By Williams J. W.

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We denote by [O,~] the resulting normed space. 1. 1) is valid for it. 3773 In order to obtain a sufficient condition for the (n,w)-well-posedness of the (DCP), we shall investigate the properties of the generators of a degenerate n-times integrated semigroup.. 1. 4) (n C {0} U N). 2) t t ' ~ "-+ l ) ! B- x : B V. ( t ) x - A / V-(s)xds, (n+ t>0, X E Xl ~-D1 . 3) 0 The family of the operators {V'(t), t > 0} forms a degenerate n-times integrated, exponentially bounded semigroup in the subspace F := (AB - A)-IB(Xa).

E. Hille and R. S. Phillips, Functional Analysis and Semigroups, Am. Math. , Providence, Rhode Island (1957). 26. N. H. Abdelazis and F. Neubrander, "Degenerate abstract Cauchy problems," Seminar Notes in Func. Anal. and Part. Diff. , 1-12 (1991-1992). 27. W. Arendt, "Vector valued Laplace transforms and Cauchy problems," Israel J. , 59, 327-352 (1987). 28. W. Arendt, "Resolvent positive operators," Proc. London Math. , 54, 321-349 (1987). 29. W. Arendt, F. Neubrander, and U. Schlotterbeck, "Interpolation of semigroups and integrated semigroups," Semigroup Forum, 45, 26-37 (1992).

15. [. V. Mel'nikova and M. A. Al'shansky, "Well-posedness of the degenerate Cauchy problem in a Banach space," Dokl. Rossiisk. Akad. Nauk, 336, No. 1 (1994). 16. I. V. Mel'nikova and S. V. Bochkareva, "C-semigroups and the regularization of the ill-posed Cauchy problem," Dokl. Rossiisk. Akad. Nauk, 329, No. 6, 270-273 (1993). 17. V. B. Osipov, "On one equation in a Banach space with a degenerate operator in the derivative," In: Collection of Works by Post-Graduates in Math. and Mech. Voronezh Univ.

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