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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
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##### 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), BirkhĂ¤user 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
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