Fourth-order problems with nonlinear boundary conditions. (English) Zbl 1068.34013

The authors consider monotone-type and nonmonotone boundary conditions for fourth-order ordinary differential equations.
In each of the two types of problems an existence result is established. The proofs (especially in monotone case) are technically complicated.


34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI


[1] Agarwal, R. P.; Meehan, M.; O’Regan, D., Fixed Point Theory and Applications (2001), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0960.54027
[2] Bai, Z., The method of lower and upper solutions for a bending of an elastic beam equation, J. Math. Anal. Appl., 248, 195-202 (2000) · Zbl 1016.34010
[3] Cabada, A., The method of lower and upper solutions for second, third, fourth and higher order boundary value problems, J. Math. Anal. Appl., 185, 302-320 (1994) · Zbl 0807.34023
[4] Cabada, A.; Pouso, R. L., Existence results for the problem \((\psi(u^\prime))^\prime = f(t, u, u^\prime)\) with nonlinear boundary conditions, Nonlinear Anal., 35, 221-231 (1999) · Zbl 0920.34029
[5] Drávek, P.; Holubová, G.; Matas, A.; Nec˘esal, P., Nonlinear models of suspension bridgesdiscussion of the results, Appl. Math., 6, 497-514 (2003)
[6] Ehme, J.; Eloe, P. W.; Henderson, J., Existence of solutions \(2\) nd order fully nonlinear generalized Sturm-Liouville boundary value problems, Math. Inequal. Appl., 4, 247-255 (2001) · Zbl 0994.34007
[7] Ehme, J.; Eloe, P. W.; Henderson, J., Upper and lower solution methods for fully nonlinear boundary value problems, J. Differential Equations, 180, 51-64 (2002) · Zbl 1019.34015
[8] Erbe, L. H., Nonlinear boundary value problems for second order differential equations, J. Differential Equations, 7, 459-472 (1970) · Zbl 0284.34017
[9] Fabry, Ch.; Habets, P., Upper and lower solutions for second-order boundary value problems with nonlinear boundary conditions, Nonlinear Anal., 10, 985-1007 (1986) · Zbl 0612.34015
[10] Franco, D.; Nieto, J. J.; O’Regan, D., Anti-periodic boundary value problem for nonlinear first order ordinary differential equations, Math. Inequal. Appl., 6, 477-485 (2003) · Zbl 1097.34015
[11] Franco, D.; O’Regan, D., A new upper and lower solutions approach for second order problems with nonlinear boundary conditions, Arch. Inequal. Appl., 1, 413-420 (2003) · Zbl 1098.34520
[12] Hartman, P., Ordinary Differential Equations (1964), Wiley: Wiley New York · Zbl 0125.32102
[14] Kiguradze, I. T.; Puz˘a, B., Some boundary-value problems for a system of ordinary differential equations, Differential Equations, 12, 1493-1500 (1977) · Zbl 0374.34013
[15] Ma, R. Y.; Zhang, J. H.; Fu, S. M., The method of lower and upper solutions for fourth-order two-point boundary value problems, J. Math. Anal. Appl., 215, 415-422 (1997) · Zbl 0892.34009
[16] Mawhin, J.; Schmitt, K., Upper and lower solutions and semilinear second order elliptic equations with nonlinear boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 97, 199-207 (1984) · Zbl 0573.35029
[17] Peletier, M. A., Generalized monotonicity from global minimization in fourth-order ordinary differential equations, Nonlinearity, 14, 1221-1238 (2001) · Zbl 1001.37053
[18] Rachunková, I.; Tomeček, J., On nonlinear boundary value problem for systems of differential equations with impulses, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., 41, 119-129 (2002) · Zbl 1069.34043
[19] Wong, P. J.Y.; Agarwal, R. P., Eigenvalues of Lidstone boundary value problems, Appl. Math. Comput., 104, 15-31 (1999) · Zbl 0933.65089
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