Inequalities based on a generalization of concavity
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- by Paul W. Eloe and Johnny Henderson PDF
- Proc. Amer. Math. Soc. 125 (1997), 2103-2107 Request permission
Abstract:
The concept of concavity is generalized to functions, $y$, satisfying $nth$ order differential inequalities, $(-1)^{n-k}y^{(n)}(t)\ge 0, 0\le t\le 1$, 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, $y$, below, is constructed, and then is employed to obtain that $y(t)\ge ||y||/4^{m}, 1/4\le t\le 3/4$, 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.References
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Additional Information
- Paul W. Eloe
- Affiliation: Department of Mathematics, University of Dayton, Dayton, Ohio 45469-2316
- MR Author ID: 63110
- ORCID: 0000-0002-6590-9931
- Email: eloe@saber.udayton.edu
- Johnny Henderson
- Affiliation: Department of Mathematics, 218 Parker Hall, Auburn University, Alabama 36849-5310
- MR Author ID: 84195
- ORCID: 0000-0001-7288-5168
- Email: hendej2@mail.auburn.edu
- Received by editor(s): July 12, 1995
- Received by editor(s) in revised form: February 6, 1996
- Communicated by: Hal L. Smith
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 125 (1997), 2103-2107
- MSC (1991): Primary 34A40; Secondary 34B27
- DOI: https://doi.org/10.1090/S0002-9939-97-03800-8
- MathSciNet review: 1376760