On the solution of ill-posed problems in locally convex spaces. (English. Russian original) Zbl 0796.34042

Tanana, V. P.

Russ. Acad. Sci., Dokl., Math. 46, No. 2, 254-257 (1993); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 326, No. 2, 233-236 (1992).

Let \(H\) be a separable Hilbert space, and \(A\) a semibounded selfadjoint linear operator, \(D(A)\subset H\). Two ill-posed problems: the calculation of \(Af\), \(f\in H\) and the Cauchy problem \(u'(t)= Au(t)\), \(t\in [0,T]\), \(u(t)\in H\), \(u(0)= u_ 0\in H\) are considered. The author proposes a new construction of locally convex spaces (spaces of solutions), which not only ensures the well-posedness of the problems, but also permits to determine their solutions.

Reviewer: S.I.Trofimchuk (Kiev)

MSC:

34G10	Linear differential equations in abstract spaces
46A04	Locally convex Fréchet spaces and (DF)-spaces
46A13	Spaces defined by inductive or projective limits (LB, LF, etc.)

Keywords:

ill-posed problems; Cauchy problem; construction of locally convex spaces; well-posedness