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
References
[1]
M. Kosti？, “Convoluted C-cosine functions and convoluted C-semigroups,” Bulletin, Classe des Sciences Mathématiques et Naturelles, Sciences Mathématiques, vol. 127, no. 28, pp. 75–92, 2003.
[2]
W. Arendt, “Vector-valued Laplace transforms and Cauchy problems,” Israel Journal of Mathematics, vol. 59, no. 3, pp. 327–352, 1987.
[3]
W. Arendt, O. El-Mennaoui, and V. Kéyantuo, “Local integrated semigroups: evolution with jumps of regularity,” Journal of Mathematical Analysis and Applications, vol. 186, no. 2, pp. 572–595, 1994.
[4]
W. Arendt, C. J. K. Batty, M. Hieber, and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Birkh？user, Basel, Switzerland, 2001.
[5]
H. Kellerman and M. Hieber, “Integrated semigroups,” Journal of Functional Analysis, vol. 84, no. 1, pp. 160–180, 1989.
[6]
F. Neubrander, “Integrated semigroups and their applications to the abstract Cauchy problem,” Pacific Journal of Mathematics, vol. 135, pp. 233–251, 1989.
[7]
M. Hieber, “Laplace transforms and -times integrated semigroups,” Forum Mathematicum, vol. 3, no. 6, pp. 595–612, 1991.
[8]
P. J. Miana, “ -times integrated semigroups and fractional derivation,” Forum Mathematicum, vol. 14, no. 1, pp. 23–46, 2002.
[9]
M. Mijatovi？, S. Pilipovi？, and F. Vajzovi？, “ -times integrated semigroups ,” Journal of Mathematical Analysis and Applications, vol. 210, no. 2, pp. 790–803, 1997.
[10]
Y. C. Li, Integral C-semigroups and C-cosine functions of operators on locally convex spaces [Ph.D. dissertation], National Central University, 1991.
[11]
Y. C. Li and S. Y. Shaw, “On generators of integrated C-semigroups and C-cosine functions,” Semigroup Forum, vol. 47, no. 1, pp. 29–35, 1993.
[12]
Y. C. Li and S. Y. Shaw, “N-times integrated C-semigroups and the abstract Cauchy problem,” Taiwanese Journal of Mathematics, vol. 1, no. 1, pp. 75–102, 1997.
[13]
C. C. Kuo and S. Y. Shaw, “On -times integrated C-semigroups and the abstract Cauchy problem,” Studia Mathematica, vol. 142, no. 3, pp. 201–217, 2000.
[14]
I. Cioranescu, “Local convoluted semigroups,” in Evolution Equations, pp. 107–122, Dekker, New York, NY, USA, 1995.
[15]
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.
[16]
I. Cioranescu and G. Lumer, “On -convoluted semigroups,” in Recent Developments in Evolution Equations, pp. 86–93, Longman Scientific & Technical, Harlow, UK, 1995.
[17]
P. C. Kunstmann, “Stationary dense operators and generation of non-dense distribution semigroups,” Journal of Operator Theory, vol. 37, no. 1, pp. 111–120, 1997.
[18]
I. V. Melnikova and A. Filinkov, Abstract Cauchy Problems: Three Approaches, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2001.
[19]
V. Keyantuo, C. Müller, and P. Vieten, “The Hille-Yosida theorem for local convoluted semigroups,” Proceedings of the Edinburgh Mathematical Society, vol. 46, no. 2, pp. 395–413, 2003.
[20]
M. Kosti？ and S. Pilipovi？, “Global convoluted semigroups,” Mathematische Nachrichten, vol. 280, no. 15, pp. 1727–1743, 2007.
[21]
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.
[22]
S. D. Eidelman and A. N. Kochubei, “Cauchy problem for fractional diffusion equations,” Journal of Differential Equations, vol. 199, no. 2, pp. 211–255, 2004.
[23]
V. Lakshmikanthan and S. Leela, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, UK, 2009.
[24]
M. M. Meerschaert, E. Nane, and P. Vellaisamy, “Fractional Cauchy problems on bounded domains,” The Annals of Probability, vol. 37, no. 3, pp. 979–1007, 2009.
[25]
R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, no. 1, pp. 1–77, 2000.
[26]
I. Podlubny, Fractional Differential Equations, Academic Press, New York, NY, USA, 1999.
[27]
E. G. Bajlekova, Fractional Evolution Equations in Banach Spaces, University Press Facilities, Eindhoven University of Technology, Eindhoven, The Netherlands, 2001.
[28]
G. da Prato and M. Iannelli, “Linear integro-differential equations in Banach spaces,” Rendiconti del Seminario Matematico dell'Università di Padova, vol. 62, pp. 207–219, 1980.
[29]
M. Li, Q. Zheng, and J. Zhang, “Regularized resolvent families,” Taiwanese Journal of Mathematics, vol. 11, no. 1, pp. 117–133, 2007.
[30]
C. Lizama, “Regularized solutions for abstract Volterra equations,” Journal of Mathematical Analysis and Applications, vol. 243, no. 2, pp. 278–292, 2000.
[31]
J. Prüss, Evolutionary Integral Equations and Applications, Birkh？user, Basel, Switzerland, 1993.
[32]
C. Chen and M. Li, “On fractional resolvent operator functions,” Semigroup Forum, vol. 80, no. 1, pp. 121–142, 2010.
[33]
L. Kexue and P. Jigen, “Fractional abstract Cauchy problems,” Integral Equations and Operator Theory, vol. 70, no. 3, pp. 333–361, 2011.
[34]
L. Kexue and P. Jigen, “Fractional resolvents and fractional evolution equations,” Applied Mathematics Letters, vol. 25, no. 5, pp. 808–812, 2012.
[35]
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.
[36]
Z. D. Mei, J. G. Peng, and Y. Zhang, “On general fractional abstract Cauchy problem,” Communications on Pure and Applied Analysis, vol. 12, no. 6, pp. 2753–2772, 2013.
[37]
C. Lizama and F. Poblete, “On a functional equation associated with -regularized resolvent families,” Abstract and Applied Analysis, vol. 2012, Article ID 495487, 23 pages, 2012.
[38]
J. Peng and K. Li, “A novel characteristic of solution operator for the fractional abstract Cauchy problem,” Journal of Mathematical Analysis and Applications, vol. 385, no. 2, pp. 786–796, 2012.
[39]
Z. D. Mei, J. G. Peng, and Y. Zhang, “A characteristic of fractional resolvents,” Fractional Calculus and Applied Analysis, vol. 16, no. 4, pp. 777–790, 2013.
[40]
M. Kosti？, “ -regularized C-resolvent families: regularity and local properties,” Abstract and Applied Analysis, vol. 2009, Article ID 858242, 27 pages, 2009.
[41]
C. Lizama and G. M. N'Guerekata, “Mild solutions for abstract fractional differential equations,” Applicable Analysis, vol. 92, no. 8, pp. 1731–1754, 2013.