July  2017, 16(4): 1471-1492. doi: 10.3934/cpaa.2017070

On the set of harmonic solutions of a class of perturbed coupled and nonautonomous differential equations on manifolds

Dipartimento di Matematica e Informatica "Ulisse Dini", Via S. Marta 3,50139 Firenze, Italy

Received  July 2016 Revised  February 2017 Published  April 2017

Fund Project: The authors have been supported by the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

We study the set of $T$-periodic solutions of a class of $T$-periodically perturbed coupled and nonautonomous differential equations on manifolds. By using degree-theoretic methods we obtain a global continuation result for the $T$-periodic solutions of the considered equations.

Citation: Luca Bisconti, Marco Spadini. On the set of harmonic solutions of a class of perturbed coupled and nonautonomous differential equations on manifolds. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1471-1492. doi: 10.3934/cpaa.2017070
References:
[1]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Review, 18 (1976), 620-709. doi: 10.1137/1018114. Google Scholar

[2]

L. Bisconti, Harmonic solutions to a class of differential-algebraic equations with separated variables, Electron. J. Differ. Equ., 2012 (2012), 15 pp. Google Scholar

[3]

L. Bisconti and M. Spadini, On a class of differential-algebraic equations with infinite delay, Electron. J. Qual. Theory Differ. Equ. , 2011, 1-21. Google Scholar

[4]

L. Bisconti and M. Spadini, Corrigendum to On a class of differential-algebraic equations with infinite delay, Electron. J. Qual. Theory Differ. Equ. , 2012, 1-5. Google Scholar

[5]

L. Bisconti and M. Spadini, Sunflower model: Time-dependent coefficients and topology of the periodic solutions set, Nonlinear Differential Equations and Applications (NoDEA), 22 (2015), 1573-1590. doi: 10.1007/s00030-015-0336-z. Google Scholar

[6]

L. Bisconti and M. Spadini, Harmonic perturbations with delay of periodic separated variables differential equations, Topological Methods in Nonlinear Analysis, 46 (2015), 261-281. doi: 10.12775/TMNA.2015.046. Google Scholar

[7]

L. Bisconti and M. Spadini, About the notion of non-T -resonance and applications to topological multiplicity results for ODEs on differentiable manifolds, Math. Methods Appl. Sci., 38 (2015), 4760-4773. doi: 10.1002/mma.3390. Google Scholar

[8]

F. Brauer and J. A. Nohel, The Qualitative Theory of Ordinary Differential Equations. An Introduction, Dover Publ. , New York, 1989.Google Scholar

[9]

T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Dover Publ. Mineola N. Y. 2005. Originally published by Academic Press, Orlando Fl. 1985. Google Scholar

[10]

J. Cronin, Differential Equations. Introduction and Qualitative Theory, 2nd edition, Monographs and Textbooks in Pure and Applied Mathematics, 180. Marcel Dekker, Inc. , New York, 1994. Google Scholar

[11]

J. Dugundji and A. Granas, Fixed Point Theory, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21593-8. Google Scholar

[12]

M. Furi and M. P. Pera, Cobifurcating branches of solutions for nonlinear eigenvalue problems in Banach spaces, Annali di Matematica Pura ed Applicata, 135 (1983), 119-132. doi: 10.1007/BF01781065. Google Scholar

[13]

M. Furi and M. P. Pera, A continuation principle for periodic solutions of forced motion equations on manifolds and application to bifurcation theory, Pacific J. Math., 160 (1993), 219-244. Google Scholar

[14]

M. FuriM. P. Pera and M. Spadini, On the uniqueness of the fixed point index on differentiable manifolds, Fixed Point Theory and Applications, 2004 (2004), 251-259. doi: 10.1155/S168718200440713X. Google Scholar

[15]

M. Furi, M. P. Pera and M. Spadini, The fixed point index of the Poincaré operator on differentiable manifolds, Handbook of Topological Fixed Point Theory, (eds. R. F. Brown, M. Furi, L. Górniewicz, B. Jiang), Springer, (2005) 741-782. doi: 10.1007/1-4020-3222-6_20. Google Scholar

[16]

M. Furi, M. P. Pera and M. Spadini, A Set of Axioms for the Degree of a Tangent Vector Field on Differentiable Manifolds, Fixed Point Theory Appl. , 2010 Art. ID 845631, 11 pp. Google Scholar

[17]

M. Furi and M. Spadini, On the fixed point index of the flow and applications to periodic solutions of differential equations on manifolds, Boll. Un. Mat. Ital. A (7), 10 (1996), 333-346. Google Scholar

[18]

