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A note on optimal control problem for a hemivariational inequality modeling fluid flow
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. (CIMAUE), Rua Romão Ramalho, 59. 7000671 Évora 
2.  College of the Bahamas, School of Mathematics, Physics and Technologies, Department of Mathematics, Oakes Field Campus, Nassau, Bahamas 
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.
References:
[1] 
A. Cabada, R. Pouso and F. Minhós, Extremal solutions to fourthorder functional boundary value problems including multipoint condition., Nonlinear Anal.: Real World Appl., (2009), 2157. Google Scholar 
[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. Google Scholar 
[3] 
J. Fialho and F. Minhós, Existence and location results for hinged beams with unbounded nonlinearities,, Nonlinear Anal., 71 (2009). Google Scholar 
[4] 
J. Fialho, F. Minhós, On higher order fully periodic boundary value problems,, J. Math. Anal. Appl., (2012), 616. Google Scholar 
[5] 
J. Graef, L. Kong and B. Yang, Existence of solutions for a higherorder multipoint boundary value problem,, Result. Math., (2009), 77. Google Scholar 
[6] 
M.R. Grossinho, F.M. Minhós, A.I. Santos, olvability of some thirdorder boundary value problems with asymmetric unbounded linearities,, Nonlinear Analysis, (2005), 1235. Google Scholar 
[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. Google Scholar 
[8] 
J. Mawhin, Topological degree methods in nonlinear boundary value problems,, Regional Conference Series in Mathematics, (1979). Google Scholar 
[9] 
F. Minhós, Existence, nonexistence and multiplicity results for some beam equations, Progress in Nonlinear Differential Equations and Their Applications,, Vol. 75, (2007), 245. Google Scholar 
[10] 
F. Minhós, On some third order nonlinear boundary value problems: existence, location and multiplicity results,, J. Math. Anal. Appl., (2008), 1342. Google Scholar 
[11] 
F. Minhós and J. Fialho, AmbrosettiProdi type results to fourth order nonlinear fully differential equations,, Proceedings of Dynamic Systems and Applications, (2008), 325. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[15] 
M. Šenkyřík, Existence of multiple solutions for a third order threepoint regular boundary value problem,, Mathematica Bohemica, (1994), 113. Google Scholar 
show all references
References:
[1] 
A. Cabada, R. Pouso and F. Minhós, Extremal solutions to fourthorder functional boundary value problems including multipoint condition., Nonlinear Anal.: Real World Appl., (2009), 2157. Google Scholar 
[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. Google Scholar 
[3] 
J. Fialho and F. Minhós, Existence and location results for hinged beams with unbounded nonlinearities,, Nonlinear Anal., 71 (2009). Google Scholar 
[4] 
J. Fialho, F. Minhós, On higher order fully periodic boundary value problems,, J. Math. Anal. Appl., (2012), 616. Google Scholar 
[5] 
J. Graef, L. Kong and B. Yang, Existence of solutions for a higherorder multipoint boundary value problem,, Result. Math., (2009), 77. Google Scholar 
[6] 
M.R. Grossinho, F.M. Minhós, A.I. Santos, olvability of some thirdorder boundary value problems with asymmetric unbounded linearities,, Nonlinear Analysis, (2005), 1235. Google Scholar 
[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. Google Scholar 
[8] 
J. Mawhin, Topological degree methods in nonlinear boundary value problems,, Regional Conference Series in Mathematics, (1979). Google Scholar 
[9] 
F. Minhós, Existence, nonexistence and multiplicity results for some beam equations, Progress in Nonlinear Differential Equations and Their Applications,, Vol. 75, (2007), 245. Google Scholar 
[10] 
F. Minhós, On some third order nonlinear boundary value problems: existence, location and multiplicity results,, J. Math. Anal. Appl., (2008), 1342. Google Scholar 
[11] 
F. Minhós and J. Fialho, AmbrosettiProdi type results to fourth order nonlinear fully differential equations,, Proceedings of Dynamic Systems and Applications, (2008), 325. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[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. Google Scholar 
[15] 
M. Šenkyřík, Existence of multiple solutions for a third order threepoint regular boundary value problem,, Mathematica Bohemica, (1994), 113. Google Scholar 
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