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Boundary value problems for operator differential equations with variable domain of definition of the operator coefficients. (Russian) Zbl 0745.34062
The authors consider the boundary value problem \[ \begin{aligned} Lu\equiv(- 1)^{[n/2]}{d^ nu\over dt^ n}+A(t)u=f(t),\quad f\in(0;T),\quad T<\infty, \\ \left.{d^ ku\over dt^ k}\right|_{t=0}=\left.{d^ lu\over dt^ l}\right|_{t=T}=0,\quad 0\leq k\leq\left[{n-1\over 2}\right],\quad 0\leq l\leq\left[{n-2\over 2}\right],\tag{1}\end{aligned} \] where \(n\) and \(f\) are functions whose values belong to a Hilbert space \(H\), \(A(t):H\to H\) is a linear closed operator whose domain of definition is a dense subset of \(H\). The existence and uniqueness of strongly generalized solutions of problem (1) and continuous dependence on the operator coefficient are investigated in this paper. The authors show the effectiveness and substantiate the necessity for the consideration of such a formulation of a problem as (1).

34G10 Linear differential equations in abstract spaces
34B05 Linear boundary value problems for ordinary differential equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35D05 Existence of generalized solutions of PDE (MSC2000)