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January  2021, 20(1): 281-300. doi: 10.3934/cpaa.2020266

Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles

Chao Wang 1,, , Qihuai Liu 2, and  Zhiguo Wang 3,

1. 

School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, 224002, China
2. 

School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin University of Electronic Technology, Guilin, 541004, China
3. 

School of Mathematical Sciences, Soochow University, Suzhou, 215006, China

* Corresponding author

Received  May 2020 Revised  August 2020 Published  January 2021 Early access  October 2020

Fund Project: The first author is supported by Natural Science Foundation of Jiangsu Province (Grant Nos. BK20171275, BK20181058) and NSFC (Grant No. 12071410). The second author is supported by NSFC (Grant No. 11771105)

In this paper, we prove the existence and multiplicity of subharmonic bouncing motions for a Hill's type sublinear oscillator with an obstacle. Furthermore, we also consider the existence, multiplicity and dense distribution of symmetric periodic bouncing solutions when the weight function is even. Based on an appropriate coordinate transformation and the method of phase-plane analysis, we can study our main results via Poincar$ \acute{e} $ map by applying some suitable fixed point theorems.

Keywords: Impact oscillator, sublinear Hill's type equation, phase-plane analysis method, subharmonic bouncing solution, Poincaré-Birkhoff twist theorem.
Mathematics Subject Classification: Primary:34C15;Secondary:34C25.
Citation: Chao Wang, Qihuai Liu, Zhiguo Wang. Periodic bouncing solutions for Hill's type sub-linear oscillators with obstacles. Communications on Pure and Applied Analysis, 2021, 20 (1) : 281-300. doi: 10.3934/cpaa.2020266
References:
[1]

D. Bonheure and C. Fabry, Periodic motions in impact oscillators with perfectly elastic bounces, Nonlinearity, 15 (2002), 1281-1297.  doi: 10.1088/0951-7715/15/4/314.

[2]

W. Ding and D. Qian, Infinitesimal periodic solutions of impact Hamiltonian systems, Science China: Math., 40 (2010), 563-574. 

[3]

W. Ding, D. Qian, C. Wang and Z. Wang, Existence of periodic solutions of sublinear hamiltonian systems, Acta Math. Appl. Sin., 32 (2016), 621-632.  doi: 10.1007/s10114-016-4162-y.

[4]

C. Fabry and P. Habets, Periodic solutions of second order differential equations with superlinear asymmetric nonlinearities, Arch. Math., 60 (1993), 266-276.  doi: 10.1007/BF01198811.

[5]

A. Fonda and A. Sfecci, Periodic bouncing solutions for nonlinear impact oscillators, Adv. Nonlinear Stud., 13 (2013), 179-189.  doi: 10.1515/ans-2013-0110.

[6]

M. Jiang, Periodic solutions of second order differential equations with an obstacle, Nonlinearity, 19 (2006), 1165. doi: 10.1088/0951-7715/19/5/007.

[7]

A. C. Lazer and P J. McKenna, Periodic bouncing for a forced linear spring with obstacle, Differ. Integral Equ., 5 (1992), 165-172. 

[8]

F. Nakajima, Even and periodic solutions of the equation $\ddot{u}+g(u) = e(t)$, J. Differ. Equ., 83 (1990), 277-299.  doi: 10.1016/0022-0396(90)90059-X.

[9]

R. Ortega, Variational and Topological Methods in the Study of Nonlinear Phenomena, Springer, New York, 2002.

[10]

R. Ortega, Asymmetric Oscillators and Twist Mappings, J. London Math. Soc., 53 (1996), 325-342.  doi: 10.1112/jlms/53.2.325.

[11]

D. Qian and P. J. Torres, Bouncing solutions of an equation with attractive singularity, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 201-213.  doi: 10.1017/S0308210500003164.

[12]

D. Qian and P. J. Torres, Periodic motions of linear impact oscillators via the successor map, SIAM J. Math. Anal., 36 (2005), 1707-1725.  doi: 10.1137/S003614100343771X.

[13]

D. Qian, Large amplitude periodic bouncing for impact oscillators with damping, Proc. Am. Math. Soc., 133 (2005), 1797-1804.  doi: 10.1090/S0002-9939-04-07759-7.

[14]

D. Qian and X. Sun, Invariant tori for asymptotically linear impact oscillators, Sci. China Math., 49 (2006), 669-687.  doi: 10.1007/s11425-006-0669-5.

[15]

X. Sun and D. Qian, Periodic bouncing solutions for attractive singular second-order equations, Nonlinear Anal., 71 (2009), 4751-4757.  doi: 10.1016/j.na.2009.03.049.

