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., (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 higher-order multi-point boundary value problem,, Result. Math., (2009), 77. Google Scholar

[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. 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, Ambrosetti-Prodi 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 three-point 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 fourth-order 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 higher-order multi-point boundary value problem,, Result. Math., (2009), 77. Google Scholar

[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. 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, Ambrosetti-Prodi 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 three-point regular boundary value problem,, Mathematica Bohemica, (1994), 113. Google Scholar

[1]

Xuewei Cui, Mei Yu. Non-existence of positive solutions for a higher order fractional equation. Discrete & Continuous Dynamical Systems - A, 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 & 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 & Continuous Dynamical Systems - A, 2018, 38 (10) : 5145-5161. doi: 10.3934/dcds.2018227
[4]

Elisa Sovrano. Ambrosetti-Prodi type result to a Neumann problem via a topological approach. Discrete & Continuous Dynamical Systems - S, 2018, 11 (2) : 345-355. doi: 10.3934/dcdss.2018019
[5]

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

Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209
[7]

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

Luis Barreira, Davor Dragičević, Claudia Valls. From one-sided dichotomies to two-sided dichotomies. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2817-2844. doi: 10.3934/dcds.2015.35.2817
[9]

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
[10]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125
[11]

Shu-Yu Hsu. Non-existence and behaviour at infinity of solutions of some elliptic equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (3) : 769-786. doi: 10.3934/dcds.2004.10.769
[12]

Fuqin Sun, Mingxin Wang. Non-existence of global solutions for nonlinear strongly damped hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 949-958. doi: 10.3934/dcds.2005.12.949
[13]

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

Li-Li Wan, Chun-Lei Tang. Existence and multiplicity of homoclinic orbits for second order Hamiltonian systems without (AR) condition. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 255-271. doi: 10.3934/dcdsb.2011.15.255
[15]

Jiafeng Liao, Peng Zhang, Jiu Liu, Chunlei Tang. Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1959-1974. doi: 10.3934/dcdss.2016080
[16]

Jitsuro Sugie, Tadayuki Hara. Existence and non-existence of homoclinic trajectories of the Liénard system. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 237-254. doi: 10.3934/dcds.1996.2.237
[17]

Elias M. Guio, Ricardo Sa Earp. Existence and non-existence for a mean curvature equation in hyperbolic space. Communications on Pure & Applied Analysis, 2005, 4 (3) : 549-568. doi: 10.3934/cpaa.2005.4.549
[18]

Bilgesu A. Bilgin, Varga K. Kalantarov. Non-existence of global solutions to nonlinear wave equations with positive initial energy. Communications on Pure & Applied Analysis, 2018, 17 (3) : 987-999. doi: 10.3934/cpaa.2018048
[19]

Jérôme Coville, Nicolas Dirr, Stephan Luckhaus. Non-existence of positive stationary solutions for a class of semi-linear PDEs with random coefficients. Networks & Heterogeneous Media, 2010, 5 (4) : 745-763. doi: 10.3934/nhm.2010.5.745
[20]

Luca Rossi. Non-existence of positive solutions of fully nonlinear elliptic equations in unbounded domains. Communications on Pure & Applied Analysis, 2008, 7 (1) : 125-141. doi: 10.3934/cpaa.2008.7.125

 Impact Factor: 

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences