Fully nonlinear fourth-order equations with functional boundary conditions. (English) Zbl 1138.34008

The authors prove an existence and location result for a nonlinear fourth-order ordinary differential equation with functional boundary conditions. The approach relies on the method of upper-lower solutions and the use of Schauder’s fixed point theorem. An application to a human spine deformation model is also included.


34B15 Nonlinear boundary value problems for ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
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[1] Cabada, A.; Grossinho, M. R.; Minhós, F., On the solvability of some discontinuous third order nonlinear differential equations with two point boundary conditions, J. Math. Anal. Appl., 285, 1, 174-190 (2003) · Zbl 1048.34033
[2] Cabada, A.; Habets, P.; Pouso, R. L., Optimal existence conditions for \(φ\)-Laplacian equations with upper and lower solutions in the reversed order, J. Differential Equations, 166, 2, 385-401 (2000) · Zbl 0999.34011
[3] Cabada, A.; Minhós, F.; Santos, A. I., Solvability for a third order discontinuous fully equation with functional boundary conditions, J. Math. Anal. Appl., 322, 735-748 (2006) · Zbl 1104.34008
[4] Cabada, A.; Pouso, R. L., Existence result for the problem \((\varphi(u^\prime))^\prime = f(t, u, u^\prime)\) with nonlinear boundary conditions, Nonlinear Anal., 35, 221-231 (1999) · Zbl 0920.34029
[5] Chen, S.; Ni, W.; Wang, C., Positive solutions of fourth order ordinary differential equation with four-point boundary conditions, Appl. Math. Lett., 19, 161-168 (2006) · Zbl 1096.34009
[6] Franco, D.; O’Regan, D.; Perán, J., Fourth-order problems with nonlinear boundary conditions, J. Comput. Appl. Math., 174, 315-327 (2005) · Zbl 1068.34013
[7] Habets, P.; Pouso, R. P., Examples of the nonexistence of a solution in the presence of upper and lower solutions, ANZIAM J., 44, 4, 591-594 (2003) · Zbl 1048.34036
[8] Jiang, D.; Fan, M.; Wan, A., A monotone method for constructing extremal solutions to second-order periodic boundary value problems, J. Comput. Appl. Math., 136, 1-2, 189-197 (2001) · Zbl 0993.34011
[9] Li, F.; Zhang, Q.; Liang, Z., Existence and multiplicity of solutions of a kind of fourth-order boundary value problem, Nonlinear Anal., 62, 803-816 (2005) · Zbl 1076.34015
[10] Lloyd, N. G., Degree Theory (1978), Cambridge Univ. Press · Zbl 0367.47001
[11] Love, A., A Treatise on the Mathematical Theory of Elasticity (1944), Dover Books on Physics and Chemistry · Zbl 0063.03651
[12] Ma, T. F.; da Silva, J., Iterative solutions for a beam equation with nonlinear boundary conditions of third order, Appl. Math. Comput., 159, 11-18 (2004) · Zbl 1095.74018
[13] Minhós, F.; Gyulov, T.; Santos, A. I., Existence and location result for a fourth order boundary value problem, Proc. of the 5th International Conference on Dynamical Systems and Differential Equations. Proc. of the 5th International Conference on Dynamical Systems and Differential Equations, Discrete Contin. Dyn. Syst., 662-671 (2005) · Zbl 1157.34310
[14] Minhós, F.; Gyulov, T.; Santos, A. I., On an elastic beam fully equation with nonlinear boundary conditions, (Agarwal, R.; Perera, K., Proc. of the Conference on Differential & Difference Equations and Applications (2006), Hindawi Publishing Corporation), 805-814 · Zbl 1129.34011
[15] Noone, G.; Ang, W. T., The inferior boundary condition of a continuous cantilever beam model of the human spine, Aust. Phys. Engrg. Sci. Med., 19, 1, 26-30 (1996)
[16] Patwardhan, A.; Bunch, W.; Meade, K.; Vandeby, R.; Knight, G., A biomechanical analog of curve progression and orthotic stabilization in idiopathic scoliosis, J. Biomech., 19, 2, 103-117 (1986)
[17] Šenkyřík, M., Fourth order boundary value problems and nonlinear beams, Appl. Anal., 59, 15-25 (1995) · Zbl 0841.34023
[18] Yuji, L.; Weigao, G., Double positive solutions of fourth order nonlinear boundary value problems, Appl. Anal., 82, 369-380 (2003) · Zbl 1037.34017
[19] Wang, M.; Cabada, A.; Nieto, J. J., Monotone method for nonlinear second order periodic boundary value problems with Carathéodory functions, Ann. Polon. Math., 58, 3, 221-235 (1993) · Zbl 0789.34027
[20] Zhang, Q.; Chen, S.; Lü, J., Upper and lower solution method for fourth-order four-point boundary value problems, J. Comput. Appl. Math., 196, 387-393 (2006) · Zbl 1102.65084
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