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A class of fourth order singular boundary value problems. (English) Zbl 1057.34006

The author deals with the singular boundary value problem for the fourth-order ordinary differential equation \[ \begin{gathered} x^{(4)}(t)= f(t,x(t),-x''(t)), \qquad t\in (0,1),\\ x(0)= x(1)= 0, \quad ax''(0)- bx'''(0)= 0, \quad cx''(1)+ dx'''(1)= 0. \end{gathered} \tag{1} \] The author presents a sufficient and necessary condition for the existence of \(C^2([0,1])\) as well as \(C^3([0,1])\) positive solutions for (1) by constructing lower and upper solutions.

MSC:

34B16 Singular nonlinear 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] O’Regan, D., Theory of singular boundary value problems, (1994), World Scientific Press Singapore · Zbl 0808.34022
[2] Taliaferro, S.D., A nonlinear singular boundary value problem, Nonlinear anal. TMA, 3, 897-904, (1979) · Zbl 0421.34021
[3] Zhang, Y., Positive solutions of singular sublinear emden – fowler boundary value problems, J. math. anal. appl., 185, 215-222, (1994) · Zbl 0823.34030
[4] Schroder, J., Fourth order two-point boundary value problems; estimates by two-sided bounds, Nonlinear anal. TMA, 8, 2, 107-114, (1984) · Zbl 0533.34019
[5] 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
[6] Aftabizadeh, A.R., Existence and uniqueness theorems for fourth order boundary value problems, J. math. anal. appl., 116, 415-426, (1986) · Zbl 0634.34009
[7] Usmani, R.A., A uniqueness theorem for a boundary value problem, Proc. am. math. soc., 77, 329-335, (1979) · Zbl 0424.34019
[8] 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
[9] O’Regan, D., Solvability of some fourth (and higher) order singular boundary value problems, J. math. anal. appl., 161, 78-116, (1991) · Zbl 0795.34018
[10] Wei, Z., Positive solutions of singular boundary value problems of fourth order differential equations, Math. sin., 42, 4, 715-722, (1999), (in Chinese) · Zbl 1022.34018
[11] Wei, Z., Positive solutions of singular boundary value problems of negative exponent Emden-Fowler equations, Acta math. sin., 41, 3, 655-662, (1998), (in Chinese) · Zbl 1027.34024
[12] Hartman, P., Ordinary differential equations, (1982), Birkhauser Boston, MA · Zbl 0125.32102
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