikhin technique. 17). 20. A function W W R ! RC belongs to the class W0 if: 1. t; 0/ D 0; t 2 R. 2. t; x/ is locally Lipschitz continuous with respect to its second argument. 3. tks ; x/ holds. Let the function W 2 W0 and x 2 PCŒR; . t// : In the investigation of the qualitative properties of solutions of differential equations, it is well known that employing several Lyapunov functions is more useful than employing a single one since each function can satisfy less rigid requirements.

T s/ ; t s; t; s 2 R: 2 T. tk0 C ; s C /; q; q 2 P; k D ˙1; ˙2; : : :. t; / ˛. t; s/ ˛. C / W. s; t/. t s/ ; ˛ 2 1CNC N . ˛ 2 We shall now prove the main theorem of this section. 1. Let the following conditions hold. 1. 6 are met. 2. The following inequality holds n X Bi < ˛: iD1 Then: 1. 1). 2. t/ is exponentially stable. Proof. 1. 8), we obtain that S' 2 AP. Let '; 2 AP. 1 it follows that S is a contracting operator in AP. 1). t0 /. t/je˛t . 1 is complete. 1. tk /; where t 2 R; ˛; ˇ; ˙1; ˙2; : : :.