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Two-parameter nonresonance condition for the existence of fourth-order boundary value problems. (English) Zbl 1071.34016
Summary: We discuss the existence of the fourth-order boundary value problem $u^{(4)}= f(t,u,u''),\quad 0< t< 1,\quad u(0)= u(1)= u''(0)= u''(1)= 0,$ where $$f: [0,1]\times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$$ is continuous, and partly solve the Del Pino and Manasevich’s conjecture on the nonresonance condition involving the two-parameter linear eigenvalue problem. We present a two-parameter nonresonance condition described by circle, too.

##### MSC:
 34B15 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] Yang, Y., Fourth-order two-point boundary value problems, Proc. amer. math. soc., 104, 175-180, (1988) · Zbl 0671.34016 [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] Agarwal, R., On fourth-order boundary value problems arising in beam analysis, Differential integral equations, 2, 91-110, (1989) · Zbl 0715.34032 [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] Li, Y., Positive solutions of fourth-order boundary value problems with two parameters, J. math. anal. appl., 281, 477-484, (2003) · Zbl 1030.34016 [12] Li, Y., Existence and method of lower and upper solutions for fourth-order nonlinear boundary value problems, Acta math. sci. ser. A, 23, 245-252, (2003), (in Chinese) · Zbl 1040.34025
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