American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Multiple critical points for a class of periodic lower semicontinuous functionals
  • DCDS Home
  • This Issue
  • Next Article
    Multibump solutions of nonlinear Schrödinger equations with steep potential well and indefinite potential
January  2013, 33(1): 27-46. doi: 10.3934/dcds.2013.33.27

On general properties of retarded functional differential equations on manifolds

Pierluigi Benevieri 1, , Alessandro Calamai 2, , Massimo Furi 1, and  Maria Patrizia Pera 1,

1. 

Dipartimento di Sistemi e Informatica, Università degli Studi di Firenze, Via S. Marta 3, I-50139 Firenze, Italy, Italy, Italy
2. 

Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, I-60131 Ancona, Italy

Received  August 2011 Revised  February 2012 Published  September 2012

We investigate general properties, such as existence and uniqueness, continuous dependence on data and continuation, of solutions to retarded functional differential equations with infinite delay on a differentiable manifold.
Keywords: RFDEs with infinite delay, Retarded functional differential equations (RFDEs), properties of solutions., initial value problems, RFDEs on manifolds.
Mathematics Subject Classification: 34K05, 34C4.
Citation: Pierluigi Benevieri, Alessandro Calamai, Massimo Furi, Maria Patrizia Pera. On general properties of retarded functional differential equations on manifolds. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 27-46. doi: 10.3934/dcds.2013.33.27
References:
[1]

P. Benevieri, A. Calamai, M. Furi and M. P. Pera, Retarded functional differential equations on manifolds and applications to motion problems for forced constrained systems, Adv. Nonlinear Stud., 9 (2009), 199-214.  Google Scholar

[2]

P. Benevieri, A. Calamai, M. Furi and M. P. Pera, A continuation result for forced oscillations of constrained motion problems with infinite delay,, to appear in Adv. Nonlinear Stud., ().   Google Scholar

[3]

P. Benevieri, A. Calamai, M. Furi and M. P. Pera, On the existence of forced oscillations for the spherical pendulum acted on by a retarded periodic force, J. Dyn. Diff. Equat., 23 (2011), 541-549. doi: 10.1007/s10884-010-9201-2.  Google Scholar

[4]

G. Bouligand, "Introduction à la Géométrie Infinitésimale Directe,'' Gauthier-Villard, Paris, 1932. Google Scholar

[5]

N. Dunford and J. T. Schwartz, "Linear Operators,'' Wiley & Sons, Inc., New York, 1957. Google Scholar

[6]

R. Gaines and J. Mawhin, "Coincidence Degree and Nonlinear Differential Equations,'' Lecture Notes in Math., Springer Verlag, Berlin, 568, 1977.  Google Scholar

[7]

V. Guillemin and A. Pollack, "Differential Topology,'' Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1974.  Google Scholar

[8]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkc. Ekvac., 21 (1978), 11-41.  Google Scholar

[9]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,'' Springer Verlag, New York, 1993.  Google Scholar

[10]

Y. Hino, S. Murakami and T. Naito, "Functional-Differential Equations with Infinite Delay,'' Lecture Notes in Math., Springer Verlag, Berlin, 1473, 1991.  Google Scholar

[11]

M. W. Hirsch, "Differential Topology,'' Graduate Texts in Math., Springer Verlag, Berlin, 33, 1976.  Google Scholar

[12]

J. Mallet-Paret, R. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional-differential equations with multiple state-dependent time lags, Topol. Methods Nonlinear Anal., 3 (1994), 101-162.  Google Scholar

[13]

J. M. Milnor, "Topology from the Differentiable Viewpoint,'' Univ. Press of Virginia, Charlottesville, 1965.  Google Scholar

[14]

J. R. Munkres, "Elementary Differential Topology,'' Princeton University Press, Princeton, New Jersey, 1966.  Google Scholar

[15]

R. D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl., 101 (1974), 263-306. doi: 10.1007/BF02417109.  Google Scholar

[16]

