Frequently hypercyclic translation semigroups

Frequent hypercyclicity for translation $C_0$-semigroups on weighted spaces of continuous functions is investigated. The results are achieved by establishing an analogy between frequent hypercyclicity for the translation semigroup and for weighted pseudo-shifts and by characterizing frequent hypercyclic weighted pseudo-shifts in spaces of vanishing sequences. Frequent hypercylic translation semigroups in weighted $L^p$-spaces are also characterized

Maximal regularity in lpl_p spaces for discrete time fractional shifted equations

In this paper, we are presenting a new method based on operator-valued Fourier multipliers to \- characterize the existence and uniqueness of $\ell_p$-solutions for discrete time fractional models in the form \Delta^{\alpha}u(n,x) = Au(n ,x) + \sum_{j=1}^k \beta_j u(n-\tau_j,x) +f(n,u(n,x)),\,\,\, n \in \mathbb{Z}, x \in \Omega \subset \mathbb{R}^N, \beta_j\in\mathbb{R}\hspace{0.1cm}\mbox{and}\hspace{0.1cm} \tau_j \in \mathbb{Z}, where $A$ is a closed linear operator defined on a Banach space $X$ and $\Delta^{\alpha}$ denotes the Gr\"unwald-Letnikov fractional derivative of order $\alpha>0.$ If $X$ is a $UMD$ space, we provide this characterization only in terms of the $R$-boundedness of the operator-valued symbol associated to the abstract model. To illustrate our results, we derive new qualitative properties of nonlinear difference equations with shiftings, including fractional versions of the logistic and Nagumo equations

Sharp values for the constants in the polynomial Bohnenblust-Hille inequality

In this paper we prove that the complex polynomial Bohnenblust-Hille constant for $2$-homogeneous polynomials in ${\mathbb C}^2$ is exactly $\sqrt[4]{\frac{3}{2}}$. We also give the exact value of the real polynomial Bohnenblust-Hille constant for $2$-homogeneous polynomials in ${\mathbb R}^2$. Finally, we provide lower estimates for the real polynomial Bohnenblust-Hille constant for polynomials in ${\mathbb R}^2$ of higher degrees.Comment: 16 page

Supercyclicity of weighted composition operators on spaces of continuous functions

