Geodesic
Differential equations where the derivative is taken with respect to a~measure
Sbornik. Mathematics, Tome 202 (2011) no. 2, pp. 243-256

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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, factorization.
Mots-clés : Riccati equation 
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     author = {N. B. Engibaryan},
     title = {Differential equations where the derivative is taken with respect to a~measure},
     journal = {Sbornik. Mathematics},
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N. B. Engibaryan. Differential equations where the derivative is taken with respect to a~measure. Sbornik. Mathematics, Tome 202 (2011) no. 2, pp. 243-256. http://geodesic.mathdoc.fr/item/SM_2011_202_2_a2/