## 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
Full Text:

### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.