[EN] Our study is focused on the dynamics of weighted composition operators defined on a locally convex space E similar to. (C( X), tp) with X being a topological Hausdorff space containing at least two different points and such that the evaluations {dx : x. X} are linearly independent in E similar to. We prove, when X is compact and E is a Banach space containing a nowhere vanishing function, that a weighted composition operator Cw,. is never weakly supercyclic on E. We also prove that if the symbol. lies in the unit ball of A(D), then every weighted composition operator can never be tp-supercyclic neither on C( D) nor on the disc algebra A(D). Finally, we obtain Ansari-Bourdon type results and conditions on the spectrum for arbitrary weakly supercyclic operators, and we provide necessary conditions for a composition operator to be weakly supercyclic on the space of holomorphic functions defined in non necessarily simply connected planar domains. As a consequence, we show that no composition operator can be weakly supercyclic neither on the space of holomorphic functions on the punctured disc nor in the punctured plane.The authors are very thankful to the referee for his/her careful reading of the manuscript and his/her valuable comments and observations. The first and the second author were supported by MEC, MTM2016-76647-P. The third author was supported by MEC, MTM2016-75963-P and GVA/2018/110.Beltrán-Meneu, MJ.; Jorda Mora, E.; Murillo Arcila, M. (2020). Supercyclicity of weighted composition operators on spaces of continuous functions. Collectanea mathematica. 71(3):493-509. https://doi.org/10.1007/s13348-019-00274-1493509713Albanese, A., Jornet, D.: A note on supercyclic operators in locally convex spaces. Mediterr. J. Math. 16, 107 (2019). https://doi.org/10.1007/s00009-019-1386-yAleman, A., Suciu, L.: On ergodic operator means in Banach spaces. Integr. Equ. Oper. Theory 85(2), 259–287 (2016)Ansari, S.: Hypercyclic and cyclic vectors. J. Funct. Anal. 128(2), 374–383 (1995)Ansari, S.I., Bourdon, P.S.: Some properties of cyclic operators. Acta Sci. Math. 63, 195–207 (1997)Bayart, F., Matheron, É.: Hyponormal operators, weighted shifts and weak forms of supercyclicity. Proc. Edinb. Math. Soc. 49, 1–15 (2006)Bayart, F., Matheron, É.: Dynamics of Linear Operators. Cambridge University Press, Cambridge (2009)Bermudo, S., Montes-Rodríguez, A., Shkarin, S.: Orbits of operators commuting with the Volterra operator. J. Math. Pures Appl. 89(2), 145–173 (2008)Bernal-Rodríguez, L., Montes-Rodríguez, A.: Universal functions for composition operators. Complex Var. Theory Appl. 27(1), 47–56 (1995)Bès, J.: Dynamics of weighted composition operators. Complex Anal. Oper. Theory 8, 159–176 (2014)Bonet, J., Peris, A.: Hypercyclic operators on non-normable Fréchet spaces. J. Funct. 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Cambridge University Press, New York (2013)Garrido, M.I., Jaramillo, J.A.: Variations on the Banach–Stone theorem. In: IV Course on Banach spaces and Operators (Laredo, 2001), Extracta Math. 17, 351–383 (2002)Gadgil, S.: Dynamics on the circle-interval dynamics and rotation number. Reson. J. Sci. Educ. 8(11), 25–36 (2003)Grosse-Erdmann, K.G., Peris, A.: Linear Chaos. Springer, Berlin (2011)Gunatillake, G.: Invertible weighted composition operators. J. Funct. Anal. 261, 831–860 (2011)Herrero, D.A.: Limits of hypercyclic and supercyclic operators. J. Funct. Anal. 99(1), 179–190 (1991)Hilden, H.M., Wallen, L.J.: Some cyclic and non-cyclic vectors of certain operators. Indiana Univ. Math. J. 23, 557–565 (1974)Kalmes, T.: Dynamics of weighted composition operators on function spaces defined by local properties. Studia Math. 249(3), 259–301 (2019)Kamali, Z., Hedayatian, K., Khani Robati, B.: Non-weakly supercyclic weighted composition operators. Abstr. Appl. Anal. Art. (2010) ID 143808Köthe, G.: Topological Vector Spaces II. Springer, New York (1979)Liang, Y.X., Zhou, Z.H.: Supercyclic tuples of the adjoint weighted composition operators on Hilbert spaces. Bull. Iran. Math. Soc. 41(1), 121–139 (2015)Milnor, J.: Dynamics in One Complex Variable, 3rd edn. Princeton University Press, Princeton (2006)Montes-Rodríguez, A., Rodríguez-Martínez, A., Shkarin, S.: Cyclic behaviour of Volterra composition operators. Proc. Lond. Math. Soc. 103(3), 535–562 (2011)Montes-Rodríguez, A., Shkarin, S.: Non-weakly supercyclic operators. J. Oper. Theory 58(1), 39–62 (2007)Moradi, A., Khani Robati, B., Hedayatian, K.: Non-weakly supercyclic classes of weighted composition operators on Banach spaces of analytic functions. Bull. Belg. Math. Soc. Simon Stevin 24(2), 227–241 (2017)Peris, A.: Multi-hypercyclic operators are hypercyclic. Math. Z. 236(4), 779–786 (2001)Sanders, R.: Weakly supercyclic operators. J. Math. Anal. 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Strong mixing measures for linear operators and frequent hypercyclicity

We construct strongly mixing invariant measures with full support for operators on F-spaces which satisfy the Frequent Hypercyclicity Criterion. For unilateral backward shifts on sequence spaces, a slight modification shows that one can even obtain exact invariant measures. (c) 2012 Elsevier Inc. All rights reserved.This work was supported in part by MEC and FEDER, Project MTM2010-14909, and by CV, Project PROMETEO/2008/101. The first author was also supported by a grant from the FPU Program of MEC. We thank the referee whose detailed report led to an improvement in the presentation of this work.Murillo Arcila, M.; Peris Manguillot, A. (2013). Strong mixing measures for linear operators and frequent hypercyclicity. Journal of Mathematical Analysis and Applications. 398(2):462-465. https://doi.org/10.1016/j.jmaa.2012.08.050S462465398

