By Ivanka Stamova, Gani Stamov
Using the speculation of impulsive differential equations, this booklet makes a speciality of mathematical types which mirror present learn in biology, inhabitants dynamics, neural networks and economics. The authors give you the simple heritage from the basic thought and provides a scientific exposition of contemporary effects on the topic of the qualitative research of impulsive mathematical types. inclusive of six chapters, the booklet provides many appropriate thoughts, making them on hand in one resource simply obtainable to researchers drawn to mathematical types and their functions. 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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Extra info for Applied Impulsive Mathematical Models
18) where f W R PCŒŒ r; 0; Rn ! Rn , Ik W Rn ! Rn ; k D ˙1; ˙2; : : :, tk < tkC1 < : : : and lim tk D ˙1. ˙1 Let '0 2 PCŒŒ r; 0; Rn . tI t0 ; '0 / is defined. 14. tI t0 ; '0 /jj < BI (d) uniformly ultimately bounded if (b) and (c) hold together. 18). 15. 4 Piecewise Continuous Lyapunov Functions and Lyapunov Functionals An interesting and fruitful technique that has gained increasing significance and has given decisive impetus to the modern development of the stability theory of impulsive functional differential equations is Lyapunov’s second method .
These notions are used throughout the book. 1 Almost Periodic Sequences In this part, we shall follow  and , and consider the main definitions and properties of almost periodic sequences. We shall consider the sequence fxk g; xk 2 Rn ; k D ˙1; ˙2; : : :, and let " > 0. 3. 11) 20 2 Basic Theory It is easy to see that if p and q are "-almost periods of fxk g, then p C q; p 2"-almost periods of the sequence fxk g. 4. e. 11) holds. ˚ « Let B˛ D x 2 Rn W jjxjj < ˛ ; ˛ > 0. 7 (). Let the following conditions hold.
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. 22 (). Let the following conditions hold: 1. 1 is met. 2.