M. Ashordia
abstract:
The system of the generalized linear ordinary differential equations
$$ d x(t)=d A(t)\cdot x(t)+df(t) $$
is considered with general $\ell(x)=c_0,$ multipoint $\sum_{j=1}^{n_0} L_jx(t_j)=c_0,$
and Cauchy-Nicoletti type $x_i(t_i)=\ell_i(x_1,\dots,x_n)+c_{0i} \;\; (i=1,\dots,n)$
boundary value conditions, 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 $[a,b]$, $c_0=(c_{0i})_{i=1}^n\in \mathbb{R}^n$, $t_i\in [a,b]$
$(i=1,\dots,n(n_0))$, $n_0$ 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, among them spectral, conditions are obtained for the
unique solvability of these problems. The obtained results are realized for the
linear impulsive system
$$
\frac{dx}{dt}=P(t) x+q(t), \;\;\; x(\tau_k+)-x(\tau_k-)=G_kx(\tau_k)+g_k
\;\;(k=1,2,\dots),
$$
where $P\in L([a,b], \mathbb{R}^{n\times n})$, $q\in L([a,b],\mathbb{R}^n)$, $G_k\in
\mathbb{R}^{n\times n}$, $g_k\in \mathbb{R}^n$ and $\tau_k\in[a,b]$
$(k=1,2,\dots)$, and linear difference system
$$
\Delta y(k-1)=G_1(k-1) y(k-1)+G_2(k)y(k)+G_3(k)y(k+1)+g_0(k)
$$
$(k=1,\dots,m_0)$, where $G_j(k)\in \mathbb{R}^{n\times n}$, $g_0(k)\in \mathbb{R}^n$
$(j=1,2,3$; $k=1,\dots,m_0)$.
Mathematics Subject Classification: 34K06, 34A37, 34B05, 34B10
Key words and phrases: Systems of linear generalized ordinary differential, impulsive and difference equations, general linear and multipoint boundary value problems, unique solvability, the Lebesgue-Stieltjes integral