Chaotic asymptotic behaviour of the solutions of the Lighthill Whitham Richards equation

[EN] The phenomenon of chaos has been exhibited in mathematical nonlinear models that describe traffic flows, see, for instance (Li and Gao in Modern Phys Lett B 18(26-27):1395-1402, 2004; Li in Phys. D Nonlinear Phenom 207(1-2):41-51, 2005). At microscopic level, Devaney chaos and distributional chaos have been exhibited for some car-following models, such as the quick-thinking-driver model and the forward and backward control model (Barrachina et al. in 2015; Conejero et al. in Semigroup Forum, 2015). We present here the existence of chaos for the macroscopic model given by the Lighthill Whitham Richards equation.The authors are supported by MEC Project MTM2013-47093-P. The second and third authors are supported by GVA, Project PROMETEOII/2013/013Conejero, JA.; Martínez Jiménez, F.; Peris Manguillot, A.; Ródenas Escribá, FDA. (2016). Chaotic asymptotic behaviour of the solutions of the Lighthill Whitham Richards equation. Nonlinear Dynamics. 84(1):127-133. https://doi.org/10.1007/s11071-015-2245-4S127133841Albanese, A.A., Barrachina, X., Mangino, E.M., Peris, A.: Distributional chaos for strongly continuous semigroups of operators. Commun. Pure Appl. Anal. 12(5), 2069–2082 (2013)Aroza, J., Peris, A.: Chaotic behaviour of birth-and-death models with proliferation. J. Differ. Equ. Appl. 18(4), 647–655 (2012)Banasiak, J., Lachowicz, M.: Chaos for a class of linear kinetic models. C. R. Acad. Sci. Paris Sér. II 329, 439–444 (2001)Banasiak, J., Lachowicz, M.: Topological chaos for birth-and-death-type models with proliferation. Math. Models Methods Appl. Sci. 12(6), 755–775 (2002)Banasiak, J., Moszyński, M.: A generalization of Desch–Schappacher–Webb criteria for chaos. Discrete Contin. Dyn. Syst. 12(5), 959–972 (2005)Banasiak, J., Moszyński, M.: Dynamics of birth-and-death processes with proliferation—stability and chaos. Discrete Contin. Dyn. Syst. 29(1), 67–79 (2011)Barrachina, X., Conejero, J.A.: Devaney chaos and distributional chaos in the solution of certain partial differential equations. Abstr. Appl. Anal. Art. ID 457019, 11 (2012)Barrachina, X., Conejero, J.A., Murillo-Arcila, M., Seoane-Sepúlveda, J.B.: Distributional chaos for the forward and backward control traffic model (2015, preprint)Bayart, F., Matheron, É.: Dynamics of Linear Operators, Cambridge Tracts in Mathematics, vol. 179. Cambridge University Press, Cambridge (2009)Bayart, F., Matheron, É.: Mixing operators and small subsets of the circle. J Reine Angew. Math. (2015, to appear)Bermúdez, T., Bonilla, A., Conejero, J.A., Peris, A.: Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces. Stud. Math. 170(1), 57–75 (2005)Bermúdez, T., Bonilla, A., Martínez-Giménez, F., Peris, A.: Li-Yorke and distributionally chaotic operators. J. Math. Anal. Appl. 373(1), 83–93 (2011)Bernardes Jr, N.C., Bonilla, A., Müller, V., Peris, A.: Distributional chaos for linear operators. J. Funct. Anal. 265(9), 2143–2163 (2013)Brackstone, M., McDonald, M.: Car-following: a historical review. Transp. Res. Part F Traffic Psychol. Behav. 2(4), 181–196 (1999)Conejero, J.A., Lizama, C., Rodenas, F.: Chaotic behaviour of the solutions of the Moore–Gibson–Thompson equation. Appl. Math. Inf. Sci. 9(5), 1–6 (2015)Conejero, J.A., Mangino, E.M.: Hypercyclic semigroups generated by Ornstein-Uhlenbeck operators. Mediterr. J. Math. 7(1), 101–109 (2010)Conejero, J.A., Müller, V., Peris, A.: Hypercyclic behaviour of operators in a hypercyclic $$C_0$$ C 0 -semigroup. J. Funct. Anal. 244, 342–348 (2007)Conejero, J.A., Murillo-Arcila, M., Seoane-Sepúlveda, J.B.: Linear chaos for the quick-thinking-driver model. Semigroup Forum (2015). doi: 10.1007/s00233-015-9704-6Conejero, J.A., Peris, A., Trujillo, M.: Chaotic asymptotic behavior of the hyperbolic heat transfer equation solutions. Int. J. Bifur. Chaos Appl. Sci. Eng. 20(9), 2943–2947 (2010)Conejero, J.A., Rodenas, F., Trujillo, M.: Chaos for the hyperbolic bioheat equation. Discrete Contin. Dyn. Syst. 35(2), 653–668 (2015)Desch, W., Schappacher, W., Webb, G.F.: Hypercyclic and chaotic semigroups of linear operators. Ergod. Theory Dyn. Syst. 17(4), 793–819 (1997)Engel, K.-J., Nagel, R.: One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194. Springer, New York (2000). With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. SchnaubeltGrosse-Erdmann, K.-G., Peris Manguillot, A.: Linear Chaos. Universitext. Springer, London (2011)Herzog, G.: On a universality of the heat equation. Math. Nachr. 188, 169–171 (1997)Li, K., Gao, Z.: Nonlinear dynamics analysis of traffic time series. Modern Phys. Lett. B 18(26–27), 1395–1402 (2004)Li, T.: Nonlinear dynamics of traffic jams. Phys. D Nonlinear Phenom. 207(1–2), 41–51 (2005)Lustri, C.: Continuum Modelling of Traffic Flow. Special Topic Report. Oxford University, Oxford (2010)Lighthill, M.J., Whitham, G.B.: On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. R. Soc. Lond. Ser. A. 229, 317–345 (1955)Maerivoet, S., De Moor, B.: Cellular automata models of road traffic. Phys. Rep. 419(1), 1–64 (2005)Mangino, E.M., Peris, A.: Frequently hypercyclic semigroups. Stud. Math. 202(3), 227–242 (2011)Murillo-Arcila, M., Peris, A.: Strong mixing measures for linear operators and frequent hypercyclicity. J. Math. Anal. Appl. 398, 462–465 (2013)Murillo-Arcila, M., Peris, A.: Strong mixing measures for $$C_0$$ C 0 -semigroups. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. 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Linear chaos for the Quick-Thinking-Driver model

