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On the correctness of linear boundary value problems for systems of generalized ordinary differential equations. (English) Zbl 0808.34015
Sufficient conditions are established for correctness of the linear boundary value problem (BVP) for the generalized differential system \((*)\) \(dx(t) = dA(t)x(t) + df(t)\), \(l(x) = c_ 0\), where \(A : [a,b] \to \mathbb{R}^{n \times n}\), \(f : [a,b] \to \mathbb{R}^ n\) are functions of bounded variation, \(c_ 0 \in \mathbb{R}\) and \(l\) is a linear continuous operator. Particularly, sufficient conditions are given which guarantee that the problem \(dx(t) = dA_ k (t)x(t) + df_ k(t)\), \(l_ k(x) = c_ k\), has a unique solution for large \(k\) and this solution \(x_ k(t) \to x_ 0(t)\) uniformly on \([a,b]\), where \(x_ 0\) is the solution of \((*)\). This result is based on application of some general results given in the monograph [Š. Schwábik, M. Tvrdý and O. Vejvoda, Differential and integral equations. Boundary value problems and adjoints, Academia, Praha (1979; Zbl 0417.45001) and completes the investigation of correctness of the BVP for ordinary differential equations given by the author’s paper [Proc. Georgian Acad. Sci. Math. 1, 129-141 (1993; Zbl 0786.34028)].
Reviewer: O.Došlý (Brno)

34B05 Linear boundary value problems for ordinary differential equations
34A30 Linear ordinary differential equations and systems
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