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Ordinary dichotomy and perturbations of the coefficient matrix of the linear impulsive differential equation. (English) Zbl 0751.34009
Let \(t_ 0<t_ 1<\cdots<t_ i<\cdots,\lim_{i\to\infty}t_ i=\infty\) be a given sequence of real numbers. Consider the linear differential equation \(dx/dt=A(t)x\), \((t\neq t_ i)\) with impulses \(x(t_ i+0)=B_ ix(t_ i)\), \(i=1,2,\ldots\). The main result establishes that the ordinary dichotomy is preserved under perturbations of the coefficient matrix \(A(t)\).
Reviewer: P.Smith (Keele)

MSC:
34A37 Ordinary differential equations with impulses
34D05 Asymptotic properties of solutions to ordinary differential equations
34D10 Perturbations of ordinary differential equations
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References:
[1] Coppel ( W.A. ) .- Dichotomies in stability theory , Lecture Notes in Math. , Springer-Verlag , 629 ( 1978 ). MR 481196 | Zbl 0376.34001 · Zbl 0376.34001
[2] Daleckii ( Ju.L. ) and Krein ( M.G. ) .- Stability of solutions of differential equations in Banach spaces , Amer. Math. Soc. Transl. , Providence, R.I. ( 1974 ). MR 352639
[3] Dishliev ( A.B. ) and Bainov ( D.D. ) .- Continuous dependence of the solution of a system of differential equations with impulses on the impulse hypersurfaces , J. Math. Anal. Appl ., 135 , n^\circ 2 ( 1988 ), pp. 369 - 382 . MR 967216 | Zbl 0674.34005 · Zbl 0674.34005
[4] Hekimova ( M.A. ) and Bainov ( D.D. ) .- Periodic solutions of singularly perturbed systems of differential equations with impulse effect , ZAMP , 36 ( 1985 ), pp. 520 - 537 . MR 801524 | Zbl 0612.34033 · Zbl 0612.34033
[5] Lakshmikantham ( V. ) and Xinzhi Liu .- Stability for impulsive differential systems in terms of two measures , Appl. Math. Comp. (to appear). MR 973495 | Zbl 0669.34056 · Zbl 0669.34056
[6] Lakshmikantham ( V. ) and Xinzhi Liu .- On quasi stability for impulsive differential systems , Nonlinear Analysis (to appear). MR 999331 | Zbl 0688.34032 · Zbl 0688.34032
[7] Massera ( J.L. ) and Schäffer ( J.J. ) .- Linear differential equations and functional analysis, I , Ann. of Math. , 67 ( 1958 ), pp. 517 - 573 . MR 96985 | Zbl 0178.17701 · Zbl 0178.17701
[8] Milev ( N.V. ) and Bainov ( D.D. ) . - Dichotomies for linear differential equations with variable structure and impulse effect , (to appear). MR 1150157
[9] Palmer ( K.J. ) .- A perturbation theorem for exponential dichotomies , Proc. Roy. Soc. Edinburgh , 106 A ( 1987 ), pp. 25 - 37 . MR 899938 | Zbl 0629.34058 · Zbl 0629.34058
[10] Simeonov ( P.S. ) and Bainov ( D.D. ) .- Stability with respect to part of the variables in systems with impulse effect , J. Math. Appl. , 117 , n^\circ 1 ( 1986 ), pp. 247 - 263 . MR 843016 | Zbl 0588.34044 · Zbl 0588.34044
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