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

Existence of solutions for nonlinear operator equations

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.

Citation: Yucheng Bu, Yujun Dong. Existence of solutions for nonlinear operator equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4429-4441. doi: 10.3934/dcds.2019180
References:
[1]

G. BonannoR. 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. ChenY. 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. BonannoR. 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. ChenY. 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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