American Institute of Mathematical Sciences

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

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

Feliz Minhós 1, and  João Fialho 2,

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.
Keywords: one-sided Nagumo condition, non-existence and multiplicity of solutions., Higher order Ambrosetti-Prodi problems, existence.
Mathematics Subject Classification: Primary: 34B15; Secondary: 34B18, 47H1.
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., 10 (2009) 2157-2170.

[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), 173-180.

[3]

J. Fialho and F. Minhós, Existence and location results for hinged beams with unbounded nonlinearities, Nonlinear Anal., 71 (2009) e1519-e1525.

[4]

J. Fialho, F. Minhós, On higher order fully periodic boundary value problems, J. Math. Anal. Appl., 395 (2012) 616-625.

[5]

J. Graef, L. Kong and B. Yang, Existence of solutions for a higher-order multi-point boundary value problem, Result. Math., 53 (2009), 77-101.

[6]

M.R. Grossinho, F.M. Minhós, A.I. Santos, olvability of some third-order boundary value problems with asymmetric unbounded linearities, Nonlinear Analysis, 62 (2005), 1235-1250.

[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., 70 (2009) 4027-4038.

[8]

J. Mawhin, Topological degree methods in nonlinear boundary value problems, Regional Conference Series in Mathematics, 40, American Mathematical Society, Providence, Rhode Island, 1979.

[9]

F. Minhós, Existence, nonexistence and multiplicity results for some beam equations, Progress in Nonlinear Differential Equations and Their Applications, Vol. 75, 245-255, Birkhäuser Verlag Basel (2007).

[10]

F. Minhós, On some third order nonlinear boundary value problems: existence, location and multiplicity results, J. Math. Anal. Appl., Vol. 339 (2008) 1342-1353.

[11]

F. Minhós and J. Fialho, Ambrosetti-Prodi type results to fourth order nonlinear fully differential equations, Proceedings of Dynamic Systems and Applications, Vol. 5, 325-332, Dynamic Publishers, Inc., USA, 2008

[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, Biology and Medicine, American Institute of Physics Conference Proceedings, Vol. 1124, 244-253, 2009

[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, Vol. 19 no 1 (1996) 26-30.

[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, Vol 19, no 2 (1986) 103-117.

[15]

M. Šenkyřík, Existence of multiple solutions for a third order three-point regular boundary value problem, Mathematica Bohemica, 119, no. 2 (1994), 113-121.

show all references

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., 10 (2009) 2157-2170.

[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), 173-180.

[3]

J. Fialho and F. Minhós, Existence and location results for hinged beams with unbounded nonlinearities, Nonlinear Anal., 71 (2009) e1519-e1525.

[4]

J. Fialho, F. Minhós, On higher order fully periodic boundary value problems, J. Math. Anal. Appl., 395 (2012) 616-625.

[5]

J. Graef, L. Kong and B. Yang, Existence of solutions for a higher-order multi-point boundary value problem, Result. Math., 53 (2009), 77-101.

[6]

M.R. Grossinho, F.M. Minhós, A.I. Santos, olvability of some third-order boundary value problems with asymmetric unbounded linearities, Nonlinear Analysis, 62 (2005), 1235-1250.

[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., 70 (2009) 4027-4038.

[8]

J. Mawhin, Topological degree methods in nonlinear boundary value problems, Regional Conference Series in Mathematics, 40, American Mathematical Society, Providence, Rhode Island, 1979.

[9]

F. Minhós, Existence, nonexistence and multiplicity results for some beam equations, Progress in Nonlinear Differential Equations and Their Applications, Vol. 75, 245-255, Birkhäuser Verlag Basel (2007).

[10]

F. Minhós, On some third order nonlinear boundary value problems: existence, location and multiplicity results, J. Math. Anal. Appl., Vol. 339 (2008) 1342-1353.

[11]

F. Minhós and J. Fialho, Ambrosetti-Prodi type results to fourth order nonlinear fully differential equations, Proceedings of Dynamic Systems and Applications, Vol. 5, 325-332, Dynamic Publishers, Inc., USA, 2008

[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, Biology and Medicine, American Institute of Physics Conference Proceedings, Vol. 1124, 244-253, 2009

[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, Vol. 19 no 1 (1996) 26-30.

