Mathematica Bohemica, Vol. 141, No. 2, pp. 183-215, 2016
On the opial type criterion for the well-posedness of the Cauchy problem for linear systems of generalized ordinary differential equations
Malkhaz Ashordia
Abstract: The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense $ d x(t)= d A_0(t)\cdot x(t)+ d f_0(t)$, $x(t_0)=\nobreak c_0$ $(t\in I)$ with a unique solution $x_0$ is considered. Necessary and sufficient conditions are obtained for a sequence of the Cauchy problems $ d x(t)= d A_k(t)\cdot x(t)+ 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 $x_k(t)\to x_0(t)$ 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.
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
Affiliations: Malkhaz Ashordia, A. Razmadze Mathematical Institute of I. Javakhishvili Tbilisi State University, 6 Tamarashvili Str., Tbilisi 0177 Georgia, and Sukhumi State University, 12 Politkovskaia Str., Tbilisi 0186 Georgia; e-mail: ashord@rmi.ge
