Differential equations where the derivative is taken with respect to a measure
N. B. Engibaryan
Institute of Mathematics, National Academy of Sciences of Armenia
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.
linear differential equations with derivative with respect to a measure, Riccati equation, factorization.
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Sbornik: Mathematics, 2011, 202:2, 243–256
MSC: Primary 34A30; Secondary 34A12, 34B05, 47A68, 47B38, 47G10, 60J25, 60J35
Received: 30.11.2009 and 28.06.2010
N. B. Engibaryan, “Differential equations where the derivative is taken with respect to a measure”, Mat. Sb., 202:2 (2011), 93–106; Sb. Math., 202:2 (2011), 243–256
Citation in format AMSBIB
\paper Differential equations where the derivative is taken with respect to a~measure
\jour Mat. Sb.
\jour Sb. Math.
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