
Previous Article
An approximation model for the densitydependent magnetohydrodynamic equations
 PROC Home
 This Issue

Next Article
A reinjected cuspidal horseshoe
The role of lower and upper solutions in the generalization of Lidstone problems
1.  Centro de Investigação em Matemática e Aplicações da U.E. (CIMACE), Rua Romão Ramalho 59, 7000671 Évora 
2.  School of Sciences and Technology. Department of Mathematics, University of Évora, Research Center in Mathematics and Applications of the University of Évora, (CIMAUE), Rua Romão Ramalho, 59, 7000671 Évora, Portugal 
\begin{equation*} u^{(iv)} (x) + f( x,u(x) ,u^{\prime}(x) ,u^{\prime \prime}(x) ,u^{\prime \prime \prime}(x) ) = 0 \end{equation*} for $x\in [ 0,1] ,$ where $f:[ 0,1] \times \mathbb{R} ^{4} \to \mathbb{R}$ is a continuous functions, coupled with the Lidstone boundary conditions, \begin{equation*} u(0) = u(1) = u^{\prime \prime}(0) = u^{\prime \prime }(1) = 0. \end{equation*}
They discuss how different definitions of lower and upper solutions can generalize existence and location results for boundary value problems with Lidstone boundary data. In addition, they replace the usual bilateral Nagumo condition by a onesided condition, allowing the nonlinearity to be unbounded$.$ An example will show that this unilateral condition generalizes the usual one and stress the potentialities of the new definitions.
References:
show all references
References:
[1] 
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 
[2] 
Luis Barreira, Davor Dragičević, Claudia Valls. From onesided dichotomies to twosided dichotomies. Discrete & Continuous Dynamical Systems  A, 2015, 35 (7) : 28172844. doi: 10.3934/dcds.2015.35.2817 
[3] 
WolfJüergen Beyn, Janosch Rieger. The implicit Euler scheme for onesided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems  B, 2010, 14 (2) : 409428. doi: 10.3934/dcdsb.2010.14.409 
[4] 
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 
[5] 
Charles Fulton, David Pearson, Steven Pruess. Characterization of the spectral density function for a onesided tridiagonal Jacobi matrix operator. Conference Publications, 2013, 2013 (special) : 247257. doi: 10.3934/proc.2013.2013.247 
[6] 
Piermarco Cannarsa, Vilmos Komornik, Paola Loreti. Onesided and internal controllability of semilinear wave equations with infinitely iterated logarithms. Discrete & Continuous Dynamical Systems  A, 2002, 8 (3) : 745756. doi: 10.3934/dcds.2002.8.747 
[7] 
Kengo Matsumoto. Kgroups of the full group actions on onesided topological Markov shifts. Discrete & Continuous Dynamical Systems  A, 2013, 33 (8) : 37533765. doi: 10.3934/dcds.2013.33.3753 
[8] 
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 
[9] 
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 
[10] 
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 
[11] 
Juntang Ding, Xuhui Shen. Upper and lower bounds for the blowup time in quasilinear reaction diffusion problems. Discrete & Continuous Dynamical Systems  B, 2018, 23 (10) : 42434254. doi: 10.3934/dcdsb.2018135 
[12] 
Christoph Kawan. Upper and lower estimates for invariance entropy. Discrete & Continuous Dynamical Systems  A, 2011, 30 (1) : 169186. doi: 10.3934/dcds.2011.30.169 
[13] 
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 
[14] 
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 
[15] 
Dongfeng Zhang, Junxiang Xu. On elliptic lower dimensional tori for Gevreysmooth Hamiltonian systems under Rüssmann's nondegeneracy condition. Discrete & Continuous Dynamical Systems  A, 2006, 16 (3) : 635655. doi: 10.3934/dcds.2006.16.635 
[16] 
Chiara Corsato, Franco Obersnel, Pierpaolo Omari, Sabrina Rivetti. On the lower and upper solution method for the prescribed mean curvature equation in Minkowski space. Conference Publications, 2013, 2013 (special) : 159169. doi: 10.3934/proc.2013.2013.159 
[17] 
Armando G. M. Neves. Upper and lower bounds on Mathieu characteristic numbers of integer orders. Communications on Pure & Applied Analysis, 2004, 3 (3) : 447464. doi: 10.3934/cpaa.2004.3.447 
[18] 
Amadeu Delshams, Vassili Gelfreich, Angel Jorba and Tere M. Seara. Lower and upper bounds for the splitting of separatrices of the pendulum under a fast quasiperiodic forcing. Electronic Research Announcements, 1997, 3: 110. 
[19] 
Inbo Sim. On the existence of nodal solutions for singular onedimensional $\varphi$Laplacian problem with asymptotic condition. Communications on Pure & Applied Analysis, 2008, 7 (4) : 905923. doi: 10.3934/cpaa.2008.7.905 
[20] 
Jinggang Tan. Positive solutions for non local elliptic problems. Discrete & Continuous Dynamical Systems  A, 2013, 33 (2) : 837859. doi: 10.3934/dcds.2013.33.837 
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]