On a weakly degenerate first-order linear differential equation in a Banach space. (English. Russian original) Zbl 1275.34082

Fomin, V. I.

Differ. Equ. 41, No. 10, 1514-1516 (2005); translation from Differ. Uravn. 41, No. 10, 1433-1435 (2005).

Consider the equation \[ \varphi(t) x'(t)= Ax(t)+ f(t)\quad\text{for }t\in (0,\infty)\tag{\(*\)} \] in a Banach space \(E\) under the following assumptions (i) \(A: D(A)\subset E\to E\) is an unbounded linear operator

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and (ii) \(\varphi\in C((0,\infty), (0,\infty))\) satisfies \(\lim_{t\to +0}{\varphi(t)\over t^\alpha}= K\) with \(0< \alpha< 1\), \(0< K<\infty\).
The author gives additional conditions on \(A\) and \(f\) such that \((*)\) has a solution bounded for \(t= 0\).

Reviewer: Klaus R. Schneider (Berlin)

MSC:

34G10	Linear differential equations in abstract spaces
34C11	Growth and boundedness of solutions to ordinary differential equations

References:

[1]	Fomin, V.I., Differents. Uravn., 2004, vol. 40, no.2, pp. 183–190.
[2]	Krein, S.G., Lineinye differentsial’nye uravneniya v banakhovom prostranstve (Linear Differential Equations in a Banach Space), Moscow: Nauka, 1967.
[3]	Balakrishnan, A.V., Applied Functional Analysis, New York: Springer, 1976. Translated under the title Prikladnoi funktsional’nyi analiz, Moscow: Nauka, 1980.
[4]	Fomin, V.I., Materialy konferentsii ”Sovremennye metody teorii funktsii i smezhnye problemy,” 26 yanv.–2 fevr. 2003 g. (Proc. Conf. ”Modern Methods of Function Theory and Related Problems,” January 26–February 2, 2003), Voronezh, 2003, pp. 270–271.