# 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
##### Keywords:
existence; multiple solutions; fourth-order BVP
Full Text:
##### References:
  Aftabizadeh, A.R., Existence and uniqueness theorems for fourth-order boundary value problems, J. math. anal. appl., 116, 415-426, (1986) · Zbl 0634.34009  Agarwal, R.P., On fourth order boundary value problems arising in beam analysis, Differential integral equations, 2, 91-110, (1989) · Zbl 0715.34032  Agarwal, R.P.; Wilson, S.J., On a fourth order boundary value problem, Util. math., 26, 297-310, (1984) · Zbl 0591.34016  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  Anello, G., A multiplicity theorem for critical points of functionals on reflexive Banach spaces, Arch. math., 82, 172-179, (2004) · Zbl 1071.35039  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  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  Bonanno, G., Some remarks on a three critical points theorem, Nonlinear anal. TMA, 54, 651-665, (2003) · Zbl 1031.49006  Chen, W.Y., A decomposition problem for operators, Xuebao of dongbei renmin university, 1, 95-98, (1957), (in Chinese)  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  Guo, D., Nonlinear functional analysis, (2001), Shandong Science and Technology Press, (in Chinese)  Gupta, C.P., Existence and uniqueness theorems for a bending of an elastic beam equation, Appl. anal., 26, 289-304, (1988) · Zbl 0611.34015  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  Liu, B., Positive solutions of fourth-order two point boundary value problems, Appl. math. comput., 148, 407-500, (2004) · Zbl 1039.34018  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  Li, Y., Positive solutions of fourth-order boundary value problems with two parameters, J. math. anal. appl., 281, 477-484, (2003) · Zbl 1030.34016  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  Li, Y., Positive solutions of fourth-order periodic boundary value problems, Nonlinear anal. TMA, 54, 1069-1078, (2003) · Zbl 1030.34025  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)  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  X.L. Liu, W.T. Li, Positive solutions of the nonlinear fourth-order beam equation with three parameters (II), Dynam. Systems Appl., in press  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  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  Ricceri, B., On a three critical points theorem, Arch. math., 75, 220-226, (2000) · Zbl 0979.35040  Ricceri, B., A general variational principle and some of its applications, J. comput. appl. math., 113, 401-410, (2000) · Zbl 0946.49001  Willem, M., Minimax theorems, (1996), Birkhäuser Boston · Zbl 0856.49001  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  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  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.