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By A M Samoilenko

For researchers in nonlinear technological know-how, this paintings comprises insurance of linear platforms, balance of ideas, periodic and virtually periodic impulsive platforms, vital units of impulsive platforms, optimum regulate in impulsive structures, and extra

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Impulsive differential equations

For researchers in nonlinear technology, this paintings contains assurance of linear platforms, balance of recommendations, periodic and virtually periodic impulsive platforms, crucial units of impulsive structures, optimum regulate in impulsive structures, and extra

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22) Suppose that the times r; are indexed by the set of natural numbers such that Ti — ► +oo for i — ► oo. Without loss of generality, we can assume that Tl > t0. 23) where X(t,tQ) = eA^-T^ Yl (E + B)eA^~T^\ rQ = t0. 22) for arbitrary matrices A and B, in other words, there is no such an elegant description of properties of solutions of an impulsive system in terms of the eigenvalues of the matrix of the system as we have for a system of ordinary differential equations. 22) is not autonomous. 22).

These motions have trajectories with countably many points of discontinuity. The point (l,%/§) has the property of being an attractor for all the motions starting in the region 5x2 + y2>8, x > 1, 0 2 « / | , become periodic by the time the trajectory gets to the point f 2*/§,0J for the first time and after that they follow along the line segment y = 0, x € ( 2 * / | , 2 ) periodically jumping from the point ( 2 * / | , 0 J to the point (2,0).

It is clear that t* > Ti(x0) and also the solution x(t) is continuous in this interval. Moreover, x* = x0 + Ii(x0) + / f(T,cp(r))dT. 39) where h= f{r,x(T))dT. 39), by Cauchy-Schwarz inequality, admits the estimate I I ( f ^ o + Ii(xo) + crh),h\ do < NC(t* - Ti(x0)), CHAPTER 1. GENERAL 22 DESCRIPTION and thus, (1 - NC)(t* - Ti(x0)) < J (^(xo + o-Ii(xo)),Ii(x0)\da. 37). 34) in the case when the functions r,-(x) are not continuously differentiable. 34), ^ e functions f(t,x) and Ii(x) satisfy the conditions of the previous lemma and the functions r,(x) satisfy the Lipschitz condition |r,-(x') - r,-(*")| < N\\x' - x'% x',x" € D.

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