Geodesic
Disjoint hypercyclic and disjoint topologically mixing properties of degenerate fractional differential equations
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2018), pp. 36-53.

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The main purpose of this paper is to analyze the classes of disjoint hypercyclic and disjoint topologically mixing abstract degenerate (multi-term) fractional differential equations in Banach and Fréchet function spaces. We focus special attention on the analysis of abstract degenerate differential equations of first and second order, when we also consider disjoint chaoticity as a linear topological dynamical property. We provide several illustrative examples and applications of our abstract results.
Keywords: disjoint hypercyclicity, disjoint topologically mixing property, abstract degenerate differential equation, fractional differential equation
Mots-clés : Fréchet space. 
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M. Kostić; V. E. Fedorov. Disjoint hypercyclic and disjoint topologically mixing properties of degenerate fractional differential equations. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 7 (2018), pp. 36-53. http://geodesic.mathdoc.fr/item/IVM_2018_7_a2/

[1] Bés J., Peris A., “Disjointness in hypercyclicity”, J. Math. Anal. Appl., 336:1 (2007), 297–315 | DOI | MR | Zbl

[2] Bernal-González L., “Disjoint hypercyclic operators”, Studia Math., 182:2 (2007), 113–131 | DOI | MR | Zbl

[3] Bès J., Martin Ö., Peris A., Shkarin S., “Disjoint mixing operators”, J. Funct. Anal., 263:5 (2013), 1283–1322 | DOI | MR

[4] Martin Ö., Disjoint hypercyclic and supercyclic composition operators, PhD Thesis, Bowling Green State Univ., 2010 | MR | Zbl

[5] Salas H. N., “Dual disjoint hypercyclic operators”, J. Math. Anal. Appl., 374:1 (2011), 106–117 | DOI | MR | Zbl

[6] Kostić M., Abstract Volterra integro-differential equations, CRC Press, Boca Raton, FL, 2015 | MR | Zbl

[7] Kostić M., “Hypercyclic and topologically mixing properties of degenerate multi-term fractional differential equations”, Differ. Equat. Dyn. Syst., 24:4 (2016), 475–498 | DOI | MR | Zbl

[8] Kostić M., “$\mathcal{D}$-hypercyclic and $\mathcal{D}$-topologically mixing properties of degenerate multi-term fractional differential equations”, Azerbaijan J. Math., 5:2 (2015), 78–95 | MR | Zbl

[9] Arendt W., Batty C. J.K., Hieber M., Neubrander F., Vector-valued Laplace transforms and Cauchy problems, Birkhäuser/Springer Basel AG, Basel, 2001 | MR | Zbl

[10] Xiao T.-J., Liang J., The Cauchy problem for higher-order abstract differential equations, Springer-Verlag, Berlin, 1998 | MR | Zbl

[11] Wong R., Zhao Y.-Q., “Exponential asymptotics of the Mittag-Leffler function”, Constr. Approx., 18:3 (2002), 355–385 | DOI | MR | Zbl

[12] Bajlekova E. G., Fractional evolution equations in Banach spaces, PhD Thesis, Eindhoven Univ. Technology, Univ. Press Facilities, 2001 | MR | Zbl

[13] Krein S. G., Lineinye differentsialnye uravneniya v banakhovom prostranstve, Nauka, M., 1967

[14] Carroll R. W., Showalter R. E., Singular and degenerate Cauchy problems, Academic Press, New York, 1976 | MR

[15] Favini A., Yagi A., Degenerate differential equations in Banach spaces, Pure and Appl. Math., Chapman and Hall/CRC, New York, 1998

[16] Melnikova I. V., Filinkov A. I., Abstract Cauchy problems: three approaches, Chapman Hall/CRC Press, Boca Raton, 2001 | MR | Zbl

[17] Sviridyuk G. A., Fedorov V. E., Linear Sobolev type equations and degenerate semigroups of operators, Inverse and Ill-Posed Problems, 42, VSP, Utrecht–Boston, 2003 | MR

[18] Ji L., Weber A., “Dynamics of the heat semigroup on symmetric spaces”, Ergodic Theory Dynam. Systems, 30:2 (2010), 457–468 | DOI | MR | Zbl

[19] Astengo F., Di Blasio B., “Dynamics of the heat semigroup in Jacobi analysis”, J. Math. Anal. Appl., 391:1 (2012), 48–56 | DOI | MR | Zbl

[20] Conejero J. A., Mangino E. M., “Hypercyclic semigroups generated by Ornstein–Uhlenbeck operators”, Mediterr. J. Math., 7:1 (2010), 101–109 | DOI | MR | Zbl

[21] Desch W., Schappacher W., Webb G. F., “Hypercyclic and chaotic semigroups of linear operators”, Ergodic Theory Dynam. Systems, 17:4 (1997), 793–819 | DOI | MR | Zbl

[22] Metafune G., “$L^{p}$-spectrum of Ornstein–Uhlenbeck operators”, Ann. Scuola Norm. Sup. Pisa, Cl. Sci. (4), 30:1 (2001), 97–124 | MR | Zbl

[23] Pramanik M., Sarkar R. P., “Chaotic dynamics of the heat semigroup on Riemannian symmetric spaces”, J. Funct. Anal., 266:5 (2014), 2867–2909 | DOI | MR | Zbl

[24] Sarkar R. P., “Chaotic dynamics of the heat semigroup on the Damek–Ricci spaces”, Israel J. Math., 198:1 (2013), 487–508 | DOI | MR | Zbl

[25] Kostić M., “The existence of distributional chaos in abstract degenerate fractional differential equations”, J. Fract. Calc. Appl., 7:2 (2016), 153–174 | MR