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

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.

### 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.)