By Ivanka Stamova, Gani Stamov
Using the idea of impulsive differential equations, this booklet makes a speciality of mathematical versions which mirror present learn in biology, inhabitants dynamics, neural networks and economics. The authors give you the uncomplicated historical past from the basic concept and provides a scientific exposition of contemporary effects on the topic of the qualitative research of impulsive mathematical versions. along with six chapters, the ebook offers many acceptable ideas, making them to be had in one resource simply available to researchers drawn to mathematical types and their functions. 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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Additional info for Applied Impulsive Mathematical Models
T/. 2. The sequence f˛k g is almost periodic and 1 < ˛k Ä 0; k D ˙1; ˙2; : : :. 3. ftk g 2 UAPS. s/ exp ) ˛. /d s for tk 1 < s Ä t Ä tk ; k D ˙1; ˙2; : : :. 6) in Chap. 3) the matrix 8 n Z t ˆ ˆ exp ˛. 1 C ˛i / exp ˆ ˆ ˆ : iDm tm 1 < s Ä tm < t k o ; tk Z t 1 < s Ä t Ä tk ; ) ˛. 4. 5. 6. The sequence f k g; k D ˙1; ˙2; : : :, is almost periodic. 7 in . 1. 6 hold. t/j < "; t 2 R; j˛kCq ˛k j < "; q 2 P; k D ˙1; ˙2; : : : I j kCq k j < "; q 2 P; k D ˙1; ˙2; : : : I q jtk rj < "1 ; q 2 P; r 2 T; k D ˙1; ˙2; : : :.
0/ has a dominant role. 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.
Let the following conditions hold. 1. 6 hold for k D 1; 2; : : :. 2. 18 are met. 3. The function V W Œt0 ; 1/ ! 0// for r Ä s Ä 0. 20. 21. Let the following conditions hold. 1. 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.