Eloe, Paul W.; Henderson, Johnny Inequalities based on a generalization of concavity. (English) Zbl 0868.34008 Proc. Am. Math. Soc. 125, No. 7, 2103-2107 (1997). Summary: The concept of concavity is generalized to functions \(y\) satisfying \(n\)th order differential inequalities, \((-1)^{n-k}y^{(n)}(t)\geq 0, 0\leq t\leq 1\), and homogeneous two-point boundary conditions, \(y(0)=\ldots =y^{(k-1)}(0)=0, y(1)=\ldots =y^{(n-k-1)}(1)=0\), for some \(k\in \{ 1,\ldots ,n-1\}\). A piecewise polynomial, which bounds the function \(y\) from below, is constructed, and then is employed to obtain that \(y(t)\geq ||y||/4^{m}, 1/4\leq t\leq 3/4\), where \(m=\max\{ k, n-k\}\) and \(||\cdot ||\) denotes the supremum norm. An analogous inequality for a related Green’s function is also obtained. These inequalities are useful in applications of certain cone theoretic fixed point theorems. Cited in 1 ReviewCited in 19 Documents MSC: 34A40 Differential inequalities involving functions of a single real variable 34B27 Green’s functions for ordinary differential equations Keywords:differential inequalities; concavity; \(n\)th order differential inequalities; homogeneous two-point boundary inequalities; related Green’s function; cone theoretic fixed point theorems × Cite Format Result Cite Review PDF Full Text: DOI References: [1] E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. · Zbl 0064.33002 [2] W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, Berlin-New York, 1971. · Zbl 0224.34003 [3] Paul W. Eloe and Johnny Henderson, Singular nonlinear boundary value problems for higher order ordinary differential equations, Nonlinear Anal. 17 (1991), no. 1, 1 – 10. · Zbl 0731.34015 · doi:10.1016/0362-546X(91)90116-I [4] Paul W. Eloe and Johnny Henderson, Positive solutions for higher order ordinary differential equations, Electron. J. Differential Equations (1995), No. 03, approx. 8 pp., issn=1072-6691, review=\MR{1316529},. · Zbl 0814.34017 [5] L. H. Erbe and Haiyan Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120 (1994), no. 3, 743 – 748. · Zbl 0802.34018 [6] J. A. Gatica, Vladimir Oliker, and Paul Waltman, Singular nonlinear boundary value problems for second-order ordinary differential equations, J. Differential Equations 79 (1989), no. 1, 62 – 78. · Zbl 0685.34017 · doi:10.1016/0022-0396(89)90113-7 [7] Ferhan Merdivenci, Green’s matrices and positive solutions of a discrete boundary value problem, PanAmer. Math. J. 5 (1995), no. 1, 25 – 42. · Zbl 0839.39002 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.