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On a type of linear differential equations in Fréchet spaces. (English) Zbl 0902.34052

Let \(\mathbb{F}\) be an arbitrarily chosen Fréchet space. \(\mathbb{F}\) can be the limit of a projective system of Banach spaces [see H. H. Schaefer, Studies in functional analysis, MAA Stud. Math. 21, 158-221 (1980; Zbl 0494.46021)]. Linear differential equations (l.d.e.) in Fréchet spaces cannot be solved in general. Even in the case if solutions exist, they are not uniquely determined by initial conditions.
The author proves that a special type of l.d.e. in Fréchet spaces admits a unique solution for any given initial condition. The main results are included in the following theorem:
With a given Fréchet space \(\mathbb{F}\) the l.d.e. \[ \dot x= Ax+ B\tag{1} \] is considered, where \(A: I= [0,1]\to (\mathbb{F})\) and \(B: I\to \mathbb{F}\) are continuous maps. If \(A\) can be factorized in the form \(A= \varepsilon A^*\), where \(A^*\) is continuous, then (1) admits a unique solution for any initial condition \((t_0, x_0)\in I\times\mathbb{F}\).
In the last part of this note, the author proves that the linear differential equations studied by N. Papaghiuc [Rev. Roum. Math. Pures Appl. 25, 83-88 (1980; Zbl 0441.34042)] and R. S. Hamilton [Bull. Am. Math. Soc., New Ser. 7, 65-222 (1982; Zbl 0499.58003)] are special cases of his main result.

MSC:

34G10 Linear differential equations in abstract spaces
34A30 Linear ordinary differential equations and systems
54D55 Sequential spaces
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References:

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