Hilbert's Projective Metric and the Norm on a Banach Space
Received:December 02, 2008  Revised:January 18, 2010
Key Word: Hilbert's projective metric   normal and solid cone   norm.
Fund ProjectL:Supported by the Natural Science Foundation of Shanxi Province (Grant No.20041003) and the Youth Science Foundation of Shanxi Province (Grant No.2010021002-1).
 Author Name Affiliation Cheng Bo ZHAI School of Mathematical Sciences, Shanxi University, Shanxi 030006, P. R. China Zhan Dong LIANG School of Mathematical Sciences, Shanxi University, Shanxi 030006, P. R. China
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In this paper, we establish some relations between the Hilbert's projective metric and the norm on a Banach space and show that the metric and the norm induce equivalent convergences at certain set. As applications, we utilize the main results to discuss the eigenvalue problems for a class of positive homogeneous operators of degree $\alpha$ and the positive solutions for a class of nonlinear algebraic system.