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Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term
1. | School of Mathematics and Computer Science, Shangrao Normal University, Shangrao 334001, China |
2. | Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China |
$$x''+α x^+-β x^-+q(x)f(x')+g(t,x)=p(t), $$ |
$q, f∈ C^1(\mathbb{R}),$ |
$g(t,x)∈ C^{0,1}(\mathbf{S}^1× \mathbb{R})$ |
$p(t)∈ C^0(\mathbf{S}^1)$ |
$\mathbf{S}^1= \mathbb{R}/2π\mathbb{Z}$ |
$α$ |
$β $ |
$$\frac{1}{\sqrt{α}}+\frac{1}{\sqrt{β}}=\frac{2}{ω}$$ |
$ω∈ \mathbb{R}^+ $ |
$f,$ |
$g$ |
$p$ |
$q, f, g$ |
$p$ |
References:
[1] |
A. Capietto and B. Liu,
Quasi-periodic solutions of a forced asymmetric oscillator at resonance, Nonlinear Analysis: Theory, Methods & Applications, 56 (2004), 105-117.
doi: 10.1016/j.na.2003.09.001. |
[2] |
A. Capietto, W. Dambrosio and B. Liu,
On the boundedness of solutions to a nonlinear singular oscillator, Z. Angew. Math. Phys., 60 (2009), 1007-1034.
doi: 10.1007/s00033-008-8094-y. |
[3] |
A. Capietto, W. Dambrosio and X. Wang,
Quasi-periodic solutions of a damped reversible oscillator at resonance, Differential and Integral Equations, 22 (2009), 1033-1046.
|
[4] |
S. Chow and M. Pei,
Aubry-Mather theorem and quasiperiodic orbits for time dependent reversible systems, Nonlinear Analysis: Theory, Methods} & \emph{Applications, 25 (1995), 905-931.
doi: 10.1016/0362-546X(95)00087-C. |
[5] |
B. Liu,
Boundedness in asymmetric oscillations, J. Math. Anal. Appl., 231 (1999), 355-373.
doi: 10.1006/jmaa.1998.6219. |
[6] |
B. Liu and J. Song,
Invariant curved of reversible mappings with small twist, Acta Math. Sinica (English Series), 20 (2004), 15-24.
doi: 10.1007/s10114-004-0316-4. |
[7] |
X. Li,
Invariant tori for semilinear reversible systems, Nonlinear Analysis, 56 (2004), 133-146.
doi: 10.1016/j.na.2003.09.004. |
[8] |
R. Ortega,
Asymmetric oscillators and twist mappings, J. London Math. Soc., 53 (1996), 325-342.
doi: 10.1112/jlms/53.2.325. |
[9] |
M. Pei,
Mather sets for superlinear Duffing equations, Science in China, Ser. A, 36 (1993), 524-537.
|
[10] |
M. Pei,
Aubry-Mather sets for finite-twist maps of a cylinder and semilinear Duffing equations, J. Differential Equations, 113 (1994), 106-127.
doi: 10.1006/jdeq.1994.1116. |
[11] |
D. Qian,
Mather sets for sublinear Duffing Equations, Chin. Ann. of Math., Ser. B, 15 (1994), 421-434.
|
[12] |
J. Si,
Invariant tori for a reversible oscillator with a nonlinear damping and periodic forcing term, Nonlinear Anal., 64 (2006), 1475-1495.
doi: 10.1016/j.na.2005.06.046. |
[13] |
X. Wang,
Invariant tori and boundedness in asymmetric oscillations, Acta Math. Sinica(English Series), 19 (2003), 765-782.
doi: 10.1007/s10114-003-0249-3. |
[14] |
X. Wang,
Aubry-Mather sets for semilinear Duffing equations, Acta Mathematica Sinica(Chinese Series), 52 (2009), 605-610.
|
[15] |
X. Wang,
Aubry-Mather sets for sublinear asymmetric Duffing equations, Science China Mathematics, 42 (2012), 13-21.
doi: 10.1360/012011-328. |
[16] |
X. Wang,
Quasi-periodic solutions for second order differential equation with superlinear asymmetric nonlinearities and nonlinear damping term, Boundary Value Problems, 101 (2015), 1-12.
doi: 10.1186/s13661-015-0370-0. |
[17] |
X. Wang,
Aubry-Mather sets in semilinear asymmetric Duffing equations, Advances in Difference Equations, 297 (2016), 1-12.
doi: 10.1186/s13662-016-1024-y. |
[18] |
Y. Wang,
Boundedness of solutions in asymmetric oscillations via the twist theorem, Acta Math. Sinica(English Series), 17 (2001), 313-318.
