Semigroups on time scales and applications to abstract Cauchy problems
Keywords
Dynamic equations in time scales, one parameter semigroups, abstract Cauchy problem, Banach spaceAbstract
In this paper, we introduce the definition of a $C_0$-semigroup on a time scale, which unifies the continuous, discrete and other cases which lie between them. Also, it extends the classical theory of operator semigroups to the quantum case. We study the relationship between the semigroup and its infinitesimal generator. We apply our theory to study the homogeneous and non homogeneous abstract Cauchy problem in Banach and Fréchet spaces.References
M. Adivar and Y. N. Raffoul, Existence results for periodic solutions of integrodynamic equations on time scales, Ann. Mat. Pura Appl. 188 (2009), no. 4, 543–559.
R. Agarwal, C. Cuevas and C. Lizama, Regularity of Difference Equations on Banach Space, Springer–Verlag, New York, 2014.
W. Arendt, C.J.K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, 2nd edition, Monographs in Mathematics, vol. 96, Birkhäuser , 2011.
F.M. Atici, D.C. Biles and A. Lebedinsky, An application of time scales to economics, Math. Comput. Modelling 43 (2006), 718–726.
M. Bohnerand G. Guseinov, Multiple integration on time scales, Dynam. Systems Appl. 14 (2005), no. 3–4, 579–606.
M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, Boston, 2001.
M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, 2003.
D. Brigo and F. Mercurio, Discrete time vs continuous time stock-price dynamics and implications for option pricing, Finance and Stochastics 4 (2000), 147–159.
F.B. Christiansen and T.M. Fenchel, Theories of populations in biological communities, Lect. Notes in Ecological Studies, vol. 20, Springer–Verlag, Berlin, 1977.
J.M. Davis, I.A. Gravagne, B.J. Jackson, R.J. Marks II and A.A. Ramos, The Laplace transform on time scales revisited, J. Math. Anal. and Appl. 332 (2007), 1291–1307.
A. Dogan, J. Graef and L. Kong, Higher-order singular multi-point boundary-value problems on time scales, Proc. Edinb. Math. Soc. (2) 54 (2011), 345–361.
K.J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, vol. 194, Springer–Verlag, New York, 2000.
L. Erbe, J. Baoguo and A. Peterson, On the asymptotic behavior of solutions of Emden–Fowler equations on time scales, Ann. Mat. Pura Appl. 191 (2012), no. 4, 205–217.
M. Federson and J.G. Mesquita, Nonperiodic averaging principles for measure functional differential equations and functional dynamic equations on time scales involving impulses, J. Differential Equations 255 (2013), 3098–3126.
A.E. Hamza and K.M. Oraby, Stability of abstract dynamic equations on time scales, Adv. Differantial Equations 2012 (2012), 15 pp.
S. Hilger, Ein Maßkettenkalkül mit Anwendung auf Zentrumsmanningfaltigkeiten, Ph.D. thesis, Universität Würzburg, 1988.
B. Karpuz, On the uniqueness of the Laplace transform on time scales, Panamer. Math. J. 21 (2011), no. 2, 101–110.
S. Keller, Asymptotisches Verhalten Invarianter Faserbündel bei Diskretisierung und Mittelwertbildung in Rahmen der Analysis auf Zeitskalin, Ph.D. thesis, Universität Augsburg, 1999.
I. Klapper and H. Qian, Remarks on discrete and continuous large-scale models of DNA dynamics, Biophys. J. 74 (1998), 2504-2514.
Y. Li and C. Wang, Pseudo almost periodic functions and pseudo almost periodic solutions to dynamic equations on time scales, Adv. Difference Equ. 2012 (2012), 24 pp.
J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces I, Springer, New York, 1996.
C. Lizama and J.G. Mesquita, Almost automorphic solutions of dynamic equations on time scales, J. Funct. Anal. 265 (2013), 2267–2311.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci., vol. 44, Springer–Verlag, New York Inc. 1983.
C. Pötzsche, S. Siegmund and F. Wirth, A spectral characterization of exponential stability for linear time-invariant systems on time scales, Discrete Contin. Dyn. Syst. 9 (2003), no. 5, 1223–1241.
A.E. Taylor and D.C. Lay, Introduction to Functional Analysis, John Wiley Sons, 2nd edition, New York, 1980.
C.C. Tisdell and A. Zaidi, Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling, Nonlinear Anal. 68 (2008), no. 11, 3504–3524.
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