The final publication is available at Springer via http://dx.doi.org/10.1007/s00233-015-9704-6In recent years, the topic of car-following has experimented an increased importance in traffic engineering and safety research. This has become a very interesting topic because of the development of driverless cars (Google driverless cars, http://en.wikipedia.org/wiki/Google_driverless_car).Driving models which describe the interaction between adjacent vehicles in the same lane have a big interest in simulation modeling, such as the Quick-Thinking-Driver model. A non-linear version of it can be given using the logistic map, and then chaos appears. We show that an infinite-dimensional version of the linear model presents a chaotic behaviour using the same approach as for studying chaos of death models of cell growth.The authors were supported by a grant from the FPU program of MEC and MEC Project MTM2013-47093-P.Conejero, JA.; Murillo Arcila, M.; Seoane-Sepúlveda, JB. (2016). Linear chaos for the Quick-Thinking-Driver model. Semigroup Forum. 92(2):486-493. https://doi.org/10.1007/s00233-015-9704-6S486493922Aroza, J., Peris, A.: Chaotic behaviour of birth-and-death models with proliferation. J. Differ. Equ. Appl. 18(4), 647–655 (2012)Banasiak, J., Lachowicz, M.: Chaos for a class of linear kinetic models. C. R. Acad. Sci. Paris Série II 329, 439–444 (2001)Banasiak, J., Lachowicz, M.: Topological chaos for birth-and-death-type models with proliferation. Math. Models Methods Appl. Sci. 12(6), 755–775 (2002)Banasiak, J., Lachowicz, M., Moszyński, M.: Topological chaos: when topology meets medicine. Appl. Math. Lett. 16(3), 303–308 (2003)Banasiak, J., Moszyński, M.: A generalization of Desch–Schappacher–Webb criteria for chaos. Discret. Contin. Dyn. Syst. 12(5), 959–972 (2005)Banasiak, J., Moszyński, M.: Dynamics of birth-and-death processes with proliferation–stability and chaos. Discret. Contin. Dyn. 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Well-posedness for degenerate third order equations with delay and applications to inverse problems

