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Extension of continuous and discrete inequalities due to Eloe and Henderson. (English) Zbl 0933.34008

If \(p\) is a fixed integer \((1,2, \dots, n-1)\) and \[ y^{(n-p)} (t)\geq 0,\quad t\in[0,1],\tag{1} \] with \[ y^{(i)}(0)=0\;(0\leq i\leq p-1),\quad y^i(1)=0\;(0\leq i\leq n-p-1),\tag{2} \] P. W. Eloe and J. Henderson [Electron. J. Differ. Equ. 1997, Paper 3, 11 p. (1997; Zbl 0888.34013), and Nonlinear Anal., Theory Meth. Appl. 28, No. 10, 1669-1680 (1997; Zbl 0871.34015)] have shown that \(y(t)\) has the lower bound \(\sup_{t\in[0,1]}|y(t) |/4^r\) for \(t\in[{1\over 4},{3\over 4}]\) and \(r=\max(p,n-p)\). The authors extend the above result and sharpen it, using a different technique. The main result of the paper is the theorem:
If \(y(t)\in C^n[0,1]\) and satisfies (1) and (2), a lower bound for \(y(t)\) is \(A*\sup_{t\in [0,1]}|y(t)|\) for \(t\in[\delta,1- \delta]\), with \(A=\min\{b(p) \min[c(p), c(n-p-1)]\), \(b(p-1) \min [c(p-1), c(n-p)]\}\), \(\delta\in(0, {1\over 2})\), \(b(x)={(n-1)^{n-1}\over n^x (n-x-1)^{n-x-1}}\), and \(c(x)= \delta^x(1- \delta)^{n-x-1}\). The authors discuss a discrete version of the problem involving an inequality for the forward differences with step equal to one. In the continuous and the discrete cases, lower bounds are obtained for Green’s functions to the analogous boundary value problems.

MSC:

34A40 Differential inequalities involving functions of a single real variable
34B27 Green’s functions for ordinary differential equations
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