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

Mathematica Bohemica (2016)

- Volume: 141, Issue: 2, page 183-215
- ISSN: 0862-7959

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topAshordia, 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 -

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