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January  2013, 33(1): 239-254. doi: 10.3934/dcds.2013.33.239

Resonant forced oscillations in systems with periodic nonlinearities

Alexander Krasnosel'skii 1,

1. 

Institute for Information Transmission Problems, 19 Bolshoi Karetny, 127994, GSP-4 Moscow, Russian Federation

Received  August 2011 Revised  February 2012 Published  September 2012

We present an approach to study degenerate ODE with periodic nonlinearities; for resonant higher order nonlinear equations $L(p)x=f(x)+b(t),\;p=d/dt$ with $2\pi$-periodic forcing $b$ and periodic $f$ we give multiplicity results, in particular, conditions of existence of infinite and unbounded sets of $2\pi$-periodic solutions.
Keywords: higher order ODE, resonance, unbounded set of solutions, method of stationary phase., Forced periodic oscillations, periodic nonlinearities.
Mathematics Subject Classification: Primary: 34C15, 34C25; Secondary: 93C1.
Citation: Alexander Krasnosel'skii. Resonant forced oscillations in systems with periodic nonlinearities. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 239-254. doi: 10.3934/dcds.2013.33.239
References:
[1]

J. M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonance, Nonlinearity, 9 (1996), 1099-1111. doi: 10.1088/0951-7715/9/5/003.  Google Scholar

[2]

V. I. Arnold, "Arnold's Problems,'' Problem 1996-5, comments by S.B.Kuksin, Springer-Verlag, New York, (2005), 580-582. Google Scholar

[3]

D. Bonheure, C. Fabry and D. Ruiz, Problems at resonance for equations with periodic nonlinearities, Nonlinear Analysis, 55 (2003), 557-581. doi: 10.1016/j.na.2003.07.005.  Google Scholar

[4]

A. Cańada and F. Roca, Existence and multiplicity of solutions of someconservative pendulum-type equations with homogeneous Dirichlet conditions, Diff. and Int. Eq., 10 (1997), 1113-1122.  Google Scholar

[5]

E. N. Dancer, On the use of asymptotics in nonlinear boundary value problems, Ann. Mat. Pura Appl., 131 (1982), 167-185. doi: 10.1007/BF01765151.  Google Scholar

[6]

A. Isidori, "Nonlinear Control Systems,'' Springer-Verlag, London, 1995.  Google Scholar

[7]

H. K. Khalil, "Nonlinear Systems,'' Prentice Hall, Upper Saddle River, New Jersey, 2002. Google Scholar

[8]

A. M. Krasnosel'skii and M. A. Krasnosel'skii, Vector fields in the direct product of spaces, and applications todifferential equations, Differential Equations, 33 (1997), 59-66.  Google Scholar

[9]

A. M. Krasnosel'skii and J. Mawhin, The index at infinity of some twice degenerate compact vector fields, Discrete Contin. Dyn. Syst., 1 (1995), 207-216. doi: 10.3934/dcds.1995.1.207.  Google Scholar

[10]

A. M. Krasnosel'skii and J. Mawhin, Periodic solutions of equations with oscillating nonlinearities, Mathematical and Computer Modelling, 32 (2000), 1445-1455 doi: 10.1016/S0895-7177(00)00216-8.  Google Scholar

[11]

M. A. Krasnosel'skii and P. P. Zabreiko, "Geometrical Methods of Nonlinear Analysis,'' Springer-Verlag, Berlin, Heidelberg, 1984.  Google Scholar

[12]

F. W. S. Olver, "Asymptotics and Special Functions,'' Academic Press, New York, 1974.  Google Scholar

[13]

G. Pólya and G. Szegö, "Aufgaben and Lehrsätze aus der Analysis,'' Springer, Berlin, B I-II (1925). Google Scholar

[14]

R. Schaaf and K. Schmitt, A class of nonlinear Sturm-Liouville problems with infinitely many solutions, Trans. Amer. Math. Soc., 306 (1988), 853-859. doi: 10.1090/S0002-9947-1988-0933322-5.  Google Scholar

show all references

References:
[1]

J. M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonance, Nonlinearity, 9 (1996), 1099-1111. doi: 10.1088/0951-7715/9/5/003.  Google Scholar

[2]

V. I. Arnold, "Arnold's Problems,'' Problem 1996-5, comments by S.B.Kuksin, Springer-Verlag, New York, (2005), 580-582. Google Scholar

[3]

D. Bonheure, C. Fabry and D. Ruiz, Problems at resonance for equations with periodic nonlinearities, Nonlinear Analysis, 55 (2003), 557-581. doi: 10.1016/j.na.2003.07.005.  Google Scholar

[4]

A. Cańada and F. Roca, Existence and multiplicity of solutions of someconservative pendulum-type equations with homogeneous Dirichlet conditions, Diff. and Int. Eq., 10 (1997), 1113-1122.  Google Scholar

[5]

E. N. Dancer, On the use of asymptotics in nonlinear boundary value problems, Ann. Mat. Pura Appl., 131 (1982), 167-185. doi: 10.1007/BF01765151.  Google Scholar

[6]

A. Isidori, "Nonlinear Control Systems,'' Springer-Verlag, London, 1995.  Google Scholar

[7]

H. K. Khalil, "Nonlinear Systems,'' Prentice Hall, Upper Saddle River, New Jersey, 2002. Google Scholar

[8]

A. M. Krasnosel'skii and M. A. Krasnosel'skii, Vector fields in the direct product of spaces, and applications todifferential equations, Differential Equations, 33 (1997), 59-66.  Google Scholar

[9]

A. M. Krasnosel'skii and J. Mawhin, The index at infinity of some twice degenerate compact vector fields, Discrete Contin. Dyn. Syst., 1 (1995), 207-216. doi: 10.3934/dcds.1995.1.207.  Google Scholar

[10]

A. M. Krasnosel'skii and J. Mawhin, Periodic solutions of equations with oscillating nonlinearities, Mathematical and Computer Modelling, 32 (2000), 1445-1455 doi: 10.1016/S0895-7177(00)00216-8.  Google Scholar

[11]

M. A. Krasnosel'skii and P. P. Zabreiko, "Geometrical Methods of Nonlinear Analysis,'' Springer-Verlag, Berlin, Heidelberg, 1984.  Google Scholar

[12]

F. W. S. Olver, "Asymptotics and Special Functions,'' Academic Press, New York, 1974.  Google Scholar

[13]

G. Pólya and G. Szegö, "Aufgaben and Lehrsätze aus der Analysis,'' Springer, Berlin, B I-II (1925). Google Scholar

[14]

R. Schaaf and K. Schmitt, A class of nonlinear Sturm-Liouville problems with infinitely many solutions, Trans. Amer. Math. Soc., 306 (1988), 853-859. doi: 10.1090/S0002-9947-1988-0933322-5.  Google Scholar

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