R. D. Nussbaum, The fixed point index and fixed point theorems, in "Topological methods for ordinary differential equations'' (Montecatini Terme, 1991), Lecture Notes in Math., Springer Verlag, Berlin, 1537 (1993), 143-205.  Google Scholar

[17]

W. M. Oliva, Functional differential equations on compact manifolds and an approximation theorem, J. Differential Equations, 5 (1969), 483-496.  Google Scholar

[18]

W. M. Oliva, Functional differential equations-generic theory, in "Dynamical Systems'' (Proc. Internat. Sympos., Brown Univ., Providence, R. I., 1974), Academic Press, New York, (1976), 195-209.  Google Scholar

[19]

W. M. Oliva and C. Rocha, Reducible volterra and levin-nohel retarded equations with infinite delay, J. Dyn. Diff. Equat., 22 (2010), 509-532. doi: 10.1007/s10884-010-9177-y.  Google Scholar

show all references

References:
[1]

P. Benevieri, A. Calamai, M. Furi and M. P. Pera, Retarded functional differential equations on manifolds and applications to motion problems for forced constrained systems, Adv. Nonlinear Stud., 9 (2009), 199-214.  Google Scholar

[2]

P. Benevieri, A. Calamai, M. Furi and M. P. Pera, A continuation result for forced oscillations of constrained motion problems with infinite delay,, to appear in Adv. Nonlinear Stud., ().   Google Scholar

[3]

P. Benevieri, A. Calamai, M. Furi and M. P. Pera, On the existence of forced oscillations for the spherical pendulum acted on by a retarded periodic force, J. Dyn. Diff. Equat., 23 (2011), 541-549. doi: 10.1007/s10884-010-9201-2.  Google Scholar

[4]

G. Bouligand, "Introduction à la Géométrie Infinitésimale Directe,'' Gauthier-Villard, Paris, 1932. Google Scholar

[5]

N. Dunford and J. T. Schwartz, "Linear Operators,'' Wiley & Sons, Inc., New York, 1957. Google Scholar

[6]

R. Gaines and J. Mawhin, "Coincidence Degree and Nonlinear Differential Equations,'' Lecture Notes in Math., Springer Verlag, Berlin, 568, 1977.  Google Scholar

[7]

V. Guillemin and A. Pollack, "Differential Topology,'' Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1974.  Google Scholar

[8]

J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkc. Ekvac., 21 (1978), 11-41.  Google Scholar

[9]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,'' Springer Verlag, New York, 1993.  Google Scholar

[10]

Y. Hino, S. Murakami and T. Naito, "Functional-Differential Equations with Infinite Delay,'' Lecture Notes in Math., Springer Verlag, Berlin, 1473, 1991.  Google Scholar

[11]

M. W. Hirsch, "Differential Topology,'' Graduate Texts in Math., Springer Verlag, Berlin, 33, 1976.  Google Scholar

[12]

J. Mallet-Paret, R. D. Nussbaum and P. Paraskevopoulos, Periodic solutions for functional-differential equations with multiple state-dependent time lags, Topol. Methods Nonlinear Anal., 3 (1994), 101-162.  Google Scholar

[13]

J. M. Milnor, "Topology from the Differentiable Viewpoint,'' Univ. Press of Virginia, Charlottesville, 1965.  Google Scholar

[14]

J. R. Munkres, "Elementary Differential Topology,'' Princeton University Press, Princeton, New Jersey, 1966.  Google Scholar

[15]

R. D. Nussbaum, Periodic solutions of some nonlinear autonomous functional differential equations, Ann. Mat. Pura Appl., 101 (1974), 263-306. doi: 10.1007/BF02417109.  Google Scholar

[16]

R. D. Nussbaum, The fixed point index and fixed point theorems, in "Topological methods for ordinary differential equations'' (Montecatini Terme, 1991), Lecture Notes in Math., Springer Verlag, Berlin, 1537 (1993), 143-205.  Google Scholar

[17]

W. M. Oliva, Functional differential equations on compact manifolds and an approximation theorem, J. Differential Equations, 5 (1969), 483-496.  Google Scholar

[18]

W. M. Oliva, Functional differential equations-generic theory, in "Dynamical Systems'' (Proc. Internat. Sympos., Brown Univ., Providence, R. I., 1974), Academic Press, New York, (1976), 195-209.  Google Scholar

[19]

W. M. Oliva and C. Rocha, Reducible volterra and levin-nohel retarded equations with infinite delay, J. Dyn. Diff. Equat., 22 (2010), 509-532. doi: 10.1007/s10884-010-9177-y.  Google Scholar

[1]

Hermen Jan Hupkes, Emmanuelle Augeraud-Véron. Well-posedness of initial value problems for functional differential and algebraic equations of mixed type. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 737-765. doi: 10.3934/dcds.2011.30.737
[2]

A. V. Rezounenko. Inertial manifolds with delay for retarded semilinear parabolic equations. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 829-840. doi: 10.3934/dcds.2000.6.829
[3]

Ismael Maroto, Carmen NÚÑez, Rafael Obaya. Dynamical properties of nonautonomous functional differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3939-3961. doi: 10.3934/dcds.2017167
[4]

Burcu Gürbüz. A computational approximation for the solution of retarded functional differential equations and their applications to science and engineering. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021069
[5]

Pietro-Luciano Buono, V.G. LeBlanc. Equivariant versal unfoldings for linear retarded functional differential equations. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 283-302. doi: 10.3934/dcds.2005.12.283
[6]

Marat Akhmet. Quasilinear retarded differential equations with functional dependence on piecewise constant argument. Communications on Pure & Applied Analysis, 2014, 13 (2) : 929-947. doi: 10.3934/cpaa.2014.13.929
[7]

Tomás Caraballo, Francisco Morillas, José Valero. On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems. Discrete & Continuous Dynamical Systems, 2014, 34 (1) : 51-77. doi: 10.3934/dcds.2014.34.51
[8]

Fuke Wu, Shigeng Hu. The LaSalle-type theorem for neutral stochastic functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 1065-1094. doi: 10.3934/dcds.2012.32.1065
[9]

Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115
[10]

Ya Wang, Fuke Wu, Xuerong Mao, Enwen Zhu. Advances in the LaSalle-type theorems for stochastic functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 287-300. doi: 10.3934/dcdsb.2019182
[11]

Pham Huu Anh Ngoc. New criteria for exponential stability in mean square of stochastic functional differential equations with infinite delay. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021040
[12]

Hernán R. Henríquez, Claudio Cuevas, Alejandro Caicedo. Asymptotically periodic solutions of neutral partial differential equations with infinite delay. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2031-2068. doi: 10.3934/cpaa.2013.12.2031
[13]

Nobuyuki Kato. Linearized stability and asymptotic properties for abstract boundary value functional evolution problems. Conference Publications, 1998, 1998 (Special) : 371-387. doi: 10.3934/proc.1998.1998.371
[14]

Ben-Yu Guo, Zhong-Qing Wang. A spectral collocation method for solving initial value problems of first order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1029-1054. doi: 10.3934/dcdsb.2010.14.1029
[15]

Min Zhu, Panpan Ren, Junping Li. Exponential stability of solutions for retarded stochastic differential equations without dissipativity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2923-2938. doi: 10.3934/dcdsb.2017157
[16]

C. M. Groothedde, J. D. Mireles James. Parameterization method for unstable manifolds of delay differential equations. Journal of Computational Dynamics, 2017, 4 (1&2) : 21-70. doi: 10.3934/jcd.2017002
[17]

Jin-Mun Jeong, Seong-Ho Cho. Identification problems of retarded differential systems in Hilbert spaces. Evolution Equations & Control Theory, 2017, 6 (1) : 77-91. doi: 10.3934/eect.2017005
[18]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107
[19]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167
[20]

Ovide Arino, Eva Sánchez. A saddle point theorem for functional state-dependent delay differential equations. Discrete & Continuous Dynamical Systems, 2005, 12 (4) : 687-722. doi: 10.3934/dcds.2005.12.687

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (90)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy