By Ivanka Stamova, Gani Stamov
Using the idea of impulsive differential equations, this publication specializes in mathematical versions which replicate present examine in biology, inhabitants dynamics, neural networks and economics. The authors give you the simple history from the elemental conception and provides a scientific exposition of modern effects with regards to the qualitative research of impulsive mathematical types. inclusive of six chapters, the ebook offers many acceptable suggestions, making them to be had in one resource simply obtainable to researchers attracted to mathematical types and their purposes. Serving as a worthwhile reference, this article is addressed to a large viewers of execs, together with mathematicians, utilized researchers and practitioners.
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Additional info for Applied Impulsive Mathematical Models
6 hold. 2. 19 are met. 38 2 Basic Theory 3. 0// for r Ä s Ä 0. 3. 0// for r Ä s Ä 0. 3. Analogous comparison results can be proved for impulsive systems [35, 179] in which minimal solutions are used. 4. Similar results can be proved in terms of functions from the classes V2 and W0 [284, 289, 290]. Next we shall consider a Bihari and Gronwall type integral inequality in a special case with impulses. 22 (). Let the following conditions hold: 1. 1 is met. 2. The functions m W R ! RC ; p W R ! tk 1 ; tk ; tk > t0 .
1. 10 are met. 2. 16) Then: 1. 12). 2. t/ is exponentially stable. Proof. t/j. 18) Now let ! 3. 19), we deduce that S' 2 AP. Let '; 2 AP. 16) it follows that S is a contraction mapping on AP. Then there exists a unique fixed point h 2 D such that Fh D h. t/. t/; 2 PCŒŒ g. C I ; / D . 13). t; /. t/j Ä j je . 16) implies that the unique positive almost periodic solution of Eq. 12) is exponentially stable. 3 A More General n-Dimensional Impulsive Biological Model In this section we shall consider a generalization of the models investigated in Sects.
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 . By means of such functions it becomes possible to solve basic problems related to the application of Lyapunov’s second method to impulsive systems.