By Ivanka Stamova, Gani Stamov
Using the speculation of impulsive differential equations, this publication makes a speciality of mathematical versions which mirror present examine in biology, inhabitants dynamics, neural networks and economics. The authors give you the easy history from the basic conception and provides a scientific exposition of contemporary effects with regards to the qualitative research of impulsive mathematical types. along with six chapters, the booklet offers many appropriate suggestions, making them to be had 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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Extra info for Applied Impulsive Mathematical Models
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. 2. t; x/ are continuous real functions, includes many mathematical ecological models.
T; x2 ; y2 /j Ä H1 jjx1 x2 jj C jjy1 y2 jj ; t ¤ tk , k D ˙1; ˙2; : : :. Let V 2 V2 ; t ¤ tk ; k D ˙1; ˙2; : : : ; x 2 PCŒR; ; y 2 PCŒR; . t//: When applying Lyapunov’s second method, there are two main approaches to investigating the qualitative properties of solutions of functional differential equations. Krasovskii  adopted a functional analysis approach. He replaced the Lyapunov function with a Lyapunov functional. The method of Lyapunov– Krasovskii functionals has been used by many researchers to investigate the stability theory of functional differential equations and their applications [2, 18, 19, 48, 64, 65, 71, 79, 121, 130, 131, 159, 164, 165, 206].
Let t0 2 R. t0 / is the space of all functions W Œt0 h; t0 ! t0 h; t0 / of the first kind and are left continuous at these points. t0 /. 1. 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 .