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]

Applications of Mathematics, 48 (2003) 497-514.  Google Scholar

[2]

Nonlinear Anal., 71 (2009) 1519-1525. Google Scholar

[3]

Nonlinear Analysis, 62 (2005), 1235-1250.  Google Scholar

[4]

Nonlinear Anal., 70 (2009) 4027-4038  Google Scholar

[5]

Nonlinear Studies, 12 (2005) 165-176.  Google Scholar

[6]

Appl. Anal.,26 (1988), 289-304.  Google Scholar

[7]

Differential and Integral Equations, vol. 4, Number 2, (1991), 397-410.  Google Scholar

[8]

SIAM Review 32 (1990) 537-578.  Google Scholar

[9]

Appl. Math. Comp., 159 (2004) 11-18.  Google Scholar

[10]

Discrete Contin. Dyn. Syst., Supp., (2005) 662-671.  Google Scholar

[11]

Nonlinear Anal., 71 (2009) 281-292  Google Scholar

[12]

Appl. Analysis, 59 (1995) 15-25.  Google Scholar

show all references

References:
[1]

Applications of Mathematics, 48 (2003) 497-514.  Google Scholar

[2]

Nonlinear Anal., 71 (2009) 1519-1525. Google Scholar

[3]

Nonlinear Analysis, 62 (2005), 1235-1250.  Google Scholar

[4]

Nonlinear Anal., 70 (2009) 4027-4038  Google Scholar

[5]

Nonlinear Studies, 12 (2005) 165-176.  Google Scholar

[6]

Appl. Anal.,26 (1988), 289-304.  Google Scholar

[7]

Differential and Integral Equations, vol. 4, Number 2, (1991), 397-410.  Google Scholar

[8]

SIAM Review 32 (1990) 537-578.  Google Scholar

[9]

Appl. Math. Comp., 159 (2004) 11-18.  Google Scholar

[10]

Discrete Contin. Dyn. Syst., Supp., (2005) 662-671.  Google Scholar

[11]

Nonlinear Anal., 71 (2009) 281-292  Google Scholar

[12]

Appl. Analysis, 59 (1995) 15-25.  Google Scholar

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