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

Departamento de Matemática Aplicada, E.I. Industriales, Universidad de Valladolid, Paseo del Cauce 59, 47011 Valladolid, Spain

* 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 and Continuous Dynamical Systems, 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.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.

[9]

N. Dunford and J. T. Schwartz, Linear Operators: Part Ⅰ General Theory, Wiley-Interscience, New York, 1988.

[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.

[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.

[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.

[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.

[14]

O. Kallenberg, Random Measures, Third Edition, $ \rm{Akademie-Verlag, Berlin, 1983.} $

[15]

I. P. LongoS. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

G. Sell, Compact sets of nonlinear operators, Funkcial. Ekvac., 11 (1968), 131-138. 

[24]

G. Sell, Topological Dynamics and Ordinary Differential Equations, $ \rm{Van Nostrand-Reinhold, London, 1971. }$

[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.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, Heidelberg, 1998. doi: 10.1007/978-3-662-12878-7.

[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.

[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.

[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.

[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.

[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.

[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.

[8]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.

[9]

N. Dunford and J. T. Schwartz, Linear Operators: Part Ⅰ General Theory, Wiley-Interscience, New York, 1988.

[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.

[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.

[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.

[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.

[14]

O. Kallenberg, Random Measures, Third Edition, $ \rm{Akademie-Verlag, Berlin, 1983.} $

[15]

I. P. LongoS. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

G. Sell, Compact sets of nonlinear operators, Funkcial. Ekvac., 11 (1968), 131-138. 

[24]

G. Sell, Topological Dynamics and Ordinary Differential Equations, $ \rm{Van Nostrand-Reinhold, London, 1971. }$

[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.

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