Applied Impulsive Mathematical Models by Ivanka Stamova, Gani Stamov

By Ivanka Stamova, Gani Stamov

Using the idea of impulsive differential equations, this ebook makes a speciality of mathematical types which mirror present study in biology, inhabitants dynamics, neural networks and economics. The authors give you the simple history from the basic idea and provides a scientific exposition of modern effects on the topic of the qualitative research of impulsive mathematical types. along with six chapters, the booklet offers many acceptable recommendations, making them to be had in one resource simply obtainable to researchers drawn to mathematical types and their purposes. Serving as a worthy reference, this article is addressed to a large viewers of execs, together with mathematicians, utilized researchers and practitioners.

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1 j We shall prove that the set of sequences ftk g is uniformly almost periodic. Let " " > 0 and p be an -almost period of the sequence f˛k g. 5 it follows that the set of sequences ftk g is uniformly almost periodic. We shall use the following properties of uniformly almost periodic sequences. 2 ([256]). Let the set of sequences ftk g; tk D tkCj tk ; k; j D ˙1; ˙2; : : :, be uniformly almost periodic. 3 ([256]). Let the set of sequences ftk g; tk D tkCj ˙1; ˙2; : : :, be uniformly almost periodic.

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. 11) where ˛ > 0; ˇ > 0; ! > 0, n 2 N, has been proposed by Mackey and Glass [216] as an appropriate model of hematopoiesis, which describes the process of production of all types of blood cells generated by a remarkable self-regulated system that is responsive to the demands put upon it.

There has been a gradual expansion both in the class of objects studied and in the mathematical problems investigated by means of the method. Gurgulla and Perestyuk were the first to apply the Lyapunov direct method for impulsive systems. In the work [128] they used classical (continuous) Lyapunov functions. 4 Piecewise Continuous Lyapunov Functions and Lyapunov Functionals 31 impulsive systems restricts the possibilities of Lyapunov’s second method. The fact that the solutions of impulsive systems are piecewise continuous functions requires the introduction of an analogue of the classical Lyapunov functions which have discontinuities of the first kind [34].

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