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 & 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

[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

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

[1]

Dong-Lun Wu, Chun-Lei Tang, Xing-Ping Wu. Existence and nonuniqueness of homoclinic solutions for second-order Hamiltonian systems with mixed nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (1) : 57-72. doi: 10.3934/cpaa.2016.15.57
[2]

Ying Lv, Yan-Fang Xue, Chun-Lei Tang. Ground state homoclinic orbits for a class of asymptotically periodic second-order Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1627-1652. doi: 10.3934/dcdsb.2020176
[3]

Qiong Meng, X. H. Tang. Solutions of a second-order Hamiltonian system with periodic boundary conditions. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1053-1067. doi: 10.3934/cpaa.2010.9.1053
[4]

Xiaoping Wang. Ground state homoclinic solutions for a second-order Hamiltonian system. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2163-2175. doi: 10.3934/dcdss.2019139
[5]

José F. Cariñena, Javier de Lucas Araujo. Superposition rules and second-order Riccati equations. Journal of Geometric Mechanics, 2011, 3 (1) : 1-22. doi: 10.3934/jgm.2011.3.1
[6]

Norimichi Hirano, Zhi-Qiang Wang. Subharmonic solutions for second order Hamiltonian systems. Discrete & Continuous Dynamical Systems, 1998, 4 (3) : 467-474. doi: 10.3934/dcds.1998.4.467
[7]

Xingyong Zhang, Xianhua Tang. Some united existence results of periodic solutions for non-quadratic second order Hamiltonian systems. Communications on Pure & Applied Analysis, 2014, 13 (1) : 75-95. doi: 10.3934/cpaa.2014.13.75
[8]

Li-Li Wan, Chun-Lei Tang. Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 255-271. doi: 10.3934/dcdsb.2011.15.255
[9]

W. Sarlet, G. E. Prince, M. Crampin. Generalized submersiveness of second-order ordinary differential equations. Journal of Geometric Mechanics, 2009, 1 (2) : 209-221. doi: 10.3934/jgm.2009.1.209
[10]

José F. Cariñena, Irina Gheorghiu, Eduardo Martínez. Jacobi fields for second-order differential equations on Lie algebroids. Conference Publications, 2015, 2015 (special) : 213-222. doi: 10.3934/proc.2015.0213
[11]

Raegan Higgins. Asymptotic behavior of second-order nonlinear dynamic equations on time scales. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 609-622. doi: 10.3934/dcdsb.2010.13.609
[12]

Jaume Llibre, Amar Makhlouf. Periodic solutions of some classes of continuous second-order differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 477-482. doi: 10.3934/dcdsb.2017022
[13]

M. Euler, N. Euler, M. C. Nucci. On nonlocal symmetries generated by recursion operators: Second-order evolution equations. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4239-4247. doi: 10.3934/dcds.2017181
[14]

Addolorata Salvatore. Multiple homoclinic orbits for a class of second order perturbed Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 778-787. doi: 10.3934/proc.2003.2003.778
[15]

Guoshan Zhang, Peizhao Yu. Lyapunov method for stability of descriptor second-order and high-order systems. Journal of Industrial & Management Optimization, 2018, 14 (2) : 673-686. doi: 10.3934/jimo.2017068
[16]

Gábor Kiss, Bernd Krauskopf. Stability implications of delay distribution for first-order and second-order systems. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 327-345. doi: 10.3934/dcdsb.2010.13.327
[17]

Rubén Figueroa, Rodrigo López Pouso, Jorge Rodríguez–López. Existence and multiplicity results for second-order discontinuous problems via non-ordered lower and upper solutions. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 617-633. doi: 10.3934/dcdsb.2019257
[18]

Johnny Henderson, Rodica Luca. Existence of positive solutions for a system of nonlinear second-order integral boundary value problems. Conference Publications, 2015, 2015 (special) : 596-604. doi: 10.3934/proc.2015.0596
[19]

Kaizhi Wang, Yong Li. Existence and monotonicity property of minimizers of a nonconvex variational problem with a second-order Lagrangian. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 687-699. doi: 10.3934/dcds.2009.25.687
[20]

Yu Chen, Yanheng Ding, Suhong Li. Existence and concentration for Kirchhoff type equations around topologically critical points of the potential. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1641-1671. doi: 10.3934/cpaa.2017079

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (171)
  • HTML views (147)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences