The role of lower and upper solutions in the generalization ofLidstone problems

João Fialho; Feliz Minhós

doi:10.3934/proc.2013.2013.217

Article Contents

2013, Volume 2013: 217-226. Doi: 10.3934/proc.2013.2013.217 open access

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

João Fialho^1, and
Feliz Minhós^2,

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

Received: August 31, 2012

Revised: January 31, 2013

Published: October 2013

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Abstract

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.

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Mathematics Subject Classification: Primary: 34B15; Secondary: 47H11.

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References

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