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？ . Obviously, -convoluted -semigroups are a generalization of classical -semigroups. The extension of Widder’s representation theorem by Arendt  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  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  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 , Keyantuo et al. , and Kosti？ and Pilipovi？  systematically studied the properties of convoluted semigroups and related them to associated abstract Cauchy problems. In , 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 , 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  the definition of -convoluted semigroup by the above sufficient conditions and proved the equivalence of the two definitions. Later, Kosti？ and Pilipovi？  gave the
I. Cioranescu and G. Lumer, “Problèmes d'évolution régularisés par un noyau général . Formule de Duhamel, prolongements, théorèmes de génération,” Comptes Rendus de l'Académie des Sciences. Série I, vol. 319, no. 12, pp. 1273–1278, 1994.
M. Kosti？ and S. Pilipovi？, “Convoluted C-cosine functions and semigroups. Relations with ultradistribution and hyperfunction sines,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1224–1242, 2008.
K. X. Li, J. G. Peng, and J. X. Jia, “Cauchy problems for fractional differential equations with Riemann-Liouville fractional derivatives,” Journal of Functional Analysis, vol. 263, no. 2, pp. 476–510, 2012.