American Institute of Mathematical Sciences

Advanced
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 and 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.

[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.

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

[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.

[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.

[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.

[8]

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

[9]

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

[10]

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

[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.

[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.

[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.

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.

[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.

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

[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.

[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.

[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.

[8]

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

[9]

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

[10]

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

[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.

[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.

[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.

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