Nonlinear Theory and Its Applications, IEICE
Online ISSN : 2185-4106
ISSN-L : 2185-4106
Special section on Recent Progress in Nonlinear Theory and Its Applications
Explicit proof of an inequality related to the Omega-matrix
Tetsuo NishiShin'ichi OishiNorikazu Takahashi
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2013 Volume 4 Issue 4 Pages 430-450


The authors recently published a paper on some properties of the solution curves for the last n-1 equations of F(x)+Ax=b where x=[x1,x2,...,xn]T is a variable, F(x):Rn → Rn is a nonlinear function of which the first and second derivatives are strictly positive, A ∈ Rn × n is an Ω-matrix, and b ∈ Rn is a constant vector. In that paper, the authors showed that any solution curve possesses neither maximal points nor inflection points with respect to x1, by making use of a fundamental property of Ω-matrices, which is expressed in the form of inequality. However, the proof was a little unclear as shown in Introduction. The objective of this paper is to give an explicit proof for the property of Ω-matrices, which makes the author's previous result more rigorous.

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© 2013 The Institute of Electronics, Information and Communication Engineers
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