×

zbMATH — the first resource for mathematics

Existence and multiplicity of solutions for fourth-order boundary value problems with parameters. (English) Zbl 1109.34015
Summary: We study the existence and multiplicity of the solutions for the fourth-order boundary value problem \[ u^{(4)}(t)+\eta u''(t)-\zeta u(t) =\lambda f\bigl(t,u(t) \bigr),\;0<t<1,\;u(0)=u(1)=u''(0)=u''(1)=0,\tag{BVP} \] where \(f:[0,1]\times\mathbb{R} \times\mathbb{R}\) is continuous, \(\zeta,\eta \in\mathbb{R}\) and \(\lambda\in\mathbb{R}^+\) are parameters. By means of the idea of the decomposition of operators, and the critical point theory, we obtain that if the pair \((\eta,\zeta)\) is on the curve \(\zeta=-\eta^2/4\) satisfying \(\eta<2\pi^2\), then the above BVP has at least one, two, three, and infinitely many solutions for \(\lambda\) being in different interval, respectively.

MSC:
34B15 Nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
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] Agarwal, R.P., On fourth order boundary value problems arising in beam analysis, Differential integral equations, 2, 91-110, (1989) · Zbl 0715.34032
[3] Agarwal, R.P.; Wilson, S.J., On a fourth order boundary value problem, Util. math., 26, 297-310, (1984) · Zbl 0591.34016
[4] Agarwal, R.P.; Chow, Y.M., Iterative methods for a fourth order boundary value problem, J. comput. appl. math., 10, 203-217, (1984) · Zbl 0541.65055
[5] Anello, G., A multiplicity theorem for critical points of functionals on reflexive Banach spaces, Arch. math., 82, 172-179, (2004) · Zbl 1071.35039
[6] Bai, Z., The method of lower and upper solutions for a bending of a elastic beam equation, J. math. anal. appl., 248, 195-202, (2000) · Zbl 1016.34010
[7] Bai, Z.; Wang, H., On positive solutions of some nonlinear fourth-order beam equations, J. math. anal. appl., 270, 357-368, (2002) · Zbl 1006.34023
[8] Bonanno, G., Some remarks on a three critical points theorem, Nonlinear anal. TMA, 54, 651-665, (2003) · Zbl 1031.49006
[9] Chen, W.Y., A decomposition problem for operators, Xuebao of dongbei renmin university, 1, 95-98, (1957), (in Chinese)
[10] Davis, J.M.; Henderson, J.; Wong, P.J.Y., General lidstone problems: multiplicity and symmetry of solutions, J. math. anal. appl., 251, 527-548, (2000) · Zbl 0966.34023
[11] Guo, D., Nonlinear functional analysis, (2001), Shandong Science and Technology Press, (in Chinese)
[12] Gupta, C.P., Existence and uniqueness theorems for a bending of an elastic beam equation, Appl. anal., 26, 289-304, (1988) · Zbl 0611.34015
[13] Hao, Z.C.; Debnath, L., On eigenvalue intervals and eigenfunctions of fourth-order singular boundary value problems, Appl. math. lett., 18, 543-553, (2005) · Zbl 1074.34079
[14] Liu, B., Positive solutions of fourth-order two point boundary value problems, Appl. math. comput., 148, 407-500, (2004) · Zbl 1039.34018
[15] Li, F.; Zhang, Q.; Liang, Z., Existence and multiplicity of solutions of a kind of fourth-order boundary value problem, Nonlinear anal., 62, 803-816, (2005) · Zbl 1076.34015
[16] Li, Y., Positive solutions of fourth-order boundary value problems with two parameters, J. math. anal. appl., 281, 477-484, (2003) · Zbl 1030.34016
[17] Li, Y., Two parameter nonresonance condition for the existence of fourth-order boundary value problems, J. math. anal. appl., 308, 121-138, (2005) · Zbl 1071.34016
[18] Li, Y., Positive solutions of fourth-order periodic boundary value problems, Nonlinear anal. TMA, 54, 1069-1078, (2003) · Zbl 1030.34025
[19] Liu, Y.S., Multiple positive solutions to fourth-order singular boundary value problems in abstract space, Electron. J. differential equations, 2004, 120, 1-13, (2004)
[20] Liu, X.L.; Li, W.T., Positive solutions of the nonlinear fourth-order beam equation with three parameters, J. math. anal. appl., 303, 150-163, (2005) · Zbl 1077.34027
[21] X.L. Liu, W.T. Li, Positive solutions of the nonlinear fourth-order beam equation with three parameters (II), Dynam. Systems Appl., in press
[22] Krasnosel’skii, M., Topological methods in the theory of nonlinear integral equations, (1956), Gostehizdat Moscow, (in Russian) (Engl. transl., Pergamon Press, New York, 1964) · Zbl 0070.33001
[23] 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
[24] Ricceri, B., On a three critical points theorem, Arch. math., 75, 220-226, (2000) · Zbl 0979.35040
[25] Ricceri, B., A general variational principle and some of its applications, J. comput. appl. math., 113, 401-410, (2000) · Zbl 0946.49001
[26] Willem, M., Minimax theorems, (1996), Birkhäuser Boston · Zbl 0856.49001
[27] Yao, Q., On the positive solutions of a nonlinear fourth-order boundary value problem with two parameters, Appl. anal., 83, 97-107, (2004) · Zbl 1051.34018
[28] Yao, Q., Existence, multiplicity and infinite solvability of positive solutions to a nonlinear fourth-order periodic boundary value problem, Nonlinear anal. TMA, 63, 237-246, (2005) · Zbl 1082.34025
[29] Yang, Y.S., Fourth-order two-point boundary value problems, Proc. amer. math. soc., 104, 1, 175-180, (1988) · Zbl 0671.34016
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.