M. Furi and M. Spadini, Periodic perturbations with delay of autonomous differential equations on manifolds, Adv. Nonlinear Stud., 9 (2009), 263-276. doi: 10.1515/ans-2009-0203. Google Scholar

[19]

M. Gerdin, Identification and Estimation for Models Described by Differential-Algebraic Equations, Dissertation, Department of Electrical Engineering Linköpings universitet, SE-581 83 Linköping, Sweden (2006).Google Scholar

[20]

A. Granas, The Leray-Schauder index and the fixed point theory for arbitrary ANRs, Bull. Soc. Math. France, 100 (1972), 209-228. Google Scholar

[21]

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

[22]

M. W. Hirsch, Differential Topology, Graduate Texts in Math. Vol. 33, Springer Verlag, Berlin, 1994. Google Scholar

[23]

M. A. Krasnosel'skiĭ, P. P. Zabreĭko, Geometrical Methods of Nonlinear Analysis, Grundlehren der Mathematischen Wissenschaften, 263. Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69409-7. Google Scholar

[24]

P. Kunkel and V. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solutions, EMS Textbooks in Mathematics, 2006. doi: 10.4171/017. Google Scholar

[25]

N. G. Lloyd, Degree Theory, Cambridge tracts in Mathematics, No. 73. Cambridge Univ. Press, Cambridge-New York-Melbourne 1978. Google Scholar

[26]

J. W. Milnor, Topology from the Differentiable Viewpoint, Univ. press of Virginia, Charlottesville, 1965. Google Scholar

[27]

R. D. Nussbaum, The fixed point index and fixed points theorems, in Topological Methods for Ordinary Differential Equations (Montecatini Terme, 1991), Lecture notes in Math. 1537, Springer, Berlin, (1993), 143-205. doi: 10.1007/BFb0085077. Google Scholar

[28]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics 57, Springer 2011. doi: 10.1007/978-1-4419-7646-8. Google Scholar

[29]

M. Spadini, Branches of harmonic solutions to periodically perturbed coupled differential equations on manifolds, Discrete and Continuous Dyn. Syst., 15 (2006), 951-964. doi: 10.3934/dcds.2006.15.951. Google Scholar

[30]

M. Spadini, A note on topological methods for a class of differential-algebraic equations, Nonlinear Anal., 73 (2010), 1065-1076. doi: 10.1016/j.na.2010.04.038. Google Scholar

[31]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, Ⅰ. Fixed-point Theorems, Translated from the German by Peter R. Wadsack. Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5. Google Scholar

show all references

References:
[1]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Review, 18 (1976), 620-709. doi: 10.1137/1018114. Google Scholar

[2]

L. Bisconti, Harmonic solutions to a class of differential-algebraic equations with separated variables, Electron. J. Differ. Equ., 2012 (2012), 15 pp. Google Scholar

[3]

L. Bisconti and M. Spadini, On a class of differential-algebraic equations with infinite delay, Electron. J. Qual. Theory Differ. Equ. , 2011, 1-21. Google Scholar

[4]

L. Bisconti and M. Spadini, Corrigendum to On a class of differential-algebraic equations with infinite delay, Electron. J. Qual. Theory Differ. Equ. , 2012, 1-5. Google Scholar

[5]

L. Bisconti and M. Spadini, Sunflower model: Time-dependent coefficients and topology of the periodic solutions set, Nonlinear Differential Equations and Applications (NoDEA), 22 (2015), 1573-1590. doi: 10.1007/s00030-015-0336-z. Google Scholar

[6]

L. Bisconti and M. Spadini, Harmonic perturbations with delay of periodic separated variables differential equations, Topological Methods in Nonlinear Analysis, 46 (2015), 261-281. doi: 10.12775/TMNA.2015.046. Google Scholar

[7]

L. Bisconti and M. Spadini, About the notion of non-T -resonance and applications to topological multiplicity results for ODEs on differentiable manifolds, Math. Methods Appl. Sci., 38 (2015), 4760-4773. doi: 10.1002/mma.3390. Google Scholar

[8]

F. Brauer and J. A. Nohel, The Qualitative Theory of Ordinary Differential Equations. An Introduction, Dover Publ. , New York, 1989.Google Scholar

[9]

T. A. Burton, Stability and Periodic Solutions of Ordinary and Functional Differential Equations, Dover Publ. Mineola N. Y. 2005. Originally published by Academic Press, Orlando Fl. 1985. Google Scholar

[10]

J. Cronin, Differential Equations. Introduction and Qualitative Theory, 2nd edition, Monographs and Textbooks in Pure and Applied Mathematics, 180. Marcel Dekker, Inc. , New York, 1994. Google Scholar

