Abstract
We contribute to the existence theory of abstract time-fractional equations by stating the sufficient conditions for generation of not exponentially bounded α-times C-regularized resolvent families (α > 1) in sequentially complete locally convex spaces. We also consider the growth order of constructed solutions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
W. Arendt, C.J.K. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace Transforms and Cauchy Problems. Birkhäuser Verlag, Basel (1990).
E. Bazhlekova, Fractional Evolution Equations in Banach Spaces. PhD Thesis, Department of Mathematics, Eindhoven University of Technology, Eindhoven (2001); http://alexandria.tue.nl/extra2/200113270.pdf.
C. Chen, M. Li, On fractional resolvent operator functions. Semigroup Forum 80, No 1 (2010), 121–142.
R. deLaubenfels, Existence Families, Functional Calculi and Evolution Equations. Lecture Notes in Mathematics 1570, Springer, New York (1994).
H.J. Haubold, A.M. Mathai, R.K. Saxena, Mittag-Leffler functions and their applications. ArXiv: 0909.0230v2 [math.CA] 2009; http://arxiv.org/PScache/arxiv/pdf/0909/0909.0230v2.pdf.
M. Hieber, A. Holderrieth, F. Neubrander, Regularized semigroups and systems of linear partial differential equations. Ann. Scuola Norm. Sup. Pisa 19,No 3 (1992), 363–379.
M. Kostić, Complex powers of non-densely defined operators. Publ. Inst. Math., Nouv. Sér. (Math. Inst. Belgrade), To appear (2011).
M. Kostić, (a, k)-regularized C-resolvent families: regularity and local properties. Abstr. Appl. Anal. 2009 Article ID 858242 (2009), 27 pages.
M. Kostić, Abstract Volterra equations in locally convex spaces. Preprint (2010), Submitted to: Science in China.
M. Kostić, Generalized Semigroups and Cosine Functions. Mathematical Institute Belgrade, 2011.
F.-B. Li, M. Li, Q. Zheng, Fractional evolution equations governed by coercive differential operators. Abstr. Appl. Anal. 2009, Article ID 438690 (2009), 14 pages.
M. Li, Q. Zheng, J. Zhang, Regularized resolvent families. Taiwanese J. Math. 11, No 1 (2007), 117–133.
M. Li, C.-C. Li, F.-B. Li, On fractional powers of generators of fractional resolvent families. J. Funct. Anal. 259, No 10 (2010), 2702–2726.
C. Lizama, Regularized solutions for abstract Volterra equations. J. Math. Anal. Appl. 243, No 2 (2000), 278–292.
C. Martínez, M. Sanz, The Theory of Fractional Powers of Operators. North-Holland Math. Stud. 187, Elseiver, Amsterdam (2001).
R. Meise, D. Vogt, Introduction to Functional Analysis. Translated from the German by M.S. Ramanujan and revised by the authors. Oxford Grad. Texts Math., Clarendon Press, New York (1997).
J. Prüss, Evolutionary Integral Equations and Applications. Monographs in Mathematics 87, Birkhäuser Verlag, Basel (1993).
B. Straub, Fractional powers of operators with polynomially bounded resolvents and semigroups generated by them. Hiroshima Math. J. 24, No 3 (1994), 529–548; http://projecteuclid.org/euclid.hmj/1206127925.
T.-J. Xiao, J. Liang, The Cauchy Problem for Higher-Order Abstract Differential Equations. Springer-Verlag, Berlin (1998).
J. Zhang, Q. Zheng, Pseudodifferential operators and regularized semigroups. Chin. J. Contemp. Math. 19, No 4 (1998), 387–394.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Kostić, M. Abstract time-fractional equations: Existence and growth of solutions. fcaa 14, 301–316 (2011). https://doi.org/10.2478/s13540-011-0018-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.2478/s13540-011-0018-4