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Convoluted $$C$$-cosine functions and convoluted $$C$$-semigroups. (English) Zbl 1067.47056
Starting with Arendt’s integrated semigroups and the $$C$$-semigroups of Da Prato and Sinestrari, later, Davies and Pang, many mathematicians contributed to the analysis of various classes of semigroups and corresponding classes of cosine functions. Such investigations were motivated by the fact that infinitesimal generators are not densely defined and, mainly, by the corresponding Cauchy problems.
For the investigations of this paper, the work of I. Cioranescu and G. Lumer [C. R. Acad. Sci., Paris, Sér. I 319, No. 12, 1273–1278 (1994; Zbl 0821.47032)] on convoluted semigroups and the work of T. Tanaka and N. Okazawa [Isr. J. Math. 69, No. 3, 257–288 (1990; Zbl 0723.47035)] on $$C$$-semigroups are crucial.
The author introduces classes of $$C$$-cosine functions and convoluted $$C$$-semigroups and analyzes their structure. In this way, he unifies the earlier theories. Thus, the known assertions for integrated $$C$$-cosine functions and integrated $$C$$-semigroups are formulated in a new setting through the convolution with a suitably chosen kernel $$K$$ instead of $$t^\alpha/\Gamma(\alpha+1)$$. In this way, the author formulates new classes of Cauchy problems and proves their well-posedness.

##### MSC:
 47D60 $$C$$-semigroups, regularized semigroups 47D03 Groups and semigroups of linear operators 47D06 One-parameter semigroups and linear evolution equations
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