By Ivanka Stamova, Gani Stamov
Using the speculation of impulsive differential equations, this ebook makes a speciality of mathematical versions which mirror present learn 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 contemporary effects with regards to the qualitative research of impulsive mathematical versions. including six chapters, the booklet provides many appropriate concepts, making them on hand in one resource simply obtainable to researchers drawn to mathematical versions and their functions. Serving as a beneficial reference, this article is addressed to a large viewers of execs, together with mathematicians, utilized researchers and practitioners.
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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. 1. 10 are met. 2. 16) Then: 1. 12). 2. t/ is exponentially stable.
24) m m m where F W R Rm C ! R I Jk W RC ! R ; k D 1; 2; : : :. m Let u0 2 RC . 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 !
1. For each m D 1; 2; : : : the sequence fxkm g; k D ˙1; ˙2; : : : ; is almost periodic. 2. There exists a limit sequence fyk g; k D ˙1; ˙2; : : :, of the sequence fxkm g; k D ˙1; ˙2; : : :, as m ! 1. Then the limit sequence fyk g; k D ˙1; ˙2; : : :, is almost periodic. 9 (). The sequence fxk g; k D ˙1; ˙2; : : :, is almost periodic if and only if for any sequence of integers fmi g; i D ˙1; ˙2; : : :, there exists a subsequence fmij g such that fxkCmij g is convergent for j ! 1 uniformly on k D ˙1; ˙2; : : :.