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Existence and multiplicity of solutions of a kind of fourth-order boundary value problem. (English) Zbl 1076.34015
Summary: Here, existence and multiplicity results on the solutions are obtained for the fourth-order boundary value problem $$a^{(4)} (t)= f(t,u(t))$$ for all $$t\in [0, 1]$$ subject to $$u(0)= u(1)= u''(0)= u (1)= 0$$, where $$f$$ is continuous. The monotone operator theory and critical point theory are used to discuss this problem.

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations
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##### References:
  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  Davis, J.M.; Eloe, P.W.; Henderson, J., Triple positive solutions and dependence on higher order derivatives, J. math. anal. appl., 237, 710-720, (1999) · Zbl 0935.34020  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  Deimling, K., Nonlinear functional analysis, (1985), Springer Berlin, Heidelberg · Zbl 0559.47040  Graef, J.R.; Qian, C.; Yang, B., Multiple symmetric positive solutions of a class of boundary value problem for higher order ordinary differential equations, Proc. am. math. soc., 131, 577-585, (2003) · Zbl 1046.34037  D. Guo, Nonlinear Functional Analysis, second ed., Shandong Science & Technology Press, 2001 (in Chinese).  Li, Y., Positive solutions of fourth-order periodic boundary value problems, Nonlinear anal., 54, 1069-1078, (2003) · Zbl 1030.34025  Li, Y., Positive solutions of fourth-order boundary value problems with two parameters, J. math. anal. appl., 281, 477-484, (2003) · Zbl 1030.34016  Liu, B., Positive solutions of fourth-order two point boundary value problems, Appl. math. comput., 148, 407-420, (2004) · Zbl 1039.34018  P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65, American Mathematical Society, Providence, RI, 1986. · Zbl 0609.58002  Struwe, M., Variational methods: applications to nonlinear partial differential equations and Hamiltonian systems, (1996), Springer Berlin, Heidelberg · Zbl 0864.49001  Taylor, A.E.; Lay, D.C., Introduction to functional analysis, (1980), Wiley New York  Yao, Q., Positive solutions for eigenvalue problems of fourth-order elastic beam equations, Appl. math. lett., 17, 237-243, (2004) · Zbl 1072.34022  Zeidler, E., Nonlinear functional analysis and its applications, III: variational methods and optimization, (1985), Springer New York · Zbl 0583.47051  Zeidler, E., Nonlinear functional analysis and its applications, I: fixed-point theorems, (1986), Springer New York  Zeidler, E., Nonlinear functional analysis and its applications, II/B: nonlinear monotone operators, (1990), Springer New York  Zhang, B.; Liu, X., Existence of multiple symmetric positive solutions of higher order lidstone problems, J. math. anal. appl., 284, 672-689, (2003) · Zbl 1048.34054
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