[EN] In this paper, we study well-posedness for the following third-order in time equation with delay <disp-formula idoperators defined on a Banach space X with domains D(A) and D(B) such that t)is the state function taking values in X and u(t): (-, 0] X defined as u(t)() = u(t+) for < 0 belongs to an appropriate phase space where F and G are bounded linear operators. Using operator-valued Fourier multiplier techniques we provide optimal conditions for well-posedness of equation (0.1) in periodic Lebesgue-Bochner spaces Lp(T,X), periodic Besov spaces Bp,qs(T,X) and periodic Triebel-Lizorkin spaces Fp,qs(T,X). A novel application to an inverse problem is given.The first, second and third authors have been supported by MEC, grant MTM2016-75963-P. The second author has been supported by AICO/2016/30. The fourth author has been supported by MEC, grant MTM2015-65825-P.Conejero, JA.; Lizama, C.; Murillo-Arcila, M.; Seoane Sepúlveda, JB. (2019). 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Distributionally chaotic families of operators on Fréchet spaces

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Communications on Pure and Applied Analysis (CPAA) following peer review. The definitive publisher-authenticated version Conejero, J. A., Kostić, M., Miana, P. J., & Murillo-Arcila, M. (2016). Distributionally chaotic families of operators on Fréchet spaces.Communications on Pure and Applied Analysis, 2016, vol. 15, no 5, p. 1915-1939, is available online at: http://dx.doi.org/10.3934/cpaa.2016022The existence of distributional chaos and distributional irregular vectors has been recently considered in the study of linear dynamics of operators and C-0-semigroups. In this paper we extend some previous results on both notions to sequences of operators, C-0-semigroups, C-regularized semigroups, and alpha-timesintegrated semigroups on Frechet spaces. We also add a study of rescaled distributionally chaotic C-0-semigroups. Some examples are provided to illustrate all these results.The first and fourth authors are supported in part by MEC Project MTM2010-14909, MTM2013-47093-P, and Programa de Investigacion y Desarrollo de la UPV, Ref. SP20120700. The second author is partially supported by grant 174024 of Ministry of Science and Technological Development, Republic of Serbia. The third author has been partially supported by Project MTM2013-42105-P, DGI-FEDER, of the MCYTS; Project E-64, D.G. Aragon, and Project UZCUD2014-CIE-09, Universidad de Zaragoza. The fourth author is supported by a grant of the FPU Program of Ministry of education of Spain.Conejero, JA.; Kostic, M.; Miana Sanz, PJ.; Murillo Arcila, M. (2016). Distributionally chaotic families of operators on Fréchet spaces. Communications on Pure and Applied Analysis. 15(5):1915-1939. https://doi.org/10.3934/cpaa.2016022S1915193915

Frequently Hypercyclic Translation Semigroups

Frequent hypercyclicity for translation $C_0$-semigroups on weighted spaces of continuous functions is studied. The results are achieved by establishing an analogy between frequent hypercyclicity for translation semigroups and for weighted pseudo-shifts and by characterizing frequently hypercyclic weighted pseudo-shifts on spaces of vanishing sequences. Frequently hypercyclic translation semigroups on weighted $L^p$-spaces are also characterized