Abstract:
This paper looks at ordinary differential equations (DE) containing the derivative of the unknown functions with respect to a measure $\mu$ which is continuous with respect to the Lebesgue measure. It is shown that the Cauchy problem for a linear normal system of DE with a $\mu$-derivative is uniquely solvable. A necessary and sufficient condition is obtained for the solvability of an equation of Riccati type with a $\mu$-derivative. It is related to a boundary-value problem for a linear system of DE. Using this condition a necessary and sufficient condition is obtained for a Volterra factorization to exist for linear operators that differ from the identity by an integral operator that is completely continuous in the space $L_p(\mu)$, $1\le p<+\infty$.
Bibliography: 12 titles.
Keywords:linear differential equations with derivative with respect to a measure, Riccati equation, factorization.
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