A reduction theorem for systems of differential equations with impulse effect in a Banach space. (English) Zbl 0860.34027

Reinfelds, Andrejs

J. Math. Anal. Appl. 203, No. 1, 187-210 (1996).

The author justifies the reduction principle for the system of weakly nonlinear abstract impulsive equations (1) \(x'_i= A_i(t)x_i+ f_i(t,x_1,x_2)\), \(\Delta x_i|_{\tau_k}= D_{ik}x_i(\tau_k-0)+ p_{ik}(x_i(\tau_k-0), x_2(\tau_k-0))\), \(i=1,2\), \(k\in\mathbb{Z}\). Here \((t,x_i)\in \mathbb{R}\times X_i\), \(X_i\) (\(i=1,2\)) are complex Banach spaces, \(A_i(t)\in L(X_i)\) \(\forall t\in \mathbb{R}\). Particularly, the existence of a unique piecewise continuous bounded integral manifold \(M\), which is given by the map \(x_2= G(t,x_1): \mathbb{R}\times X_1\to X_2\), is established and the integral distance between an arbitrary solution and \(M\) is estimated (Theorem 1). This result allows to prove the global strong dynamic equivalence between (1) and some much simpler (decomposed into two parts) impulsive system (Theorem 2).

Reviewer: E.Trofimtchouk (Kiev)

MSC:

34G20	Nonlinear differential equations in abstract spaces
34A37	Ordinary differential equations with impulses

Keywords:

reduction principle; weakly nonlinear abstract impulsive equations; Banach spaces; unique piecewise continuous bounded integral manifold