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    Boundedness and Stability of Impulsively Perturbed Systems in a Banach Space

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    Consider a linear impulsive equation in a Banach space x˙(t)+A(t)x(t)=f(t), t0,\dot{x}(t)+A(t)x(t) = f(t), ~t \geq 0, x(τi+0)=Bix(τi0)+αi,x(\tau_i +0)= B_i x(\tau_i -0) + \alpha_i, with limiτi=\lim_{i \rightarrow \infty} \tau_i = \infty . Suppose each solution of the corresponding semi-homogeneous equation x˙(t)+A(t)x(t)=0,\dot{x}(t)+A(t)x(t) = 0, (2) is bounded for any bounded sequence {αi}\{ \alpha_i \}. The conditions are determined ensuring (a) the solution of the corresponding homogeneous equation has an exponential estimate; (b) each solution of (1),(2) is bounded on the half-line for any bounded ff and bounded sequence {αi}\{ \alpha_i \} ; (c) limtx(t)=0\lim_{t \rightarrow \infty}x(t)=0 for any f,αif, \alpha_i tending to zero; (d) exponential estimate of ff implies a similar estimate for xx.Comment: 19 pages, LaTex-fil
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