# American Institute of Mathematical Sciences

September  2019, 39(9): 5491-5520. doi: 10.3934/dcds.2019224

## Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics

* Corresponding author: Rafael Obaya

Received  January 2019 Published  May 2019

Fund Project: Partly supported by MINECO/FEDER under project MTM2015-66330-P and EU Marie-Skłodowska-Curie ITN Critical Transitions in Complex Systems (H2020-MSCA-ITN-2014 643073 CRITICS)

We study some already introduced and some new strong and weak topologies of integral type to provide continuous dependence on continuous initial data for the solutions of non-autonomous Carathéodory delay differential equations. As a consequence, we obtain new families of continuous skew-product semiflows generated by delay differential equations whose vector fields belong to such metric topological vector spaces of Lipschitz Carathéodory functions. Sufficient conditions for the equivalence of all or some of the considered strong or weak topologies are also given. Finally, we also provide results of continuous dependence of the solutions as well as of continuity of the skew-product semiflows generated by Carathéodory delay differential equations when the considered phase space is a Sobolev space.

Citation: Iacopo P. Longo, Sylvia Novo, Rafael Obaya. Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5491-5520. doi: 10.3934/dcds.2019224
##### References:
 [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, Heidelberg, 1998. doi: 10.1007/978-3-662-12878-7. Google Scholar [2] Z. Artstein, Topological dynamics of an ordinary differential equation, J. Differential Equations, 23 (1977), 216-223. doi: 10.1016/0022-0396(77)90127-9. Google Scholar [3] Z. Artstein, Topological dynamics of ordinary differential equations and Kurzweil equations, J. Differential Equations, 23 (1977), 224-243. doi: 10.1016/0022-0396(77)90128-0. Google Scholar [4] Z. Artstein, The limiting equations of nonautonomous ordinary differential equations, J. Differential Equations, 25 (1977), 184-202. doi: 10.1016/0022-0396(77)90199-1. Google Scholar [5] B. Aulbach and T. Wanner, Integral manifolds for Carathéodory type differential equations in Banach spaces, Six Lectures on Dynamical Systems (B. Aulbach & F. Colonius eds), World Scientific, Singapore, 1996, 45–119. doi: 10.1142/9789812812865_0002. Google Scholar [6] T. Caraballo and X. Han, Applied Non-Autonomous And Random Dynamical Systems. Applied Dynamical Systems, SpringerBriefs in Mathematics. Springer, Cham, 2016. doi: 10.1007/978-3-319-49247-6. Google Scholar [7] A. Carvalho, J. A. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-au-tono-mous Dynamical Systems, Springer-Verlag New York, 2013. doi: 10.1007/978-1-4614-4581-4. Google Scholar [8] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. Google Scholar [9] N. Dunford and J. T. Schwartz, Linear Operators: Part Ⅰ General Theory, Wiley-Interscience, New York, 1988. Google Scholar [10] J. K. Hale and M. A. Cruz, Existence, uniqueness and continuous dependence for hereditary systems, Ann. Mat. Pura Appl., (4) 85 (1970), 63–81. doi: 10.1007/BF02413530. Google Scholar [11] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, Berlin, Heidelberg, New York, 1993. doi: 10.1007/978-1-4612-4342-7. Google Scholar [12] A. J. Heunis, Continuous dependence of the solutions of an ordinary differential equation, J. Differential Equations, 54 (1984), 121-138. doi: 10.1016/0022-0396(84)90155-4. Google Scholar [13] R. Johnson, R. Obaya, S. Novo, C. Nuñez and R. Fabbri, Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory And Control, Developments in Mathematics, 36, $\rm{Springer, Switzerland, 2016.}$ doi: 10.1007/978-3-319-29025-6. Google Scholar [14] O. Kallenberg, Random Measures, Third Edition, $\rm{Akademie-Verlag, Berlin, 1983.}$ Google Scholar [15] I. P. Longo, S. Novo and R. Obaya, Topologies of $L^p_loc$ type for Carathéodory functions with applications in non-autonomous differential equations, J. Differential Equations, 263 (2017), 7187-7220. doi: 10.1016/j.jde.2017.08.006. Google Scholar [16] I. P. Longo, S. Novo and R. Obaya, Weak topologies for Carathéodory differential equations: Continuous dependence, exponential dichotomy and attractors, J. Dynam. Differential Equations, (2018). doi: 10.1007/s10884-018-9710-y. Google Scholar [17] R. K. Miller and G. Sell, Volterra Integral Equations and Topological Dynamics, Mem. Amer. Math. Soc., 102, Amer. Math. Soc., Providence, 1970. doi: 10.1090/memo/0102. Google Scholar [18] R. K. Miller and G. Sell, Existence, uniqueness and continuity of solutions of integral equations, Addendum: Ibid., 87 (1970), 281-286. doi: 10.1007/BF02411981. Google Scholar [19] L. W. Neustadt, On the solutions of certain integral-like operator equations. Existence, uniqueness and dependence theorems, Arch. Rational Mech. Anal., 38 (1970), 131-160. doi: 10.1007/BF00249976. Google Scholar [20] Z. Opial, Continuous parameter dependence in linear systems of differential equations, J. Differential Equations, 3 (1967), 571-579. doi: 10.1016/0022-0396(67)90017-4. Google Scholar [21] C. Pötzsche and M. Rasmussen, Computation of integral manifolds for Carathéodory differential equations, J. Numer. Anal., 30 (2010), 401-430. doi: 10.1093/imanum/drn059. Google Scholar [22] R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8. Google Scholar [23] G. Sell, Compact sets of nonlinear operators, Funkcial. Ekvac., 11 (1968), 131-138. Google Scholar [24] G. Sell, Topological Dynamics and Ordinary Differential Equations, $\rm{Van Nostrand-Reinhold, London, 1971. }$ Google Scholar [25] W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc., 136 (1998), x+93 pp. doi: 10.1090/memo/0647. Google Scholar

