Liu, Yansheng Multiple positive solutions of nonlinear singular boundary value problem for fourth-order equations. (English) Zbl 1073.34018 Appl. Math. Lett. 17, No. 7, 747-757 (2004). The existence of at least two positive solutions of the following fourth-order boundary value problem \[ x^{(4)}(t)=f(t,x(t),x''(t)),\quad 0<t<1;\qquad x(0)=x(1)=x''(0)=x''(1)=0, \] is discussed. Here, \(f\) may be singular at \(t=0\), \(t=1\), \(x=0\) and \(x''=0\). The proofs are based on Krasnoselskii’s fixed-point theorem on cone expansion and compression, a priori estimates on positive solutions and some properties of the spectral radius of a positive completely continuous operator. Reviewer: Mirosława Zima (Rzeszow) Cited in 13 Documents MSC: 34B16 Singular nonlinear boundary value problems for ordinary differential equations 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations Keywords:singularity; cone; positive solution; fourth-order equation PDF BibTeX XML Cite \textit{Y. Liu}, Appl. Math. Lett. 17, No. 7, 747--757 (2004; Zbl 1073.34018) Full Text: DOI References: [1] Henderson, J.; Wang, H., Positive solutions for nonlinear eigenvalue problems, J. Math. Anal. Appl., 208, 252-259 (1997) · Zbl 0876.34023 [2] Agarwal, R. P.; Wong, F.; Lian, W., Positive solutions for nonlinear singular boundary value problems, Appl. Math. Lett., 12, 2, 115-120 (1999) · Zbl 0934.34015 [3] O’Regan, D., Solvability of some fourth (and higher) order singular boundary value problems, J. Math. Anal. Appl., 161, 78-116 (1991) · Zbl 0795.34018 [4] Zhang, B.; Kong, L., Positive solutions of fourth order singular boundary value problems, Nonlinear Studies, 7, 70-77 (2000) · Zbl 1017.34016 [5] Ma, R.; Wang, H., On the existence of positive solutions of fourth order differential equations, Appl. Anal., 59, 225-231 (1995) · Zbl 0841.34019 [6] Nussbaum, R. D., Eigenvectors of nonlinear positive operators and the linear Krein-Rutman theorem, (Fixed Point Theory. Fixed Point Theory, LNM, 886 (1980), Springer-Verlag), 309-330 [7] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press New York · Zbl 0661.47045 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.