[11]

J. Dugundji and A. Granas, Fixed Point Theory, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21593-8. Google Scholar

[12]

M. Furi and M. P. Pera, Cobifurcating branches of solutions for nonlinear eigenvalue problems in Banach spaces, Annali di Matematica Pura ed Applicata, 135 (1983), 119-132. doi: 10.1007/BF01781065. Google Scholar

[13]

M. Furi and M. P. Pera, A continuation principle for periodic solutions of forced motion equations on manifolds and application to bifurcation theory, Pacific J. Math., 160 (1993), 219-244. Google Scholar

[14]

M. FuriM. P. Pera and M. Spadini, On the uniqueness of the fixed point index on differentiable manifolds, Fixed Point Theory and Applications, 2004 (2004), 251-259. doi: 10.1155/S168718200440713X. Google Scholar

[15]

M. Furi, M. P. Pera and M. Spadini, The fixed point index of the Poincaré operator on differentiable manifolds, Handbook of Topological Fixed Point Theory, (eds. R. F. Brown, M. Furi, L. Górniewicz, B. Jiang), Springer, (2005) 741-782. doi: 10.1007/1-4020-3222-6_20. Google Scholar

[16]

M. Furi, M. P. Pera and M. Spadini, A Set of Axioms for the Degree of a Tangent Vector Field on Differentiable Manifolds, Fixed Point Theory Appl. , 2010 Art. ID 845631, 11 pp. Google Scholar

[17]

M. Furi and M. Spadini, On the fixed point index of the flow and applications to periodic solutions of differential equations on manifolds, Boll. Un. Mat. Ital. A (7), 10 (1996), 333-346. Google Scholar

[18]

M. Furi and M. Spadini, Periodic perturbations with delay of autonomous differential equations on manifolds, Adv. Nonlinear Stud., 9 (2009), 263-276. doi: 10.1515/ans-2009-0203. Google Scholar

[19]

M. Gerdin, Identification and Estimation for Models Described by Differential-Algebraic Equations, Dissertation, Department of Electrical Engineering Linköpings universitet, SE-581 83 Linköping, Sweden (2006).Google Scholar

[20]

A. Granas, The Leray-Schauder index and the fixed point theory for arbitrary ANRs, Bull. Soc. Math. France, 100 (1972), 209-228. Google Scholar

[21]

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

[22]

M. W. Hirsch, Differential Topology, Graduate Texts in Math. Vol. 33, Springer Verlag, Berlin, 1994. Google Scholar

[23]

M. A. Krasnosel'skiĭ, P. P. Zabreĭko, Geometrical Methods of Nonlinear Analysis, Grundlehren der Mathematischen Wissenschaften, 263. Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69409-7. Google Scholar

[24]

P. Kunkel and V. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solutions, EMS Textbooks in Mathematics, 2006. doi: 10.4171/017. Google Scholar

[25]

N. G. Lloyd, Degree Theory, Cambridge tracts in Mathematics, No. 73. Cambridge Univ. Press, Cambridge-New York-Melbourne 1978. Google Scholar

[26]

J. W. Milnor, Topology from the Differentiable Viewpoint, Univ. press of Virginia, Charlottesville, 1965. Google Scholar

[27]

R. D. Nussbaum, The fixed point index and fixed points theorems, in Topological Methods for Ordinary Differential Equations (Montecatini Terme, 1991), Lecture notes in Math. 1537, Springer, Berlin, (1993), 143-205. doi: 10.1007/BFb0085077. Google Scholar

[28]

H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics 57, Springer 2011. doi: 10.1007/978-1-4419-7646-8. Google Scholar

[29]

M. Spadini, Branches of harmonic solutions to periodically perturbed coupled differential equations on manifolds, Discrete and Continuous Dyn. Syst., 15 (2006), 951-964. doi: 10.3934/dcds.2006.15.951. Google Scholar

[30]

M. Spadini, A note on topological methods for a class of differential-algebraic equations, Nonlinear Anal., 73 (2010), 1065-1076. doi: 10.1016/j.na.2010.04.038. Google Scholar

[31]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, Ⅰ. Fixed-point Theorems, Translated from the German by Peter R. Wadsack. Springer-Verlag, New York, 1986. doi: 10.1007/978-1-4612-4838-5. Google Scholar

Figure 1.  Initial points of $2\pi$-periodic solutions of (2): Nontrivial $2\pi$-periodic solutions correspond to points with $\lambda>0$
Figure 2.  The system of Example 3.5
Figure 3.  Projections of the set of starting triples of (2)
Figure 4.  The mechanical system of Example 6.2
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