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Unboundedness of solutions for perturbed asymmetric oscillators
1. | School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, China |
$x'' + f(x )x' + ax^+ - bx^-$ $+ g(x)=p(t), $
where $x^+ =\max\{x,0\}, x^-$ $=\max\{-x,0\}$, $a$ and $b$ are two positive constants, $f(x)$ is a continuous function and $ p(t)$ is a $2\pi $-periodic continuous function, $g(x)$ is locally Lipschitz continuous and bounded. We discuss the existence of periodic solutions and unbounded solutions under two classes of conditions: the resonance case $\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}}\in Q$ and the nonresonance case $\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{b}} \notin Q$. Unlike many existing results in the literature where the function $g(x)$ is required to have asymptotic limits at infinity, our main results here allow $g(x)$ be oscillatory without asymptotic limits.
References:
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B. Liu, Boundedness in asymmetric oscillations,, J. Math. Anal. Appl., 231 (1999), 355.
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X. Li and Z. H. Zhang, Unbounded solutions and periodic solutions for second order differential equations with asymmetric nonlinearity,, Proc. Amer. Math. Soc., 135 (2007), 2769.
doi: 10.1090/S0002-9939-07-08928-9. |
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N. J. Lloyd, "Degree Theory,", Cambridge University Press, (1978).
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S. W. Ma and J. H. Wu, A small twist theorem and boundedness of solutions for semilinear Duffing equations at resonance,, Nonlinear Anal., 67 (2007), 200.
doi: 10.1016/j.na.2006.04.023. |
[12] |
L. X. Wang and S. W. Ma, Boundedness and unboundedness of solutions for asymmetric oscillators at resonance,, Preprint., (). |
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Z. H. Wang, Coexistence of unbounded solutions and periodic solutions of Liénard equations with asymmetric nonlinearities at resonance,, Sci. China Ser. A, 50 (2007), 1205.
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doi: 10.1090/S0002-9939-02-06601-7. |
show all references
References:
[1] |
J. M. Alonso and R. Ortega, Unbounded solutions of semilinear equations at resonance,, Nonlinearity, 9 (1996), 1099.
doi: 10.1088/0951-7715/9/5/003. |
[2] |
J. M. Alonso and R. Ortega, Roots of unity and unbounded motions of an asymmetric oscillator,, J. Differential Equations, 143 (1998), 201.
doi: 10.1006/jdeq.1997.3367. |
[3] |
W. Dambrosio, A note on the existence of unbounded solutions to a perturbed asymmetric oscillator,, Nonlinear Anal., 50 (2002), 333.
doi: 10.1016/S0362-546X(01)00765-9. |
[4] |
E. N. Dancer, Boundary-value problems for weakly nonlinear ordinary differential equations,, Bull. Austral. Math. Soc., 15 (1976), 321.
doi: 10.1017/S0004972700022747. |
[5] |
E. N. Dancer, On the Dirichlet problem for weakly nonlinear elliptic partial differential equations,, Proc. Roy. Soc. Edinburgh Sect.A, 76 (1976), 283.
|
[6] |
S. Fučik, "Sovability of Nonlinear Equations and Boundary Value Problems,", D. Reidel Publishing Co., (1980).
|
[7] |
M. Kunze, T. Küpper and B. Liu, Boundedness and unboundedness solutions of reversible oscillators at resonance,, Nonlinearity, 14 (2001), 1105.
doi: 10.1088/0951-7715/14/5/311. |
[8] |
B. Liu, Boundedness in asymmetric oscillations,, J. Math. Anal. Appl., 231 (1999), 355.
doi: 10.1006/jmaa.1998.6219. |
[9] |
X. Li and Z. H. Zhang, Unbounded solutions and periodic solutions for second order differential equations with asymmetric nonlinearity,, Proc. Amer. Math. Soc., 135 (2007), 2769.
doi: 10.1090/S0002-9939-07-08928-9. |
[10] |
N. J. Lloyd, "Degree Theory,", Cambridge University Press, (1978).
|
[11] |
S. W. Ma and J. H. Wu, A small twist theorem and boundedness of solutions for semilinear Duffing equations at resonance,, Nonlinear Anal., 67 (2007), 200.
doi: 10.1016/j.na.2006.04.023. |
[12] |
L. X. Wang and S. W. Ma, Boundedness and unboundedness of solutions for asymmetric oscillators at resonance,, Preprint., (). |
[13] |
Z. H. Wang, Coexistence of unbounded solutions and periodic solutions of Liénard equations with asymmetric nonlinearities at resonance,, Sci. China Ser. A, 50 (2007), 1205.
doi: 10.1007/s11425-007-0070-z. |
[14] |
Z. H. Wang, Irrational rotation numbers and unboundedness of solutions of the second order differential equations with asymmetric nonlinearities,, Proc. Amer. Math. Soc., 131 (2003), 523.
doi: 10.1090/S0002-9939-02-06601-7. |
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