×

On a theorem of R. H. Martin on certain Cauchy problems for ordinary differential equations. (English) Zbl 0578.34003

In J. Lond. Math. Soc., II. Ser. 10, 61-65 (1975; Zbl 0305.34092), R. H. Martin showed a result concerning the Cauchy problem \(x=B(t,x)+C(t,x)\), \(x(0)=x_ 0\), where B and C are two suitable functions defined on [0,1]\(\times X\), X a closed and convex subset of a Banach space F, with B satisfying an assumption of dissipative type and C an assumption of compactness of the range. In the present note the author improves Martin’s theorem assuming that C verifies the following assumption: there is a Lebesgue measurable subset J of [0,1] such that \(m(J)=0\) and C(t,X) is relatively compact for any \(t\in [0,1]\setminus J\). The assumption on B remains the same. Moreover it could be observed that Martin supposed that C is uniformly continuous on [0,1]\(\times X\) and the present author also improves this assumption by supposing that for any \(\epsilon >0\) there is a Lebesgue measurable subset \(I_{\epsilon}\) of [0,1] with Lebesgue measure \(m(I_{\epsilon})<\epsilon\) so that C is uniformly continuous on \(([0,1]\setminus I_{\epsilon})\times X\).

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations

Citations:

Zbl 0305.34092
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] G. Emmanuele: Existence of solutions of ordinary differential equations involving dissipative and compact operators in Gelfand-Phillips space (to appear). · Zbl 0605.34002 · doi:10.1016/0022-247X(86)90177-0
[2] R. H. Martin: Remarks on ordinary differential equations involving dissipative and compact operators. J. London Math. Soc., 10, 61-65 (1975). · Zbl 0305.34092 · doi:10.1112/jlms/s2-10.1.61
[3] R. H. Martin: Nonlinear Operators and Differential Equations in Banach Spaces. Wiley and Sons (1976). · Zbl 0333.47023
[4] P. Volkmann: Ein Existenzsatz fur gewohnliche differentialgleichungen in Banachraume. Proc. Amer. Math. Soc, 80, 297-300 (1980). JSTOR: · Zbl 0506.34051 · doi:10.2307/2042966
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.