[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, Vol 19, no 2 (1986) 103-117.

[15]

M. Šenkyřík, Existence of multiple solutions for a third order three-point regular boundary value problem, Mathematica Bohemica, 119, no. 2 (1994), 113-121.

[1]

Xuewei Cui, Mei Yu. Non-existence of positive solutions for a higher order fractional equation. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1379-1387. doi: 10.3934/dcds.2019059
[2]

F. R. Pereira. Multiple solutions for a class of Ambrosetti-Prodi type problems for systems involving critical Sobolev exponents. Communications on Pure and Applied Analysis, 2008, 7 (2) : 355-372. doi: 10.3934/cpaa.2008.7.355
[3]

Delia Schiera. Existence and non-existence results for variational higher order elliptic systems. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5145-5161. doi: 10.3934/dcds.2018227
[4]

Rubén Figueroa, Rodrigo López Pouso, Jorge Rodríguez–López. Existence and multiplicity results for second-order discontinuous problems via non-ordered lower and upper solutions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 617-633. doi: 10.3934/dcdsb.2019257
[5]

Yingshu Lü. Symmetry and non-existence of solutions to an integral system. Communications on Pure and Applied Analysis, 2018, 17 (3) : 807-821. doi: 10.3934/cpaa.2018041
[6]

Imene Bendahou, Zied Khemiri, Fethi Mahmoudi. On spikes concentrating on lines for a Neumann superlinear Ambrosetti-Prodi type problem. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2367-2391. doi: 10.3934/dcds.2020118
[7]

Elisa Sovrano. Ambrosetti-Prodi type result to a Neumann problem via a topological approach. Discrete and Continuous Dynamical Systems - S, 2018, 11 (2) : 345-355. doi: 10.3934/dcdss.2018019
[8]

Weiwei Ao, Mengdie Fu, Chao Liu. Boundary concentrations on segments for a Neumann Ambrosetti-Prodi problem. Discrete and Continuous Dynamical Systems, 2022, 42 (10) : 4991-5015. doi: 10.3934/dcds.2022083
[9]

Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209
[10]

J. F. Toland. Non-existence of global energy minimisers in Stokes waves problems. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 3211-3217. doi: 10.3934/dcds.2014.34.3211
[11]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125
[12]

Shu-Yu Hsu. Non-existence and behaviour at infinity of solutions of some elliptic equations. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 769-786. doi: 10.3934/dcds.2004.10.769
[13]

Fuqin Sun, Mingxin Wang. Non-existence of global solutions for nonlinear strongly damped hyperbolic systems. Discrete and Continuous Dynamical Systems, 2005, 12 (5) : 949-958. doi: 10.3934/dcds.2005.12.949
[14]

Luis Barreira, Davor Dragičević, Claudia Valls. From one-sided dichotomies to two-sided dichotomies. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2817-2844. doi: 10.3934/dcds.2015.35.2817
[15]

Eun Bee Choi, Yun-Ho Kim. Existence of nontrivial solutions for equations of $p(x)$-Laplace type without Ambrosetti and Rabinowitz condition. Conference Publications, 2015, 2015 (special) : 276-286. doi: 10.3934/proc.2015.0276
[16]

Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409
[17]

Yulong Li, Aleksey S. Telyakovskiy, Emine Çelik. Analysis of one-sided 1-D fractional diffusion operator. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1673-1690. doi: 10.3934/cpaa.2022039
[18]

Lauren M. M. Bonaldo, Elard J. Hurtado, Olímpio H. Miyagaki. Multiplicity results for elliptic problems involving nonlocal integrodifferential operators without Ambrosetti-Rabinowitz condition. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3329-3353. doi: 10.3934/dcds.2022017
[19]

Jitsuro Sugie, Tadayuki Hara. Existence and non-existence of homoclinic trajectories of the Liénard system. Discrete and Continuous Dynamical Systems, 1996, 2 (2) : 237-254. doi: 10.3934/dcds.1996.2.237
[20]

Elias M. Guio, Ricardo Sa Earp. Existence and non-existence for a mean curvature equation in hyperbolic space. Communications on Pure and Applied Analysis, 2005, 4 (3) : 549-568. doi: 10.3934/cpaa.2005.4.549

 Impact Factor: 

Tools

Metrics

  • PDF downloads (56)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy