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January  2013, 33(1): 27-46. doi: 10.3934/dcds.2013.33.27

On general properties of retarded functional differential equations on manifolds

 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.
Citation: Pierluigi Benevieri, Alessandro Calamai, Massimo Furi, Maria Patrizia Pera. On general properties of retarded functional differential equations on manifolds. Discrete and 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. [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. [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. [4] G. Bouligand, "Introduction à la Géométrie Infinitésimale Directe,'' Gauthier-Villard, Paris, 1932. [5] N. Dunford and J. T. Schwartz, "Linear Operators,'' Wiley & Sons, Inc., New York, 1957. [6] R. Gaines and J. Mawhin, "Coincidence Degree and Nonlinear Differential Equations,'' Lecture Notes in Math., Springer Verlag, Berlin, 568, 1977. [7] V. Guillemin and A. Pollack, "Differential Topology,'' Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1974. [8] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkc. Ekvac., 21 (1978), 11-41. [9] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,'' Springer Verlag, New York, 1993. [10] Y. Hino, S. Murakami and T. Naito, "Functional-Differential Equations with Infinite Delay,'' Lecture Notes in Math., Springer Verlag, Berlin, 1473, 1991. [11] M. W. Hirsch, "Differential Topology,'' Graduate Texts in Math., Springer Verlag, Berlin, 33, 1976. [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. [13] J. M. Milnor, "Topology from the Differentiable Viewpoint,'' Univ. Press of Virginia, Charlottesville, 1965. [14] J. R. Munkres, "Elementary Differential Topology,'' Princeton University Press, Princeton, New Jersey, 1966. [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. [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. [17] W. M. Oliva, Functional differential equations on compact manifolds and an approximation theorem, J. Differential Equations, 5 (1969), 483-496. [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. [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.

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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. [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. [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. [4] G. Bouligand, "Introduction à la Géométrie Infinitésimale Directe,'' Gauthier-Villard, Paris, 1932. [5] N. Dunford and J. T. Schwartz, "Linear Operators,'' Wiley & Sons, Inc., New York, 1957. [6] R. Gaines and J. Mawhin, "Coincidence Degree and Nonlinear Differential Equations,'' Lecture Notes in Math., Springer Verlag, Berlin, 568, 1977. [7] V. Guillemin and A. Pollack, "Differential Topology,'' Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1974. [8] J. K. Hale and J. Kato, Phase space for retarded equations with infinite delay, Funkc. Ekvac., 21 (1978), 11-41. [9] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,'' Springer Verlag, New York, 1993. [10] Y. Hino, S. Murakami and T. Naito, "Functional-Differential Equations with Infinite Delay,'' Lecture Notes in Math., Springer Verlag, Berlin, 1473, 1991. [11] M. W. Hirsch, "Differential Topology,'' Graduate Texts in Math., Springer Verlag, Berlin, 33, 1976. [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. [13] J. M. Milnor, "Topology from the Differentiable Viewpoint,'' Univ. Press of Virginia, Charlottesville, 1965. [14] J. R. Munkres, "Elementary Differential Topology,'' Princeton University Press, Princeton, New Jersey, 1966. [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. [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. [17] W. M. Oliva, Functional differential equations on compact manifolds and an approximation theorem, J. Differential Equations, 5 (1969), 483-496. [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. [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.
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