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Positive solutions of a fourth order boundary value problem. (English) Zbl 1042.34051
Summary: The existence of positive solutions of the nonlinear fourth-order problem \[ u^{(4)}(x)=\lambda a(x) f(u(x)),\quad u(0)= u'(0)= u'(1)= u'''(1)= 0, \] is studied, where \(a:[0,1]\to\mathbb{R}\) may change sign, \(f(0)> 0\), and \(\lambda> 0\) is sufficiently small. Our approach is based on the Leray-Schauder fixed-point theorem.
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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[1] Gupta, C. P., Existence and uniqueness theorem for the bending of an elastic beam equation, Applicable Anal., 1988,26(4):289–304. · Zbl 0611.34015 · doi:10.1080/00036818808839715
[2] Hai, D. D., Positive solutions to a class of elliptic boundary value problems, J. Math. Anal. Appl., 1998,227:195–199. · Zbl 0915.35043 · doi:10.1006/jmaa.1998.6095
[3] Castaneda Nelson, Ma Ruyun, Positive solutions to a second order threepoint boundary value problem, Applicable Anal., 2000,76(3–4):231–239. · Zbl 1031.34026 · doi:10.1080/00036810008840879
[4] Dunninger, D. R., Multiplicity of positive solutions for a nonlinear fourth order equation, Annales Polonici Mathematici, 2001,77(2):161–168. · Zbl 0989.34014 · doi:10.4064/ap77-2-3
[5] Graef, J. R., Yong, Bo., Positive solutions of a boundary value problem for fourth order nonlinear differential equations, Proceedings of Dynamic Systems and Applications, 2000,3:217–224. · Zbl 0998.34021
[6] Ma Ruyun, Wang Haiyan, On the existence of positive solutions of fourth-order ordinary differential equations, Applicable Anal., 1995,59:225–231. · Zbl 0841.34019 · doi:10.1080/00036819508840401
[7] Dell Pino, M. R., Manásevich, R. F., Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition, Proc. Amer. Math. Soc., 1991,112(1):81–86. · Zbl 0725.34020 · doi:10.2307/2048482
[8] Zhang Bingen, Kong Lingju, Positive solutions of fourth-order singular boundary value problems, Nonolinear Studies, 2000,7(1):70–77 · Zbl 1017.34016
[9] Jain, P. K., Functional Analysis, University of Delhi: Narosa Publishing House, 1998. · Zbl 0930.00093
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