By Ivanka Stamova, Gani Stamov
Using the idea of impulsive differential equations, this ebook makes a speciality of mathematical types which mirror present examine in biology, inhabitants dynamics, neural networks and economics. The authors give you the easy heritage from the elemental 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 ebook offers many acceptable ideas, making them to be had in one resource simply obtainable to researchers attracted to mathematical types and their purposes. Serving as a priceless reference, this article is addressed to a large viewers of pros, together with mathematicians, utilized researchers and practitioners.
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Extra resources for Applied Impulsive Mathematical Models
TI t0 ; u0 / is defined. We introduce the following partial ordering on Rm : for the vectors u; v 2 Rm we shall say that u v if uj vj for each j D 1; 2; : : : ; m and u > v if uj > vj for each j D 1; 2; : : : ; m. 21. t0 ; u0 / ! / Q ! /. 24) is defined as follows. 22. The function m W Rm C ! 23. The function F W R Rm C ! e. u1 ; u2 ; : : : ; ui 1 ; uiC1 ; : : : ; um /. m In the case when the function F W R Rm is continuous and C ! t0 ; u0 / 2 Œt0 ; 1/ Rm C lie between two singular solutions – the maximal and the minimal ones.
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 . 3. 6 Coincidence Degree Lemmas In Chap. 4, we shall investigate the existence of positive periodic solutions of different classes of Lotka–Volterra models. Our main results are based on coincidence degree theory .
The sequence f k g is almost periodic, and 1 < k Ä 0, k D 1; 2; : : :. 9. 10. The sequence fık g; k D 1; 2; : : :, is almost periodic and sup jık j Ä Ä. kD1;2;::: In the proof of the main theorem we shall use the following lemmas. 3. 10 hold. t/j < "; t 2 RC ; 2 T; j kCq k j < "; q 2 P; k D 1; 2; : : :; jıkCq ık j < "; q 2 P; k D 1; 2; : : :; q jtk rj < "1 ; r 2 P; 2 T; k D 1; 2; : : :. 4. 10 hold. Then: 1. t; s/ Ä e ; t s; t; s 2 RC : 2. t s/ ; Á N . We now prove the next theorem. 2. Let the following conditions hold.
Applied Impulsive Mathematical Models by Ivanka Stamova, Gani Stamov
Categories: System Theory