Abstract
Let A be an n×n essentially nonnegative matrix and consider the linear differential system . We show that there exists a constant h(A)>0 such that the trajectory emanating from x0
reaches
at a finite time t0
=t(x
0)⩾0 if and only if the sequence of points generated by a finite differences approximation from x0
, with time-step 0<h<h(A), reaches
at a finite index k0
=k(x
0)⩾0. This generalizes and strengthens earlier results of two of the authors, where some additional spectral restrictions were imposed on A. Our proof makes use of the existence of a basis of nonnegative vectors to the Perron eigenspace.
*Research supported in part by US Air Force Research Grant No. AFOSR-88-0047 and by NSF Grant No. DMS-8901860. This author would also like to thank NSERC for making it possible for him to visit Ronald J. Stern in Montreal.
**Research supported by the Natural Sciences and Engineering Council of Canada, grant No. A4641
***Research supported in part by US Air Force Research Grant No. AFOSR-88-0047 and by NSF Grant No. DMS-8901860.
*Research supported in part by US Air Force Research Grant No. AFOSR-88-0047 and by NSF Grant No. DMS-8901860. This author would also like to thank NSERC for making it possible for him to visit Ronald J. Stern in Montreal.
**Research supported by the Natural Sciences and Engineering Council of Canada, grant No. A4641
***Research supported in part by US Air Force Research Grant No. AFOSR-88-0047 and by NSF Grant No. DMS-8901860.
Notes
*Research supported in part by US Air Force Research Grant No. AFOSR-88-0047 and by NSF Grant No. DMS-8901860. This author would also like to thank NSERC for making it possible for him to visit Ronald J. Stern in Montreal.
**Research supported by the Natural Sciences and Engineering Council of Canada, grant No. A4641
***Research supported in part by US Air Force Research Grant No. AFOSR-88-0047 and by NSF Grant No. DMS-8901860.