By Ivanka Stamova, Gani Stamov

ISBN-10: 3319280600

ISBN-13: 9783319280608

ISBN-10: 3319280619

ISBN-13: 9783319280615

Using the speculation of impulsive differential equations, this booklet specializes in mathematical types which mirror present examine in biology, inhabitants dynamics, neural networks and economics. The authors give you the uncomplicated history from the elemental thought and provides a scientific exposition of modern effects with regards to the qualitative research of impulsive mathematical types. which include six chapters, the ebook offers many acceptable ideas, making them on hand in one resource simply obtainable to researchers drawn to mathematical types 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

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.

V1 ; V2 ; : : : ; Vm / such that Vj 2 V0 , j D 1; 2; : : : ; m. In the presence of delays, we shall use the corresponding modifications and generalizations. 5 Impulsive Differential Inequalities In this section we present the main comparison results and integral inequalities we will use. The essence of the comparison method is in studying the relations between the given system and a comparison system so that some properties of the solutions of the comparison system should imply the corresponding properties of the solutions of the system under consideration.

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 [284]. 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; : : :. 2. 3 hold. Then: 1. t; s/j Ä e ; t s; t; s 2 R: 2. t s/ : Proof. 1 C ˛k / Ä 1. 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/.

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Applied Impulsive Mathematical Models by Ivanka Stamova, Gani Stamov

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