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Positive solutions of fourth-order boundary value problems with two parameters. (English) Zbl 1030.34016

Summary: Here, the existence of positive solutions are obtained for the fourth-order boundary value problem \[ u^{(4)}+ \beta u''-\alpha u=f(t, u),\;0<t<1,\quad u(0)=u(1)= u''(0)=u''(1)=0, \] where \(f:[0,1] \times\mathbb{R}^+ \to\mathbb{R}^+\) is continuous, \(\alpha,\beta \in\mathbb{R}\) and satisfy \(\beta<2 \pi^2\), \(\alpha\geq- \beta^2/4\), \(\alpha/\pi^4+\beta/ \pi^2<1\).

MSC:

34B08 Parameter dependent boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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References:

[1] Aftabizadeh, A. R., Existence and uniqueness theorems for fourth-order boundary value problems, J. Math. Anal. Appl., 116, 415-426 (1986) · Zbl 0634.34009
[2] Agarwal, R., On fourth-order boundary value problems arising in beam analysis, Differential Integral Equations, 2, 91-110 (1989) · Zbl 0715.34032
[3] Gupta, C. P., Existence and uniqueness theorems for a bending of an elastic beam equation, Appl. Anal., 26, 289-304 (1988) · Zbl 0611.34015
[4] Gupta, C. P., Existence and uniqueness results for the bending of an elastic beam equation at resonance, J. Math. Anal. Appl., 135, 208-225 (1988) · Zbl 0655.73001
[5] Yang, Y., Fourth-order two-point boundary value problems, Proc. Amer. Math. Soc., 104, 175-180 (1988) · Zbl 0671.34016
[6] Del Pino, M. A.; Manasevich, R. F., Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition, Proc. Amer. Math. Soc., 112, 81-86 (1991) · Zbl 0725.34020
[7] De Coster, C.; Fabry, C.; Munyamarere, F., Nonresonance conditions for fourth-order nonlinear boundary value problems, Internat. J. Math. Math. Sci., 17, 725-740 (1994) · Zbl 0810.34017
[8] Ma, R.; Wang, H., On the existence of positive solutions of fourth-order ordinary differential equations, Appl. Anal., 59, 225-231 (1995) · Zbl 0841.34019
[9] Ma, R.; Zhang, J.; Fu, S., 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
[10] 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
[11] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press New York · Zbl 0661.47045
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