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On the general and multipoint boundary value problems for linear systems of generalized ordinary differential equations, linear impulse and linear difference systems. (English) Zbl 1098.34010
Summary: The system of the generalized linear ordinary differential equations \[ dx(t)=dA(t)\cdot x(t)+df(t) \] is considered with general \(\ell(x)=c_0\), multipoint \(\sum^{n_0}_{j=1}L_jx(t_j)=c_0\), and Cauchy-Nicoletti-type boundary value conditions \(x_i(t_i)=\ell_i(x_1, \dots,x_n)+c_{0i}\), \(i=1,\dots,n\), where \(A:[a,b]\to \mathbb{R}^{n\times n}\) and \(f:[a,b]\to \mathbb{R}^n\) are, respectively, matrix- and vector-functions with bounded total variation components on the closed interval \([a,b]\), \(c_0= (c_{0i})^n_{i=1}\in\mathbb{R}^n\), \(t_i\in[a,b]\), \(i=1,\dots,n(n_0)\), \(n_0\) is a fixed natural number, \(L_j\in\mathbb{R}^{n\times n}\), \(j=1,\dots,n_0\), \(x_i\) is the \(i\)th component of \(x\), and \(\ell\) and \(\ell_i\), \(i=1,\dots, n\), are linear operators. Effective sufficient conditions, among them spectral, are obtained for the unique solvability of these problems. The results obtained are realized for the linear impulsive system \[ \frac{dx} {dt}=P(t)x+q(t),\;x(\tau_k+) -x(\tau_k-)=G_kx(\tau_k)+g_k,\;k=1,2, \dots, \] with \(P\in L([a,b],\mathbb{R}^{n \times n})\), \(q\in L([a,b], \mathbb{R}^n)\), \(G_k\in\mathbb{R}^{n\times n}\), \(g_k\in\mathbb{R}^n\) and \(\tau_k\in [a,b]\) \((k=1,2,\dots)\), and for the linear difference system \[ \Delta y(k-1)= G_(k-1)y(k-1)+G_2(k)y(k)+G_3(k)y(k+1)+g_0(k)\quad k=1,\dots,m_0, \] with \(G_j(k)\in\mathbb{R}^{n\times n}\), \(g_0(k)\in\mathbb{R}^n\), \(j=1,2,3; k=1,\dots,m_0\).

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34K06 Linear functional-differential equations
34A37 Ordinary differential equations with impulses
34B05 Linear boundary value problems for ordinary differential equations
34B37 Boundary value problems with impulses for ordinary differential equations