zbMATH — the first resource for mathematics

Inequalities based on a generalization of concavity. (English) Zbl 0868.34008
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.

34A40 Differential inequalities involving functions of a single real variable
34B27 Green’s functions for ordinary differential equations
Full Text: DOI
[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
[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
[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. 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.