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Separation for ordinary differential equation with matrix coefficient. (English) Zbl 0924.34055

The authors consider the differential expression \[ L[u]=(-1)^mD^{2m}u(x)+ q(x)u(x) \tag{1} \] with \(u\in L_p(\mathbb{R})^l\cap W^{2m}_{p,loc} (\mathbb{R})^l\) and \(q\) is an \(l\times l\) Hermitian matrix.
The differential expression (1) has been studied when \(m=1\), \(p=2\) and \(l=1\), \(p\in (1,\infty)\) and the case when \(m=1\), \(p\in (1,\infty)\), \(l\in \mathbb{N}\).
In this paper, they study the separation of the differential expression (1) in the Banach space \(L_p(\mathbb{R})^l\) for any \(p\in(1,\infty)\) and any arbitrary natural numbers \(m\) and \(l\). They prove existence and uniqueness of solutions to the differential equation \[ (L+\beta E)u(x)=f(x), \quad f(x) \in L_p(\mathbb{R})^l, \] where \(E\) is the indentity operator and \(\beta\geq 1\).

MSC:

34G10 Linear differential equations in abstract spaces
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