# American Institute of Mathematical Sciences

2013, 2013(special): 217-226. doi: 10.3934/proc.2013.2013.217

## The role of lower and upper solutions in the generalization of Lidstone problems

 1 Centro de Investigação em Matemática e Aplicações da U.E. (CIMA-CE), Rua Romão Ramalho 59, 7000-671 Évora 2 School of Sciences and Technology. Department of Mathematics, University of Évora, Research Center in Mathematics and Applications of the University of Évora, (CIMA-UE), Rua Romão Ramalho, 59, 7000-671 Évora, Portugal

Received  September 2012 Revised  February 2013 Published  November 2013

In this the authors consider the nonlinear fully equation
\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 one-sided 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.
Citation: João Fialho, Feliz Minhós. The role of lower and upper solutions in the generalization of Lidstone problems. Conference Publications, 2013, 2013 (special) : 217-226. doi: 10.3934/proc.2013.2013.217
##### References:
 [1] P. Drábek, G. Holubová, A. Matas, P. Nečessal, Nonlinear models of suspension bridges: discussion of results, Applications of Mathematics, 48 (2003) 497-514. [2] J. Fialho, F. Minhós, Existence and location results for hinged beams with unbounded nonlinearities, Nonlinear Anal., 71 (2009) 1519-1525. [3] M.R. Grossinho, F.M. Minhós, A.I. Santos, Solvability of some third-order boundary value problems with asymmetric unbounded linearities, Nonlinear Analysis, 62 (2005), 1235-1250. [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) 4027-4038 [5] M.R. Grossinho, F. Minhós, Upper and lower solutions for some higher order boundary value problems, Nonlinear Studies, 12 (2005) 165-176. [6] C. P. Gupta, Existence and uniqueness theorems for the bending of an elastic beam equation, Appl. Anal.,26 (1988), 289-304. [7] C. P. Gupta, Existence and uniqueness theorems for a fourth order boundary value problem of Sturm-Liouville type, Differential and Integral Equations, vol. 4, Number 2, (1991), 397-410. [8] A.C. Lazer, P.J. Mckenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Review 32 (1990) 537-578. [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) 11-18. [10] F. Minhós, T. Gyulov, A. I. Santos, Existence and location result for a fourth order boundary value problem, Discrete Contin. Dyn. Syst., Supp., (2005) 662-671. [11] F. Minhós, T. Gyulov, A. I. Santos, Lower and upper solutions for a fully nonlinear beam equations, Nonlinear Anal., 71 (2009) 281-292 [12] M. Šenkyřík, Fourth order boundary value problems and nonlinear beams, Appl. Analysis, 59 (1995) 15-25.

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##### References:
 [1] P. Drábek, G. Holubová, A. Matas, P. Nečessal, Nonlinear models of suspension bridges: discussion of results, Applications of Mathematics, 48 (2003) 497-514. [2] J. Fialho, F. Minhós, Existence and location results for hinged beams with unbounded nonlinearities, Nonlinear Anal., 71 (2009) 1519-1525. [3] M.R. Grossinho, F.M. Minhós, A.I. Santos, Solvability of some third-order boundary value problems with asymmetric unbounded linearities, Nonlinear Analysis, 62 (2005), 1235-1250. [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) 4027-4038 [5] M.R. Grossinho, F. Minhós, Upper and lower solutions for some higher order boundary value problems, Nonlinear Studies, 12 (2005) 165-176. [6] C. P. Gupta, Existence and uniqueness theorems for the bending of an elastic beam equation, Appl. Anal.,26 (1988), 289-304. [7] C. P. Gupta, Existence and uniqueness theorems for a fourth order boundary value problem of Sturm-Liouville type, Differential and Integral Equations, vol. 4, Number 2, (1991), 397-410. [8] A.C. Lazer, P.J. Mckenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Review 32 (1990) 537-578. [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) 11-18. [10] F. Minhós, T. Gyulov, A. I. Santos, Existence and location result for a fourth order boundary value problem, Discrete Contin. Dyn. Syst., Supp., (2005) 662-671. [11] F. Minhós, T. Gyulov, A. I. Santos, Lower and upper solutions for a fully nonlinear beam equations, Nonlinear Anal., 71 (2009) 281-292 [12] M. Šenkyřík, Fourth order boundary value problems and nonlinear beams, Appl. Analysis, 59 (1995) 15-25.
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