doi: 10.1007/s101140000043. |
[19] |
X. Yang,
Boundedness of solutions for sublinear reversible systems, Applied Mathematics and Computation, 158 (2004), 389-396.
doi: 10.1016/j.amc.2003.08.092. |
[20] |
X. Yang and K. Lo,
Quasi-periodic solutions in nonlinear asymmetric oscillations, Zeitschrift für Analysis und ihre Anwendungen, 26 (2007), 207-220.
doi: 10.4171/ZAA/1319. |
show all references
References:
[1] |
A. Capietto and B. Liu,
Quasi-periodic solutions of a forced asymmetric oscillator at resonance, Nonlinear Analysis: Theory, Methods & Applications, 56 (2004), 105-117.
doi: 10.1016/j.na.2003.09.001. |
[2] |
A. Capietto, W. Dambrosio and B. Liu,
On the boundedness of solutions to a nonlinear singular oscillator, Z. Angew. Math. Phys., 60 (2009), 1007-1034.
doi: 10.1007/s00033-008-8094-y. |
[3] |
A. Capietto, W. Dambrosio and X. Wang,
Quasi-periodic solutions of a damped reversible oscillator at resonance, Differential and Integral Equations, 22 (2009), 1033-1046.
|
[4] |
S. Chow and M. Pei,
Aubry-Mather theorem and quasiperiodic orbits for time dependent reversible systems, Nonlinear Analysis: Theory, Methods} & \emph{Applications, 25 (1995), 905-931.
doi: 10.1016/0362-546X(95)00087-C. |
[5] |
B. Liu,
Boundedness in asymmetric oscillations, J. Math. Anal. Appl., 231 (1999), 355-373.
doi: 10.1006/jmaa.1998.6219. |
[6] |
B. Liu and J. Song,
Invariant curved of reversible mappings with small twist, Acta Math. Sinica (English Series), 20 (2004), 15-24.
doi: 10.1007/s10114-004-0316-4. |
[7] |
X. Li,
Invariant tori for semilinear reversible systems, Nonlinear Analysis, 56 (2004), 133-146.
doi: 10.1016/j.na.2003.09.004. |
[8] |
R. Ortega,
Asymmetric oscillators and twist mappings, J. London Math. Soc., 53 (1996), 325-342.
doi: 10.1112/jlms/53.2.325. |
[9] |
M. Pei,
Mather sets for superlinear Duffing equations, Science in China, Ser. A, 36 (1993), 524-537.
|
[10] |
M. Pei,
Aubry-Mather sets for finite-twist maps of a cylinder and semilinear Duffing equations, J. Differential Equations, 113 (1994), 106-127.
doi: 10.1006/jdeq.1994.1116. |
[11] |
D. Qian,
Mather sets for sublinear Duffing Equations, Chin. Ann. of Math., Ser. B, 15 (1994), 421-434.
|
[12] |
J. Si,
Invariant tori for a reversible oscillator with a nonlinear damping and periodic forcing term, Nonlinear Anal., 64 (2006), 1475-1495.
doi: 10.1016/j.na.2005.06.046. |
[13] |
X. Wang,
Invariant tori and boundedness in asymmetric oscillations, Acta Math. Sinica(English Series), 19 (2003), 765-782.
doi: 10.1007/s10114-003-0249-3. |
[14] |
X. Wang,
Aubry-Mather sets for semilinear Duffing equations, Acta Mathematica Sinica(Chinese Series), 52 (2009), 605-610.
|
[15] |
X. Wang,
Aubry-Mather sets for sublinear asymmetric Duffing equations, Science China Mathematics, 42 (2012), 13-21.
doi: 10.1360/012011-328. |
[16] |
X. Wang,
Quasi-periodic solutions for second order differential equation with superlinear asymmetric nonlinearities and nonlinear damping term, Boundary Value Problems, 101 (2015), 1-12.
doi: 10.1186/s13661-015-0370-0. |
[17] |
X. Wang,
Aubry-Mather sets in semilinear asymmetric Duffing equations, Advances in Difference Equations, 297 (2016), 1-12.
doi: 10.1186/s13662-016-1024-y. |
[18] |
Y. Wang,
Boundedness of solutions in asymmetric oscillations via the twist theorem, Acta Math. Sinica(English Series), 17 (2001), 313-318.
doi: 10.1007/s101140000043. |
[19] |
X. Yang,
Boundedness of solutions for sublinear reversible systems, Applied Mathematics and Computation, 158 (2004), 389-396.
doi: 10.1016/j.amc.2003.08.092. |
[20] |
X. Yang and K. Lo,
Quasi-periodic solutions in nonlinear asymmetric oscillations, Zeitschrift für Analysis und ihre Anwendungen, 26 (2007), 207-220.
doi: 10.4171/ZAA/1319. |
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