Positive solutions of nonlinear beam equations with time and space singularities. (English) Zbl 1219.34033

Existence and multiplicity of positive solutions of a nonlinear fourth-order ordinary differential equation under linear two-point boundary conditions is proved under appropriate conditions on the nonlinear term \(f=f(t,u,u')\), which may be singular on time and/or space. The case in which the nonlinearity \(f\) does not depend on \(u'\) is treated separately. Height functions are introduced in order to set convenient conditions that allow the use of the Guo-Krasnosel’skii fixed point theorem of cone expansion-compression type.


34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
Full Text: DOI


[1] Agarwal, R.P.; O’Regan, D., Nonlinear superlinear singular and nonsingular second order boundary value problems, J. differential equations, 143, 60-95, (1998) · Zbl 0902.34015
[2] Agarwal, R.P.; O’Regan, D., Twin solutions to singular boundary value problems, Proc. amer. math. soc., 128, 2085-2094, (2000) · Zbl 0946.34020
[3] Agarwal, R.P.; O’Regan, D., Multiplicity results for singular conjugate, focal and \((n, p)\) problems, J. differential equations, 170, 142-156, (2001) · Zbl 0978.34018
[4] Agarwal, R.P.; O’Regan, D., Existence theory for single and multiple solutions to singular positone boundary value problems, J. differential equations, 175, 393-414, (2001) · Zbl 0999.34018
[5] Curtain, R.F.; Pritchard, A.J., Functional analysis in modern applied mathematics, (1977), Academic Press London, New York, San Francisco · Zbl 0448.46002
[6] Dalmasso, R., Uniqueness of positive solutions for some nonlinear fourth order equations, J. math. anal. appl., 201, 152-168, (1996) · Zbl 0856.34024
[7] Eloe, P.W.; Henderson, J., Singular nonlinear \((k, n - k)\) conjugate boundary value problems, J. differential equations, 133, 136-151, (1997) · Zbl 0870.34031
[8] Gupta, G.P., Existence and uniqueness theorems for the bending of an elastic beam equation, Appl. anal., 26, 289-304, (1988) · Zbl 0611.34015
[9] Gupta, G.P., A nonlinear boundary value problem associated with the static equilibrium of an elastic beam supported by sliding clamps, Int. J. math. math. sci., 12, 697-711, (1989) · Zbl 0685.34016
[10] Lan, K.Q., Multiple positive solutions of conjugate boundary value problems with singularities, Appl. math. comput., 147, 461-474, (2004) · Zbl 1054.34032
[11] Lin, X.; Jiang, D.; Li, X., Existence and uniqueness of solutions for singular fourth-order boundary value problems, J. comput. appl. math., 196, 155-161, (2006) · Zbl 1107.34307
[12] Liu, Z.; Ume, J.S.; Kang, S.M., Positive solutions of a singular nonlinear third order two-point boundary value problem, J. math. anal. appl., 326, 589-601, (2007) · Zbl 1111.34022
[13] 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
[14] Wei, Z., A class of fourth order singular boundary value problems, Appl. math. comput., 153, 865-884, (2004) · Zbl 1057.34006
[15] Yang, B., Positive solutions for the beam equation under certain boundary conditions, Electron. J. differential equations, 2005, 78, 1-8, (2005) · Zbl 1075.34025
[16] Yao, Q., Positive solutions to a class of elastic beam equations with semipositone nonlinearity, Ann. polon. math., 97, 35-50, (2010) · Zbl 1190.34023
[17] Yao, Q., Local existence of multiple positive solutions to a singular cantilever beam equation, J. math. anal. appl., 363, 138-154, (2010) · Zbl 1191.34031
[18] Zhang, X., Existence and iteration of monotone positive solutions for an elastic beam equation with a corner, Nonlinear anal. real world appl., 10, 2097-2103, (2009) · Zbl 1163.74478
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.