On positive solutions of some nonlinear fourth-order beam equations. (English) Zbl 1006.34023

Summary: The existence, uniqueness and multiplicity of positive solutions to the following boundary value problems is considered \[ u^{(4)}(t)- \lambda f(t, u(t))= 0,\quad\text{for }0< t< 1,\quad u(0)= u(1)= u''(0)= u''(1)= 0, \] where \(\lambda> 0\) is a constant, \(f: [0,1]\times [0,+\infty)\to [0,+\infty)\) are continuous.


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
Full Text: DOI


[1] Agarwal, R.P., On fourth-order boundary value problems arising in beam analysis, Differential integral equations, 2, 91-110, (1989) · Zbl 0715.34032
[2] Bai, Z.B., 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] Ma, R.Y.; Wang, H.Y., On the existence of positive solutions of fourth-order ordinary differential equations, Appl. anal., 59, 225-231, (1995) · Zbl 0841.34019
[4] 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
[5] Yao, Q.L.; Bai, Z.B., Existence of solutions of B.V.P. for u(4)(t)−λh(t)f(u(t))=0, Chinese ann. math. ser. A, 20, 575-578, (1999), in Chinese · Zbl 0948.34502
[6] Liu, Z.L.; Li, F.Y., Multiple positive solutions of nonlinear two point boundary value problem, J. math. anal. appl., 203, 610-625, (1996) · Zbl 0878.34016
[7] G.B. Gustafson, K. Schmitt, Method of nonlinear analysis in the theory of differential equations, Lecture notes, University of Utah (1975)
[8] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press New York · Zbl 0661.47045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.