[16]

Z. Wang, C. Ruan and D. Qian, Existence and multiplicity of subharmonic bouncing solutions for sub-linear impact oscillators, Nonlinear Anal., 27 (2010), 17-30. 

[17]

Z. Wang, Q. Liu and D. Qian, Existence of quasi-periodic solutions and Littlewood's boundedness problem of sub-linear impact oscillators, Nonlinear Anal., 76 (2011), 5606-5617.  doi: 10.1016/j.na.2011.05.046.

[18]

C. Wang, D. Qian and Q. Liu, Impact oscillators of Hill'S type with indefinite weight: periodic and chaotic dynamics, Discrete Contin. Dyn. Syst., 36 (2016), 2305-2328.  doi: 10.3934/dcds.2016.36.2305.

show all references

References:
[1]

D. Bonheure and C. Fabry, Periodic motions in impact oscillators with perfectly elastic bounces, Nonlinearity, 15 (2002), 1281-1297.  doi: 10.1088/0951-7715/15/4/314.

[2]

W. Ding and D. Qian, Infinitesimal periodic solutions of impact Hamiltonian systems, Science China: Math., 40 (2010), 563-574. 

[3]

W. Ding, D. Qian, C. Wang and Z. Wang, Existence of periodic solutions of sublinear hamiltonian systems, Acta Math. Appl. Sin., 32 (2016), 621-632.  doi: 10.1007/s10114-016-4162-y.

[4]

C. Fabry and P. Habets, Periodic solutions of second order differential equations with superlinear asymmetric nonlinearities, Arch. Math., 60 (1993), 266-276.  doi: 10.1007/BF01198811.

[5]

A. Fonda and A. Sfecci, Periodic bouncing solutions for nonlinear impact oscillators, Adv. Nonlinear Stud., 13 (2013), 179-189.  doi: 10.1515/ans-2013-0110.

[6]

M. Jiang, Periodic solutions of second order differential equations with an obstacle, Nonlinearity, 19 (2006), 1165. doi: 10.1088/0951-7715/19/5/007.

[7]

A. C. Lazer and P J. McKenna, Periodic bouncing for a forced linear spring with obstacle, Differ. Integral Equ., 5 (1992), 165-172. 

[8]

F. Nakajima, Even and periodic solutions of the equation $\ddot{u}+g(u) = e(t)$, J. Differ. Equ., 83 (1990), 277-299.  doi: 10.1016/0022-0396(90)90059-X.

[9]

R. Ortega, Variational and Topological Methods in the Study of Nonlinear Phenomena, Springer, New York, 2002.

[10]

R. Ortega, Asymmetric Oscillators and Twist Mappings, J. London Math. Soc., 53 (1996), 325-342.  doi: 10.1112/jlms/53.2.325.

[11]

D. Qian and P. J. Torres, Bouncing solutions of an equation with attractive singularity, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 201-213.  doi: 10.1017/S0308210500003164.

[12]

D. Qian and P. J. Torres, Periodic motions of linear impact oscillators via the successor map, SIAM J. Math. Anal., 36 (2005), 1707-1725.  doi: 10.1137/S003614100343771X.

[13]

D. Qian, Large amplitude periodic bouncing for impact oscillators with damping, Proc. Am. Math. Soc., 133 (2005), 1797-1804.  doi: 10.1090/S0002-9939-04-07759-7.

[14]

D. Qian and X. Sun, Invariant tori for asymptotically linear impact oscillators, Sci. China Math., 49 (2006), 669-687.  doi: 10.1007/s11425-006-0669-5.

[15]

X. Sun and D. Qian, Periodic bouncing solutions for attractive singular second-order equations, Nonlinear Anal., 71 (2009), 4751-4757.  doi: 10.1016/j.na.2009.03.049.

[16]

Z. Wang, C. Ruan and D. Qian, Existence and multiplicity of subharmonic bouncing solutions for sub-linear impact oscillators, Nonlinear Anal., 27 (2010), 17-30. 

[17]

Z. Wang, Q. Liu and D. Qian, Existence of quasi-periodic solutions and Littlewood's boundedness problem of sub-linear impact oscillators, Nonlinear Anal., 76 (2011), 5606-5617.  doi: 10.1016/j.na.2011.05.046.

[18]

C. Wang, D. Qian and Q. Liu, Impact oscillators of Hill'S type with indefinite weight: periodic and chaotic dynamics, Discrete Contin. Dyn. Syst., 36 (2016), 2305-2328.  doi: 10.3934/dcds.2016.36.2305.

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