Existence of positive solutions of a fourth-order boundary value problem. (English) Zbl 1082.34023

Summary: We consider the fourth-order boundary value problem \[ u''''= f(t,u,u''),\;0<t<1 \quad u(0)=u(1)=u''(0)=u''(1)=0, \] where \(f(t,u,p)= au-bp+o(|(u,p)|)\) near \((0,0)\), and \(f(t,u,p)=cu-dp+o(|(u,p)|)\) near \(\infty\). We give conditions on the constants \(a,b,c,d\) that guarantee the existence of positive solutions. The proof of our main result is based upon global bifurcation techniques.


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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[1] Amann, H., Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev., 18, 4, 620-709 (1976) · Zbl 0345.47044
[2] Ambrosetti, A.; Hess, P., Positive solutions of asymptotically linear elliptic eigenvalue problems, J. Math. Anal. Appl., 73, 2, 411-422 (1980) · Zbl 0433.35026
[3] Bai, Z.; Wang, H., On positive solutions of some nonlinear fourth-order beam equations, J. Math. Anal. Appl., 270, 2, 357-368 (2002) · Zbl 1006.34023
[4] Dancer, E. N., Global solution branches for positive mappings, Arch. Rat. Mech. Anal., 52, 181-192 (1973) · Zbl 0275.47043
[5] Del, M. R., Pino and R.F. Manásevich, Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition, Proc. Am. Math. Soc., 112, 1, 81-86 (1991) · Zbl 0725.34020
[6] Erbe, L. H.; Wang, H., On the existence of positive solutions of ordinary differential equations, Proc. Am. Math. Soc., 120, 3, 743-748 (1994) · Zbl 0802.34018
[7] Gupta, C. P., Existence and uniqueness theorems for the bending of an elastic beam equation, Applicable Anal., 26, 4, 289-304 (1988) · Zbl 0611.34015
[8] Liu, B., Positive solutions of fourth order two-point boundary value problems, Appl. Math. Comput., 148, 407-420 (2004) · Zbl 1039.34018
[9] Liu, Z.; Li, F., Multiple positive solutions of nonlinear two-point boundary value problems, J. Math. Anal. Appl., 203, 3, 610-625 (1996) · Zbl 0878.34016
[10] Ma, R.; Wang, H., On the existence of positive solutions of fourth-order ordinary differential equations, Appl. Anal., 59, 1-4, 225-231 (1995) · Zbl 0841.34019
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