# American Institute of Mathematical Sciences

2013, 2013(special): 555-564. doi: 10.3934/proc.2013.2013.555

## Existence and multiplicity of solutions in fourth order BVPs with unbounded nonlinearities

 1 Departamento de Matemática. Universidade de Évora, Centro de Investigação em Matemática e Aplicaçoes da U.E. (CIMA-UE), Rua Romão Ramalho, 59. 7000-671 Évora 2 College of the Bahamas, School of Mathematics, Physics and Technologies, Department of Mathematics, Oakes Field Campus, Nassau, Bahamas

Received  September 2012 Revised  April 2013 Published  November 2013

In this work the authors present some existence, non-existence and location results of the problem composed of the fourth order fully nonlinear equation \begin{equation*} u^{\left( 4\right) }\left( x\right) +f( x,u\left( x\right) ,u^{\prime }\left( x\right) ,u^{\prime \prime }\left( x\right) ,u^{\prime \prime \prime }\left( x\right) ) =s\text{ }p(x) \end{equation*} for $x\in \left[ a,b\right] ,$ where $f:\left[ a,b\right] \times \mathbb{R} ^{4}\rightarrow \mathbb{R},$ $p:\left[ a,b\right] \rightarrow \mathbb{R}^{+}$ are continuous functions and $s$ a real parameter, with the boundary conditions \begin{equation*} u\left( a\right) =A,\text{ }u^{\prime }\left( a\right) =B,\text{ }u^{\prime \prime \prime }\left( a\right) =C,\text{ }u^{\prime \prime \prime }\left( b\right) =D,\text{ } \end{equation*} for $A,B,C,D\in \mathbb{R}.$ In this work they use an Ambrosetti-Prodi type approach, with some new features: the existence part is obtained in presence of nonlinearities not necessarily bounded, and in the multiplicity result it is not assumed a speed growth condition or an asymptotic condition, as it is usual in the literature for these type of higher order problems.
The arguments used apply lower and upper solutions technique and topological degree theory.
An application is made to a continuous model of the human spine, used in aircraft ejections, vehicle crash situations, and some forms of scoliosis.
Citation: Feliz Minhós, João Fialho. Existence and multiplicity of solutions in fourth order BVPs with unbounded nonlinearities. Conference Publications, 2013, 2013 (special) : 555-564. doi: 10.3934/proc.2013.2013.555
##### References:
 [1] A. Cabada, R. Pouso and F. Minhós, Extremal solutions to fourth-order functional boundary value problems including multipoint condition., Nonlinear Anal.: Real World Appl., (2009), 2157. [2] C. Fabry, J. Mawhin and M. N. Nkashama, A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations., Bull. London Math. Soc., 18 (1986), 173. [3] J. Fialho and F. Minhós, Existence and location results for hinged beams with unbounded nonlinearities,, Nonlinear Anal., 71 (2009). [4] J. Fialho, F. Minhós, On higher order fully periodic boundary value problems,, J. Math. Anal. Appl., (2012), 616. [5] J. Graef, L. Kong and B. Yang, Existence of solutions for a higher-order multi-point boundary value problem,, Result. Math., (2009), 77. [6] M.R. Grossinho, F.M. Minhós, A.I. Santos, olvability of some third-order boundary value problems with asymmetric unbounded linearities,, Nonlinear Analysis, (2005), 1235. [7] M.R. Grossinho, F. Minhós and 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., (2009), 4027. [8] J. Mawhin, Topological degree methods in nonlinear boundary value problems,, Regional Conference Series in Mathematics, (1979). [9] F. Minhós, Existence, nonexistence and multiplicity results for some beam equations, Progress in Nonlinear Differential Equations and Their Applications,, Vol. 75, (2007), 245. [10] F. Minhós, On some third order nonlinear boundary value problems: existence, location and multiplicity results,, J. Math. Anal. Appl., (2008), 1342. [11] F. Minhós and J. Fialho, Ambrosetti-Prodi type results to fourth order nonlinear fully differential equations,, Proceedings of Dynamic Systems and Applications, (2008), 325. [12] F. Minhós, Location results: an under used tool in higher order boundary value problems,, International Conference on Boundary Value Problems: Mathematical Models in Engineering, (1124), 244. [13] G. Noone and W.T.Ang, The inferior boundary condition of a continuous cantilever beam model of the human spine,, Australian Physical & Engineering Sciences in Medicine, 19 (1996), 26. [14] A. Patwardhan, W. Bunch, K. Meade, R. Vandeby and G. Knight, A biomechanical analog of curve progression and orthotic stabilization in idiopathic scoliosis,, J. Biomechanics, 19 (1986), 103. [15] M. Šenkyřík, Existence of multiple solutions for a third order three-point regular boundary value problem,, Mathematica Bohemica, (1994), 113.

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##### References:
 [1] A. Cabada, R. Pouso and F. Minhós, Extremal solutions to fourth-order functional boundary value problems including multipoint condition., Nonlinear Anal.: Real World Appl., (2009), 2157. [2] C. Fabry, J. Mawhin and M. N. Nkashama, A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations., Bull. London Math. Soc., 18 (1986), 173. [3] J. Fialho and F. Minhós, Existence and location results for hinged beams with unbounded nonlinearities,, Nonlinear Anal., 71 (2009). [4] J. Fialho, F. Minhós, On higher order fully periodic boundary value problems,, J. Math. Anal. Appl., (2012), 616. [5] J. Graef, L. Kong and B. Yang, Existence of solutions for a higher-order multi-point boundary value problem,, Result. Math., (2009), 77. [6] M.R. Grossinho, F.M. Minhós, A.I. Santos, olvability of some third-order boundary value problems with asymmetric unbounded linearities,, Nonlinear Analysis, (2005), 1235. [7] M.R. Grossinho, F. Minhós and 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., (2009), 4027. [8] J. Mawhin, Topological degree methods in nonlinear boundary value problems,, Regional Conference Series in Mathematics, (1979). [9] F. Minhós, Existence, nonexistence and multiplicity results for some beam equations, Progress in Nonlinear Differential Equations and Their Applications,, Vol. 75, (2007), 245. [10] F. Minhós, On some third order nonlinear boundary value problems: existence, location and multiplicity results,, J. Math. Anal. Appl., (2008), 1342. [11] F. Minhós and J. Fialho, Ambrosetti-Prodi type results to fourth order nonlinear fully differential equations,, Proceedings of Dynamic Systems and Applications, (2008), 325. [12] F. Minhós, Location results: an under used tool in higher order boundary value problems,, International Conference on Boundary Value Problems: Mathematical Models in Engineering, (1124), 244. [13] G. Noone and W.T.Ang, The inferior boundary condition of a continuous cantilever beam model of the human spine,, Australian Physical & Engineering Sciences in Medicine, 19 (1996), 26. [14] A. Patwardhan, W. Bunch, K. Meade, R. Vandeby and G. Knight, A biomechanical analog of curve progression and orthotic stabilization in idiopathic scoliosis,, J. Biomechanics, 19 (1986), 103. [15] M. Šenkyřík, Existence of multiple solutions for a third order three-point regular boundary value problem,, Mathematica Bohemica, (1994), 113.
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