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Uniqueness of positive solutions for a class of fourth-order boundary value problems. (English) Zbl 1222.34027
Summary: The purpose of this paper is to investigate the existence and uniqueness of positive solutions for the following fourth-order boundary value problem: $$y^{(4)}(t) = f(t, y(t)),\quad t \in [0 ,1],\quad y(0) = y(1) = y'(0) = y'(1) = 0.$$ Moreover, under certain assumptions, we prove that the above boundary value problem has a unique symmetric positive solution. Finally, we present some examples and compare our results with the ones obtained in recent papers. Our analysis relies on a fixed-point theorem in partially ordered metric spaces.

34B18Positive solutions of nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
Full Text: DOI
[1] A. Cabada, J. A. Cid, and L. Sánchez, “Positivity and lower and upper solutions for fourth order boundary value problems,” Nonlinear Analysis, vol. 67, no. 5, pp. 1599-1612, 2007. · Zbl 1125.34010 · doi:10.1016/j.na.2006.08.002
[2] J. Ehme, P. W. Eloe, and J. Henderson, “Upper and lower solution methods for fully nonlinear boundary value problems,” Journal of Differential Equations, vol. 180, no. 1, pp. 51-64, 2002. · Zbl 1019.34015 · doi:10.1006/jdeq.2001.4056
[3] D. Franco, D. O’Regan, and J. Perán, “Fourth-order problems with nonlinear boundary conditions,” Journal of Computational and Applied Mathematics, vol. 174, no. 2, pp. 315-327, 2005. · Zbl 1068.34013 · doi:10.1016/j.cam.2004.04.013
[4] F. Minhós, T. Gyulov, and A. I. Santos, “Existence and location result for a fourth order boundary value problem,” Discrete and Continuous Dynamical System Suplement, vol. 2005, pp. 662-671, 2005. · Zbl 1157.34310
[5] Z. Bai, “The method of lower and upper solutions for a bending of an elastic beam equation,” Journal of Mathematical Analysis and Applications, vol. 248, no. 1, pp. 195-202, 2000. · Zbl 1016.34010 · doi:10.1006/jmaa.2000.6887
[6] Y. Guo and Y. Gao, “The method of upper and lower solutions for a Lidstone boundary value problem,” Czechoslovak Mathematical Journal, vol. 55, no. 3, pp. 639-652, 2005. · Zbl 1081.34019 · doi:10.1007/s10587-005-0051-8 · eudml:30974
[7] P. Habets and L. Sánchez, “A monotone method for fourth order boundary value problems involving a factorizable linear operator,” Portugaliae Mathematica, vol. 64, no. 3, pp. 255-279, 2007. · Zbl 1137.34315 · doi:10.4171/PM/1786
[8] J. Caballero, J. Harjani, and K. Sadarangani, “Existence and uniqueness of positive and non-decreasing solutions for a class of singular fractional boundary value problem,” Boundary Value Problems, vol. 2009, Article ID 421310, 2009. · Zbl 1182.34005 · doi:10.1155/2009/421310 · eudml:45610
[9] Z. Bai and H. Wang, “On positive solutions of some nonlinear fourth order beam equations,” Journal of Mathematical Analysis and Applications, vol. 270, no. 2, pp. 357-368, 2002. · Zbl 1006.34023 · doi:10.1016/S0022-247X(02)00071-9
[10] X. L. Liu and W. T. Li, “Existence and multiplicity of solutions for fourth-order boundary value problems with parameters,” Journal of Mathematical Analysis and Applications, vol. 327, no. 1, pp. 362-375, 2007. · Zbl 1135.34007 · doi:10.1016/j.mcm.2006.11.018
[11] J. R. L. Webb, G. Infante, and D. Franco, “Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions,” Proceedings of the Royal Society of Edinburgh, vol. 138, no. 2, pp. 427-446, 2008. · Zbl 1167.34004 · doi:10.1017/S0308210506001041
[12] C. P. Gupta, “Existence and uniqueness theorems for some fourth order fully quasilinear boundary value problems,” Applicable Analysis, vol. 36, no. 3-4, pp. 157-169, 1990. · Zbl 0713.34025 · doi:10.1080/00036819008839930
[13] G. E. Hernández and R. Manasevich, “Existence and multiplicity of solutions of a fourth order equation,” Applicable Analysis, vol. 54, no. 3-4, pp. 237-250, 1994. · Zbl 0836.34023 · doi:10.1080/00036819408840280
[14] P. Korman, “Uniqueness and exact multiplicity of solutions for a class of fourth-order semilinear problems,” Proceedings of the Royal Society of Edinburgh, vol. 134, no. 1, pp. 179-190, 2004. · Zbl 1060.34014 · doi:10.1017/S0308210500003140
[15] R. Ma, “Existence of positive solutions of a fourth-order boundary value problem,” Applied Mathematics and Computation, vol. 168, no. 2, pp. 1219-1231, 2005. · Zbl 1082.34023 · doi:10.1016/j.amc.2004.10.014
[16] B. P. Rynne, “Infinitely many solutions of superlinear fourth order boundary value problems,” Topological Methods in Nonlinear Analysis, vol. 19, no. 2, pp. 303-312, 2002. · Zbl 1017.34015
[17] A. Amini-Harandi and H. Emami, “A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations,” Nonlinear Analysis, vol. 72, no. 5, pp. 2238-2242, 2010. · Zbl 1197.54054 · doi:10.1016/j.na.2009.10.023
[18] J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223-239, 2005. · Zbl 1095.47013 · doi:10.1007/s11083-005-9018-5
[19] M. Pei and S. K. Chang, “Monotone iterative technique and symmetric positive solutions for a fourth-order boundary value problem,” Mathematical and Computer Modelling, vol. 51, no. 9-10, pp. 1260-1267, 2010. · Zbl 1206.65188 · doi:10.1016/j.mcm.2010.01.009