July  2009, 23(3): 991-1008. doi: 10.3934/dcds.2009.23.991

On the boundedness of solutions of the equation $u''+(1+f(t))u=0$

1. 

Università IUAV di Venezia, Tolentini, S. Croce 191, 30135 Venezia, Italy

Received  January 2008 Revised  August 2008 Published  November 2008

Sufficient conditions on the function $\,f\,$ are given which ensure the boundedness of the solutions of the second order linear differential equation $\,u''+(1+f(t))\,u=0\,$ as $\, t\rightarrow +\infty\,$. To do this, a suitable class of quadratic forms is introduced.
Citation: Renato Manfrin. On the boundedness of solutions of the equation $u''+(1+f(t))u=0$. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 991-1008. doi: 10.3934/dcds.2009.23.991
[1]

Tomáš Bárta. Exact rate of decay for solutions to damped second order ODE's with a degenerate potential. Evolution Equations & Control Theory, 2018, 7 (4) : 531-543. doi: 10.3934/eect.2018025

[2]

Mikhaël Balabane, Mustapha Jazar, Philippe Souplet. Oscillatory blow-up in nonlinear second order ODE's: The critical case. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 577-584. doi: 10.3934/dcds.2003.9.577

[3]

Jun Cao, Der-Chen Chang, Dachun Yang, Sibei Yang. Boundedness of second order Riesz transforms associated to Schrödinger operators on Musielak-Orlicz-Hardy spaces. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1435-1463. doi: 10.3934/cpaa.2014.13.1435

[4]

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

[5]

Shaohua Chen. Boundedness and blowup solutions for quasilinear parabolic systems with lower order terms. Communications on Pure & Applied Analysis, 2009, 8 (2) : 587-600. doi: 10.3934/cpaa.2009.8.587

[6]

Alberto Boscaggin, Fabio Zanolin. Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 89-110. doi: 10.3934/dcds.2013.33.89

[7]

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

[8]

Marc Henrard. Homoclinic and multibump solutions for perturbed second order systems using topological degree. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 765-782. doi: 10.3934/dcds.1999.5.765

[9]

Daniel Franco, Donal O'Regan. Existence of solutions to second order problems with nonlinear boundary conditions. Conference Publications, 2003, 2003 (Special) : 273-280. doi: 10.3934/proc.2003.2003.273

[10]

Robert Stegliński. On homoclinic solutions for a second order difference equation with p-Laplacian. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 487-492. doi: 10.3934/dcdsb.2018033

[11]

Xiangjin Xu. Multiple solutions of super-quadratic second order dynamical systems. Conference Publications, 2003, 2003 (Special) : 926-934. doi: 10.3934/proc.2003.2003.926

[12]

Xuelei Wang, Dingbian Qian, Xiying Sun. Periodic solutions of second order equations with asymptotical non-resonance. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4715-4726. doi: 10.3934/dcds.2018207

[13]

Yuan Guo, Xiaofei Gao, Desheng Li. Structure of the set of bounded solutions for a class of nonautonomous second order differential equations. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1607-1616. doi: 10.3934/cpaa.2010.9.1607

[14]

Marie-Françoise Bidaut-Véron, Marta García-Huidobro, Cecilia Yarur. Large solutions of elliptic systems of second order and applications to the biharmonic equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 411-432. doi: 10.3934/dcds.2012.32.411

[15]

John R. Graef, Lingju Kong, Min Wang. Existence of homoclinic solutions for second order difference equations with $p$-laplacian. Conference Publications, 2015, 2015 (special) : 533-539. doi: 10.3934/proc.2015.0533

[16]

Xiaojun Chang, Yong Li. Rotating periodic solutions of second order dissipative dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 643-652. doi: 10.3934/dcds.2016.36.643

[17]

Chiara Zanini, Fabio Zanolin. Periodic solutions for a class of second order ODEs with a Nagumo cubic type nonlinearity. Discrete & Continuous Dynamical Systems - A, 2012, 32 (11) : 4045-4067. doi: 10.3934/dcds.2012.32.4045

[18]

Inara Yermachenko, Felix Sadyrbaev. Types of solutions and multiplicity results for second order nonlinear boundary value problems. Conference Publications, 2007, 2007 (Special) : 1061-1069. doi: 10.3934/proc.2007.2007.1061

[19]

C. Rebelo. Multiple periodic solutions of second order equations with asymmetric nonlinearities. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 25-34. doi: 10.3934/dcds.1997.3.25

[20]

Zhiming Guo, Xiaomin Zhang. Multiplicity results for periodic solutions to a class of second order delay differential equations. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1529-1542. doi: 10.3934/cpaa.2010.9.1529

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (12)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]