
Previous Article
A necessary and sufficient condition for oscillation of second order sublinear delay dynamic equations
 PROC Home
 This Issue

Next Article
Gradient systems on networks
Periodic solutions for some fully nonlinear fourth order differential equations
1.  Departamento de Matemática. Universidade de Évora, Centro de Investigação em Matemática e Aplicaçoes da U.E. (CIMAUE), Rua Romão Ramalho, 59. 7000671 Évora 
$u^((i))(a) = u^((i))(b), i=0,1,2,3,$
The difficulties in the odd derivatives are overcome due to the following arguments: the control on the third derivative is done by a Nagumotype condition and the bounds on the first derivative are obtained by lower and upper solutions, not necessarily ordered.
By this technique, not only it is proved the existence of a periodic solution, but also, some qualitative properties of the solution can be obtained.
[1] 
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) : 89110. doi: 10.3934/dcds.2013.33.89 
[2] 
Alberto Cabada, João Fialho, Feliz Minhós. Non ordered lower and upper solutions to fourth order problems with functional boundary conditions. Conference Publications, 2011, 2011 (Special) : 209218. doi: 10.3934/proc.2011.2011.209 
[3] 
Massimo Tarallo, Zhe Zhou. Limit periodic upper and lower solutions in a generic sense. Discrete & Continuous Dynamical Systems  A, 2018, 38 (1) : 293309. doi: 10.3934/dcds.2018014 
[4] 
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) : 40454067. doi: 10.3934/dcds.2012.32.4045 
[5] 
John R. Graef, Lingju Kong, Min Wang. Existence of multiple solutions to a discrete fourth order periodic boundary value problem. Conference Publications, 2013, 2013 (special) : 291299. doi: 10.3934/proc.2013.2013.291 
[6] 
Chunhua Jin, Jingxue Yin, Zejia Wang. Positive periodic solutions to a nonlinear fourthorder differential equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 12251235. doi: 10.3934/cpaa.2008.7.1225 
[7] 
João Fialho, Feliz Minhós. The role of lower and upper solutions in the generalization of Lidstone problems. Conference Publications, 2013, 2013 (special) : 217226. doi: 10.3934/proc.2013.2013.217 
[8] 
Luisa Malaguti, Cristina Marcelli. Existence of bounded trajectories via upper and lower solutions. Discrete & Continuous Dynamical Systems  A, 2000, 6 (3) : 575590. doi: 10.3934/dcds.2000.6.575 
[9] 
John B. Greer, Andrea L. Bertozzi. $H^1$ Solutions of a class of fourth order nonlinear equations for image processing. Discrete & Continuous Dynamical Systems  A, 2004, 10 (1/2) : 349366. doi: 10.3934/dcds.2004.10.349 
[10] 
Takahiro Hashimoto. Existence and nonexistence of nontrivial solutions of some nonlinear fourth order elliptic equations. Conference Publications, 2003, 2003 (Special) : 393402. doi: 10.3934/proc.2003.2003.393 
[11] 
Craig Cowan. Uniqueness of solutions for elliptic systems and fourth order equations involving a parameter. Communications on Pure & Applied Analysis, 2016, 15 (2) : 519533. doi: 10.3934/cpaa.2016.15.519 
[12] 
Armengol Gasull, Hector Giacomini, Joan Torregrosa. Explicit upper and lower bounds for the traveling wave solutions of FisherKolmogorov type equations. Discrete & Continuous Dynamical Systems  A, 2013, 33 (8) : 35673582. doi: 10.3934/dcds.2013.33.3567 
[13] 
Nakao Hayashi, Chunhua Li, Pavel I. Naumkin. Upper and lower time decay bounds for solutions of dissipative nonlinear Schrödinger equations. Communications on Pure & Applied Analysis, 2017, 16 (6) : 20892104. doi: 10.3934/cpaa.2017103 
[14] 
Ana Maria Bertone, J.V. Goncalves. Discontinuous elliptic problems in $R^N$: Lower and upper solutions and variational principles. Discrete & Continuous Dynamical Systems  A, 2000, 6 (2) : 315328. doi: 10.3934/dcds.2000.6.315 
[15] 
Wen Zhang, Xianhua Tang, Bitao Cheng, Jian Zhang. Signchanging solutions for fourth order elliptic equations with Kirchhofftype. Communications on Pure & Applied Analysis, 2016, 15 (6) : 21612177. doi: 10.3934/cpaa.2016032 
[16] 
Horst Osberger. Longtime behavior of a fully discrete Lagrangian scheme for a family of fourth order equations. Discrete & Continuous Dynamical Systems  A, 2017, 37 (1) : 405434. doi: 10.3934/dcds.2017017 
[17] 
Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Discrete & Continuous Dynamical Systems  A, 2008, 21 (3) : 763800. doi: 10.3934/dcds.2008.21.763 
[18] 
C. Rebelo. Multiple periodic solutions of second order equations with asymmetric nonlinearities. Discrete & Continuous Dynamical Systems  A, 1997, 3 (1) : 2534. doi: 10.3934/dcds.1997.3.25 
[19] 
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) : 15291542. doi: 10.3934/cpaa.2010.9.1529 
[20] 
Jaume Llibre, Amar Makhlouf. Periodic solutions of some classes of continuous secondorder differential equations. Discrete & Continuous Dynamical Systems  B, 2017, 22 (2) : 477482. doi: 10.3934/dcdsb.2017022 
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]