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Mat. Sb., 2011, Volume 202, Number 2, Pages 93–106 (Mi msb7756)  

Differential equations where the derivative is taken with respect to a measure

N. B. Engibaryan

Institute of Mathematics, National Academy of Sciences of Armenia

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.

DOI: https://doi.org/10.4213/sm7756

Full text: PDF file (514 kB)
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English version:
Sbornik: Mathematics, 2011, 202:2, 243–256

Bibliographic databases:

UDC: 517.91+517.518.1
MSC: Primary 34A30; Secondary 34A12, 34B05, 47A68, 47B38, 47G10, 60J25, 60J35
Received: 30.11.2009 and 28.06.2010

Citation: 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
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