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##### References:
 [1] L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, Heidelberg, 1998. doi: 10.1007/978-3-662-12878-7. Google Scholar [2] Z. Artstein, Topological dynamics of an ordinary differential equation, J. Differential Equations, 23 (1977), 216-223. doi: 10.1016/0022-0396(77)90127-9. Google Scholar [3] Z. Artstein, Topological dynamics of ordinary differential equations and Kurzweil equations, J. Differential Equations, 23 (1977), 224-243. doi: 10.1016/0022-0396(77)90128-0. Google Scholar [4] Z. Artstein, The limiting equations of nonautonomous ordinary differential equations, J. Differential Equations, 25 (1977), 184-202. doi: 10.1016/0022-0396(77)90199-1. Google Scholar [5] B. Aulbach and T. Wanner, Integral manifolds for Carathéodory type differential equations in Banach spaces, Six Lectures on Dynamical Systems (B. Aulbach & F. Colonius eds), World Scientific, Singapore, 1996, 45–119. doi: 10.1142/9789812812865_0002. Google Scholar [6] T. Caraballo and X. Han, Applied Non-Autonomous And Random Dynamical Systems. Applied Dynamical Systems, SpringerBriefs in Mathematics. Springer, Cham, 2016. doi: 10.1007/978-3-319-49247-6. Google Scholar [7] A. Carvalho, J. A. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-au-tono-mous Dynamical Systems, Springer-Verlag New York, 2013. doi: 10.1007/978-1-4614-4581-4. Google Scholar [8] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. Google Scholar [9] N. Dunford and J. T. Schwartz, Linear Operators: Part Ⅰ General Theory, Wiley-Interscience, New York, 1988. Google Scholar [10] J. K. Hale and M. A. Cruz, Existence, uniqueness and continuous dependence for hereditary systems, Ann. Mat. Pura Appl., (4) 85 (1970), 63–81. doi: 10.1007/BF02413530. Google Scholar [11] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, 99, Springer-Verlag, Berlin, Heidelberg, New York, 1993. doi: 10.1007/978-1-4612-4342-7. Google Scholar [12] A. J. Heunis, Continuous dependence of the solutions of an ordinary differential equation, J. Differential Equations, 54 (1984), 121-138. doi: 10.1016/0022-0396(84)90155-4. Google Scholar [13] R. Johnson, R. Obaya, S. Novo, C. Nuñez and R. Fabbri, Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory And Control, Developments in Mathematics, 36, $\rm{Springer, Switzerland, 2016.}$ doi: 10.1007/978-3-319-29025-6. Google Scholar [14] O. Kallenberg, Random Measures, Third Edition, $\rm{Akademie-Verlag, Berlin, 1983.}$ Google Scholar [15] I. P. Longo, S. Novo and R. Obaya, Topologies of $L^p_loc$ type for Carathéodory functions with applications in non-autonomous differential equations, J. Differential Equations, 263 (2017), 7187-7220. doi: 10.1016/j.jde.2017.08.006. Google Scholar [16] I. P. Longo, S. Novo and R. Obaya, Weak topologies for Carathéodory differential equations: Continuous dependence, exponential dichotomy and attractors, J. Dynam. Differential Equations, (2018). doi: 10.1007/s10884-018-9710-y. Google Scholar [17] R. K. Miller and G. Sell, Volterra Integral Equations and Topological Dynamics, Mem. Amer. Math. Soc., 102, Amer. Math. Soc., Providence, 1970. doi: 10.1090/memo/0102. Google Scholar [18] R. K. Miller and G. Sell, Existence, uniqueness and continuity of solutions of integral equations, Addendum: Ibid., 87 (1970), 281-286. doi: 10.1007/BF02411981. Google Scholar [19] L. W. Neustadt, On the solutions of certain integral-like operator equations. Existence, uniqueness and dependence theorems, Arch. Rational Mech. Anal., 38 (1970), 131-160. doi: 10.1007/BF00249976. Google Scholar [20] Z. Opial, Continuous parameter dependence in linear systems of differential equations, J. Differential Equations, 3 (1967), 571-579. doi: 10.1016/0022-0396(67)90017-4. Google Scholar [21] C. Pötzsche and M. Rasmussen, Computation of integral manifolds for Carathéodory differential equations, J. Numer. Anal., 30 (2010), 401-430. doi: 10.1093/imanum/drn059. Google Scholar [22] R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358. doi: 10.1016/0022-0396(78)90057-8. Google Scholar [23] G. Sell, Compact sets of nonlinear operators, Funkcial. Ekvac., 11 (1968), 131-138. Google Scholar [24] G. Sell, Topological Dynamics and Ordinary Differential Equations, $\rm{Van Nostrand-Reinhold, London, 1971. }$ Google Scholar [25] W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc., 136 (1998), x+93 pp. doi: 10.1090/memo/0647. Google Scholar
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