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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 $$G_j(k)\in\mathbb{R}^{n\times n}$$, $$g_0(k)\in\mathbb{R}^n$$, $$j=1,2,3; k=1,\dots,m_0$$.

##### MSC:
 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