American Institute of Mathematical Sciences

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August  2019, 39(8): 4429-4441. doi: 10.3934/dcds.2019180

Existence of solutions for nonlinear operator equations

Yucheng Bu 1,2, and  Yujun Dong 1,,

1. 

School of Mathematical Science, Nanjing Normal University, Nanjing 210023, China
2. 

Danyang Normal School, Zhenjiang College, Zhenjiang 212300, China

* Corresponding author: Yujun Dong at yjdong@njnu.edu.cn

Received  July 2018 Published  May 2019

We investigate solutions for nonlinear operator equations and obtain some abstract existence results by linking methods. Some well-known theorems about periodic solutions for second-order Hamiltonian systems by M. Schechter are special cases of these results.

Keywords: Operator equations, critical points, linking, existence of solution, second-order Hamiltonian systems.
Mathematics Subject Classification: Primary: 47J05, 49J35; Secondary: 35B38, 58E30.
Citation: Yucheng Bu, Yujun Dong. Existence of solutions for nonlinear operator equations. Discrete & Continuous Dynamical Systems, 2019, 39 (8) : 4429-4441. doi: 10.3934/dcds.2019180
References:
[1]

G. Bonanno, R. Livrea and M. Schechter, Multiple solutions of second order Hamiltonian systems, Electron. J. Qual. Theory Differ. Equ., 33 (2017), 1-15.  doi: 10.14232/ejqtde.2017.1.33.  Google Scholar

[2]

Y. Chen, Y. Dong and Y. Shan, Existence of solutions for sublinear or superlinear operator equations, Sci. China Math., 58 (2015), 1653-1664.  doi: 10.1007/s11425-014-4966-0.  Google Scholar

[3] Y. Dong, Index Theory for Hamiltonian Systems and Multiple Solutions Problems, Science Press, Beijing, 2014.   Google Scholar
[4]

L. Li and M. Schechter, Existence of solutions for second order Hamiltonian systems, Nonlinear Anal. Real World Appl., 27 (2016), 283-296.  doi: 10.1016/j.nonrwa.2015.08.001.  Google Scholar

[5]

J. Pipan and M. Schechter, Non-autonomous second order Hamiltonian systems, J. Differential Equations, 257 (2014), 351-373.  doi: 10.1016/j.jde.2014.03.016.  Google Scholar

[6]

M. Schechter, Periodic solutions of second-order nonautonomous dynamical systems, Bound. Value Probl., 2006 (2006), Art. ID 25104, 9pp. doi: 10.1155/bvp/2006/25104.  Google Scholar

[7]

M. Schechter, Periodic non-autonomous second order dynamical systems, J. Differential Equations, 223 (2006), 290-302.  doi: 10.1016/j.jde.2005.02.022.  Google Scholar

[8]

M. Schechter, Minmax Systems and Critical Point Theory, Birkh$\ddot{a}$user, Boston, 2009. Google Scholar

[9]

M. Schechter, Linking Methods in Critical Point Theory, Birkh$\ddot{a}$user, Boston, 1999. Google Scholar

[10]

M. Schechter, Nonautonomous second order Hamiltonian systems, Pacific J. Math., 251 (2011), 431-452.  doi: 10.2140/pjm.2011.251.431.  Google Scholar

[11]

Z. Wang and J. Zhang, New existence results on periodic solutions of non-autonomous second order Hamiltonian systems, Appl. Math. Lett., 79 (2018), 43-50.  doi: 10.1016/j.aml.2017.11.016.  Google Scholar

[12]

X. Zhang and X. Tang, Some united existence results of periodic solutions for non-quadratic second order Hamiltonian systems, Commun. Pure Appl. Anal., 13 (2014), 75-95.  doi: 10.3934/cpaa.2014.13.75.  Google Scholar

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W. Zou and S. Li, On Schechter's linking theorems, J. Funct. Anal., 258 (2010), 3347-3361.  doi: 10.1016/j.jfa.2009.11.005.  Google Scholar

show all references

References:
[1]

G. Bonanno, R. Livrea and M. Schechter, Multiple solutions of second order Hamiltonian systems, Electron. J. Qual. Theory Differ. Equ., 33 (2017), 1-15.  doi: 10.14232/ejqtde.2017.1.33.  Google Scholar

[2]

Y. Chen, Y. Dong and Y. Shan, Existence of solutions for sublinear or superlinear operator equations, Sci. China Math., 58 (2015), 1653-1664.  doi: 10.1007/s11425-014-4966-0.  Google Scholar

[3] Y. Dong, Index Theory for Hamiltonian Systems and Multiple Solutions Problems, Science Press, Beijing, 2014.   Google Scholar
[4]

L. Li and M. Schechter, Existence of solutions for second order Hamiltonian systems, Nonlinear Anal. Real World Appl., 27 (2016), 283-296.  doi: 10.1016/j.nonrwa.2015.08.001.  Google Scholar

[5]

J. Pipan and M. Schechter, Non-autonomous second order Hamiltonian systems, J. Differential Equations, 257 (2014), 351-373.  doi: 10.1016/j.jde.2014.03.016.  Google Scholar

[6]

M. Schechter, Periodic solutions of second-order nonautonomous dynamical systems, Bound. Value Probl., 2006 (2006), Art. ID 25104, 9pp. doi: 10.1155/bvp/2006/25104.  Google Scholar

[7]

M. Schechter, Periodic non-autonomous second order dynamical systems, J. Differential Equations, 223 (2006), 290-302.  doi: 10.1016/j.jde.2005.02.022.  Google Scholar

[8]

M. Schechter, Minmax Systems and Critical Point Theory, Birkh$\ddot{a}$user, Boston, 2009. Google Scholar

[9]

M. Schechter, Linking Methods in Critical Point Theory, Birkh$\ddot{a}$user, Boston, 1999. Google Scholar

[10]

M. Schechter, Nonautonomous second order Hamiltonian systems, Pacific J. Math., 251 (2011), 431-452.  doi: 10.2140/pjm.2011.251.431.  Google Scholar

[11]

Z. Wang and J. Zhang, New existence results on periodic solutions of non-autonomous second order Hamiltonian systems, Appl. Math. Lett., 79 (2018), 43-50.  doi: 10.1016/j.aml.2017.11.016.  Google Scholar

[12]

X. Zhang and X. Tang, Some united existence results of periodic solutions for non-quadratic second order Hamiltonian systems, Commun. Pure Appl. Anal., 13 (2014), 75-95.  doi: 10.3934/cpaa.2014.13.75.  Google Scholar

[13]

W. Zou and S. Li, On Schechter's linking theorems, J. Funct. Anal., 258 (2010), 3347-3361.  doi: 10.1016/j.jfa.2009.11.005.  Google Scholar

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