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Convoluted Fractional -Semigroups and Fractional Abstract Cauchy Problems

DOI: 10.1155/2014/357821

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We present the notion of convoluted fractional -semigroup, which is the generalization of convoluted -semigroup in the Banach space setting. We present two equivalent functional equations associated with convoluted fractional -semigroup. Moreover, the well-posedness of the corresponding fractional abstract Cauchy problems is studied. 1. Introduction Let be a Banach space and let ( denotes the space of bounded linear operators from to itself) be injective. Let , where is locally integrable on . A strongly continuous operator family is called a -convoluted -semigroup if , , , and there holds If, in addition, , , implies , is called a nondegenerate -convoluted -semigroup. For more details, we refer to Kosti? [1]. Obviously, -convoluted -semigroups are a generalization of classical -semigroups. The extension of Widder’s representation theorem by Arendt [2] stimulated the development of the theory of -times integrated semigroups, which is the special case that , (see [3–6] for being an integer and [7–9] for being noninteger). Li and Shaw [10–12] introduced exponentially bounded -times integrated -semigroups which are the special case, where , and they studied their connection with the associated abstract Cauchy problem. Kuo and Shaw [13] were concerned with the case with being noninteger and they called it -times integrated -semigroups. Convoluted semigroups, which are the special case that , were introduced by Cioranescu and Lumer [14–16]. It is a generalization of integrated semigroups. Moreover, Kunstmann [17] showed that there exists a global convoluted semigroup whose generator is not stationary dense and therefore it cannot be the generator of a local integrated semigroup. After that, Melnikova and Filinkov [18], Keyantuo et al. [19], and Kosti? and Pilipovi? [20] systematically studied the properties of convoluted semigroups and related them to associated abstract Cauchy problems. In [1], Kosti? presented the notion -convoluted -semigroup and found a sufficient condition for a nondegenerate strongly continuous linear operator family to be a -Convoluted -semigroup; that is, , , , , , and . Moreover, Kosti? showed that if the sufficient condition holds, the abstract Cauchy problem is well-posed. In Proposition 2.3 of [3], it was proved that if the following abstract Cauchy problem is well-posed, then the above sufficient conditions hold. Motivated by such facts, Kosti? gave in [1] the definition of -convoluted semigroup by the above sufficient conditions and proved the equivalence of the two definitions. Later, Kosti? and Pilipovi? [21] gave the

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