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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:
[1] 
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M. R. Grossinho, F. Minhós, A. I. Santos, A note on a class of problems for a higher order fully nonlinear equation under one sided Nagumo type condition,, Nonlinear Anal., 70 (2009), 4027. Google Scholar 
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M.R. Grossinho, F. Minhós, Upper and lower solutions for some higher order boundary value problems,, Nonlinear Studies, 12 (2005), 165. Google Scholar 
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T.F. Ma, J. da Silva, Iterative solutions for a beam equation with nonlinear boundary conditions of third order,, Appl. Math. Comp., 159 (2004), 11. Google Scholar 
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F. Minhós, T. Gyulov, A. I. Santos, Existence and location result for a fourth order boundary value problem,, Discrete Contin. Dyn. Syst., (2005), 662. Google Scholar 
[11] 
F. Minhós, T. Gyulov, A. I. Santos, Lower and upper solutions for a fully nonlinear beam equations,, Nonlinear Anal., (2009), 281. Google Scholar 
[12] 
M. Šenkyřík, Fourth order boundary value problems and nonlinear beams,, Appl. Analysis, 59 (1995), 15. Google Scholar 
show all references
References:
[1] 
P. Drábek, G. Holubová, A. Matas, P. Nečessal, Nonlinear models of suspension bridges: discussion of results,, Applications of Mathematics, 48 (2003), 497. Google Scholar 
[2] 
J. Fialho, F. Minhós, Existence and location results for hinged beams with unbounded nonlinearities,, Nonlinear Anal., 71 (2009), 1519. Google Scholar 
[3] 
M.R. Grossinho, F.M. Minhós, A.I. Santos, Solvability of some thirdorder boundary value problems with asymmetric unbounded linearities,, Nonlinear Analysis, 62 (2005), 1235. Google Scholar 
[4] 
M. R. Grossinho, F. Minhós, A. I. Santos, A note on a class of problems for a higher order fully nonlinear equation under one sided Nagumo type condition,, Nonlinear Anal., 70 (2009), 4027. Google Scholar 
[5] 
M.R. Grossinho, F. Minhós, Upper and lower solutions for some higher order boundary value problems,, Nonlinear Studies, 12 (2005), 165. Google Scholar 
[6] 
C. P. Gupta, Existence and uniqueness theorems for the bending of an elastic beam equation,, Appl. Anal., 26 (1988), 289. Google Scholar 
[7] 
C. P. Gupta, Existence and uniqueness theorems for a fourth order boundary value problem of SturmLiouville type,, Differential and Integral Equations, 4 (1991), 397. Google Scholar 
[8] 
A.C. Lazer, P.J. Mckenna, Largeamplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis,, SIAM Review 32 (1990) 537578., 32 (1990), 537. Google Scholar 
[9] 
T.F. Ma, J. da Silva, Iterative solutions for a beam equation with nonlinear boundary conditions of third order,, Appl. Math. Comp., 159 (2004), 11. Google Scholar 
[10] 
F. Minhós, T. Gyulov, A. I. Santos, Existence and location result for a fourth order boundary value problem,, Discrete Contin. Dyn. Syst., (2005), 662. Google Scholar 
[11] 
F. Minhós, T. Gyulov, A. I. Santos, Lower and upper solutions for a fully nonlinear beam equations,, Nonlinear Anal., (2009), 281. Google Scholar 
[12] 
M. Šenkyřík, Fourth order boundary value problems and nonlinear beams,, Appl. Analysis, 59 (1995), 15. Google Scholar 
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