×

Local existence of multiple positive solutions to a singular cantilever beam equation. (English) Zbl 1191.34031

The authors study existence and multiplicity of positive solutions to the fourth-order problem
\[ u^{(4)}=q(t)f(t, u(t), u'(t)),\quad u(0)=u'(0)=u''(1)=u'''(1)=0, \]
where \(q:(0,1)\rightarrow [0, \infty)\) is continuous with \(q(t)>0\) in \((0,1)\), \(f\) is allowed to be singular at \(t=0,~t=1\),  \(u=0,~ v=0\). The main tool is the Guo-Krasnoselskii fixed point theorem in cones.
Reviewer: Ruyun Ma (Lanzhou)

MSC:

34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
47N20 Applications of operator theory to differential and integral equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Agarwal, R.P., Boundary value problems for higher order differential equations, (1986), World Scientific Singapore · Zbl 0598.65062
[2] Aftabizadeh, A.R., Existence and uniqueness theorems for fourth-order boundary value problems, J. math. anal. appl., 116, 415-426, (1986) · Zbl 0634.34009
[3] Gupta, C.P., Existence and uniqueness theorems for the bending of an elastic beam equation, Appl. anal., 26, 289-304, (1988) · Zbl 0611.34015
[4] Agarwal, R.P.; Henderson, J., Supperlinear and sublinear focal boundary value problems, Appl. anal., 60, 189-200, (1996) · Zbl 0872.34006
[5] Wong, P.J.; Agarwal, R.P., On two-point right focal eigenvalue problems, Z. anal. anwend., 17, 691-713, (1998) · Zbl 0915.34019
[6] Agarwal, R.P.; O’Regan, D., Right focal singular boundary value problems, Z. angew. math. mech., 79, 363-373, (1999) · Zbl 0930.34013
[7] Agarwal, R.P.; O’Regan, D.; Lakshmikantham, V., Singular \((p, n - p)\) focal and \((n, p)\) higher order boundary value problems, Nonlinear anal., 42, 215-228, (2000) · Zbl 0977.34017
[8] Agarwal, R.P.; O’Regan, D., Twin solutions to singular boundary value problems, Proc. amer. math. soc., 128, 2085-2094, (2000) · Zbl 0946.34020
[9] Agarwal, R.P., Multiplicity results for singular conjugate, focal and \((n, p)\) problems, J. differential equations, 170, 142-156, (2001) · Zbl 0978.34018
[10] Ma, R., Multiple positive solutions for a semipositone fourth-order boundary value problem, Hiroshima math. J., 33, 217-227, (2003) · Zbl 1048.34048
[11] Yao, Q., Monotonically iterative method of nonlinear cantilever beam equations, Appl. math. comput., 205, 432-437, (2008) · Zbl 1154.74021
[12] Yao, Q., Existence and multiplicity of positive solutions to a class of nonlinear cantilever beam equations, J. systems sci. math. sci., 29, 63-69, (2009), (in Chinese) · Zbl 1199.34123
[13] Agarwal, R.P.; Wong, F.H., Existence of positive solutions for non-positive higher-order BVPs, J. comput. appl. math., 88, 3-14, (1998) · Zbl 0993.34017
[14] Agarwal, R.P.; Wong, F.H., Existence of solutions to \((k, n - k - 2)\) boundary value problems, Appl. math. comput., 104, 33-50, (1999) · Zbl 0929.34020
[15] Agarwal, R.P.; O’Regan, D., Twin solutions to singular Dirichlet problems, J. math. anal. appl., 240, 433-445, (1999) · Zbl 0946.34022
[16] Yao, Q.; Bai, Z., Existence of positive solutions of BVP for \(u^{(4)}(t) - \lambda h(t) f(u(t)) = 0\), Chinese ann. math. ser. A, 20, 575-578, (1999), (in Chinese) · Zbl 0948.34502
[17] Wong, P.J.Y.; Agarwal, R.P., Multiple solutions for a system of \((n_i, p_i)\) boundary value problems, Z. anal. anwend., 19, 511-528, (2000) · Zbl 1160.34313
[18] Anderson, D.R.; Davis, J.M., Multiple solutions and eigenvalues for third-order right focal boundary value problems, J. math. anal. appl., 267, 135-157, (2002) · Zbl 1003.34021
[19] Yao, Q., The existence and multiplicity of positive solutions for a third-order three-point boundary value problem, Acta math. appl. sin. engl. ser., 19, 117-122, (2003) · Zbl 1048.34031
[20] Torres, P.J., Existence of one-signed periodic solutions of some second-order differential equations via a Krasnoselskii fixed point theorem, J. differential equations, 190, 643-662, (2003) · Zbl 1032.34040
[21] Yao, Q., Positive solutions for eigenvalue problems of fourth-order elastic beam equations, Appl. math. lett., 17, 237-243, (2004) · Zbl 1072.34022
[22] Lan, K.Q., Multiple positive solutions of conjugate boundary value problems with singularities, Appl. math. comput., 147, 461-474, (2004) · Zbl 1054.34032
[23] Yao, Q., Existence of n solutions and/or positive solutions to a semipositone elastic beam equation, Nonlinear anal., 66, 138-150, (2007) · Zbl 1113.34013
[24] 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
[25] Webb, J.R.L.; Infante, G.; Franco, D., Positive solutions of nonlinear fourth-order boundary value problems with local and nonlocal boundary conditions, Proc. roy. soc. Edinburgh sect. A, 138, 427-446, (2008) · Zbl 1167.34004
[26] Yao, Q., Positive solutions of nonlinear elastic beam equation rigidly fastened on the left and simply supported on the right, Nonlinear anal., 69, 1570-1580, (2008) · Zbl 1217.34039
[27] Yao, Q., Existence and multiplicity of positive solutions to a singular elastic beam equation rigidly fixed at both ends, Nonlinear anal., 69, 2683-2694, (2008) · Zbl 1157.34018
[28] Yao, Q., Positive solution of singular third-order three-point boundary value problems, J. math. anal. appl., 354, 207-212, (2009) · Zbl 1169.34314
[29] 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
[30] Stanek, S., Positive solutions of singular positone Dirichlet boundary value problems, Math. comput. modelling, 33, 341-351, (2001) · Zbl 0996.34019
[31] Yao, Q., Positive solution of a class of singular sublinear two-point boundary value problems, Acta math. appl. sin., 24, 522-526, (2001), (in Chinese) · Zbl 0998.34017
[32] Wei, Z., A class of fourth order singular boundary value problems, Appl. math. comput., 153, 865-884, (2004) · Zbl 1057.34006
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.