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Constructive proof of existence for a class of fourth-order nonlinear BVPs. (English) Zbl 1189.34038
Summary: A new existence proof of solutions for a class of fourth-order nonlinear boundary value problems is proposed. The proof of the main results is based on the reproducing kernel theorem. It is worthwhile to point out that the method presented in this paper can be applied for the existence proof of diverse kinds of boundary conditions.

34B15Nonlinear boundary value problems for ODE
Full Text: DOI
[1] 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 · doi:10.1016/S0022-247X(02)00071-9
[2] Li, F.; Zhang, Q.; Liang, Z.: Existence and multiplicity of solutions of a kind of fourth-order boundary value problem, Nonlinear anal. TMA 62, 803-816 (2005) · Zbl 1076.34015 · doi:10.1016/j.na.2005.03.054
[3] Liu, B.: Positive solutions of fourth-order two point boundary value problems, Appl. math. Comput. 148, 407-420 (2004) · Zbl 1039.34018 · doi:10.1016/S0096-3003(02)00857-3
[4] Han, Guodong; Xu, Zongben: Multiple solutions of some nonlinear fourth-order beam equations, Nonlinear anal. TMA 68, 3646-3656 (2008) · Zbl 1145.34008 · doi:10.1016/j.na.2007.04.007
[5] Carri√£o, P. C.; Faria, L. F. O.; Miyagaki, O. H.: Periodic solutions for extended Fisher--Kolmogorov and Swift-Hohenberg equations by truncature techniques, Nonlinear analysis TMA 67, 3076-3083 (2007) · Zbl 1128.34026 · doi:10.1016/j.na.2006.09.061
[6] Bai, Zhanbing: The upper and lower solution method for some fourth-order boundary value problems, Nonlinear analysis TMA 67, 1704-1709 (2007) · Zbl 1122.34010 · doi:10.1016/j.na.2006.08.009
[7] Feng, H.: Existence and uniqueness of solutions for a fourth-order boundary value problem, Nonlinear analysis TMA (2008)
[8] Huanmin Yao, The research of algorithms for some singular differential equations of higher even-order, Ph.D. Thesis, Department of Mathematics, Harbin Institute of Technology, 2008
[9] Du, Juan; Cui, Minggen: Constructive approximation of solution for fourth-order nonlinear boundary value problems, Math. methods appl. Sci. 32, 723-737 (2009) · Zbl 1170.34015 · doi:10.1002/mma.1064