Existence Theory for Single Positive Solution to Fourth-Order Boundary Value Problems

By fixed point theorem of a mixed monotone operator, we study boundary value problems to nonlinear singular fourth-order differential equations, and provide sufficient conditions for the existence and uniqueness of positive solution. The nonlinear term in the differential equation may be singular.

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Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

He, Y. (2014) Existence Theory for Single Positive Solution to Fourth-Order Boundary Value Problems. Advances in Pure Mathematics, 4, 480-486. doi: 10.4236/apm.2014.48053.

 [1] Jiang, D.Q., Liu, H. and Xu, X. (2005) Nonresonant Singular Fourth-Order Boundary Value Problems. Applied Mathematics Letters, 18, 69-75. http://dx.doi.org/10.1016/j.aml.2003.05.016 [2] Jiang, D.Q. (2000) Multiple Positive Solutions to Singular Boundary Value Problems for Superlinear Higher Order ODEs. Computers Mathematics with Applications, 40, 249-259. http://dx.doi.org/10.1016/S0898-1221(00)00158-9 [3] Liu, Y. (2003) Structure of a Class of Singular Boundary Value Problem with Superlinear Effect. Journal of Mathematical Analysis and Applications, 284, 64-75. http://dx.doi.org/10.1016/S0022-247X(03)00214-2 [4] Kong, L. and Wang, J. (2001) The Green’s Function for (k, n ? k) Conjugate Boundary Value Problems and Its Applications. Journal of Mathematical Analysis and Applications, 255, 404-422. http://dx.doi.org/10.1006/jmaa.2000.7158 [5] Zill, D.G. and Cullen, M.R. (2001) Differential Equations with Boundary-Value Problems. 5th Edition, Brooks Cole, Belmont. [6] Liu, Y. (2004) Multiple Positive Solutions of Nonlinear Singular Boundary Value Problem for Fourth Order Equations. Applied Mathematics Letters, 17, 747-757. http://dx.doi.org/10.1016/j.aml.2004.06.001 [7] Guo, D. (2000) The Order Methods in Nonlinear Analysis. Shandong Technology and Science Press, Jinan.