On the opial type criterion for the well-posedness of the Cauchy problem for linear systems of generalized ordinary differential equations

Ashordia, Malkhaz. "On the opial type criterion for the well-posedness of the Cauchy problem for linear systems of generalized ordinary differential equations." Mathematica Bohemica 141.2 (2016): 183-215. <http://eudml.org/doc/276997>.

@article{Ashordia2016,
abstract = {The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense $\{\rm d\} x(t)=\{\rm d\} A_0(t)\cdot x(t)+\{\rm d\} f_0(t)$, $x(t_\{0\})=c_0$$(t\in I)$ with a unique solution $x_0$ is considered. Necessary and sufficient conditions are obtained for a sequence of the Cauchy problems $\{\rm d\} x(t)=\{\rm d\} A_k(t)\cdot x(t)+\{\rm d\} f_k(t)$, $x(t_\{k\})=c_k$$(k=1,2,\dots )$ to have a unique solution $x_k$ for any sufficiently large $k$ such that $x_k(t)\rightarrow x_0(t)$ uniformly on $I$. Presented results are analogous to the sufficient conditions due to Z. Opial for linear ordinary differential systems. Moreover, efficient sufficient conditions for the problem of well-posedness are given.},
author = {Ashordia, Malkhaz},
journal = {Mathematica Bohemica},
keywords = {linear system of generalized ordinary differential equations in the Kurzweil sense; Cauchy problem; well-posedness; Opial type necessary condition; Opial type sufficient condition; efficient sufficient condition},
language = {eng},
number = {2},
pages = {183-215},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the opial type criterion for the well-posedness of the Cauchy problem for linear systems of generalized ordinary differential equations},
url = {http://eudml.org/doc/276997},
volume = {141},
year = {2016},
}

TY - JOUR
AU - Ashordia, Malkhaz
TI - On the opial type criterion for the well-posedness of the Cauchy problem for linear systems of generalized ordinary differential equations
JO - Mathematica Bohemica
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 141
IS - 2
SP - 183
EP - 215
AB - The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense ${\rm d} x(t)={\rm d} A_0(t)\cdot x(t)+{\rm d} f_0(t)$, $x(t_{0})=c_0$$(t\in I)$ with a unique solution $x_0$ is considered. Necessary and sufficient conditions are obtained for a sequence of the Cauchy problems ${\rm d} x(t)={\rm d} A_k(t)\cdot x(t)+{\rm d} f_k(t)$, $x(t_{k})=c_k$$(k=1,2,\dots )$ to have a unique solution $x_k$ for any sufficiently large $k$ such that $x_k(t)\rightarrow x_0(t)$ uniformly on $I$. Presented results are analogous to the sufficient conditions due to Z. Opial for linear ordinary differential systems. Moreover, efficient sufficient conditions for the problem of well-posedness are given.
LA - eng
KW - linear system of generalized ordinary differential equations in the Kurzweil sense; Cauchy problem; well-posedness; Opial type necessary condition; Opial type sufficient condition; efficient sufficient condition
UR - http://eudml.org/doc/276997
ER -