By Franklin Mendivil
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Extra resources for Fractals, Graphs, and Fields
50 © 2004 by CRC Press LLC © 2004 by Chapman & Hall/CRC 26 D. Zeilberger Continuous analysis and geometry are just degenerate approximations to the discrete world, made necessary by the very limited resources of the human intellect. While discrete analysis is conceptually simpler (and truer) than continuous analysis, technically it is (usually) much more difficult. e. discrete, analysis. When we watch a movie we have the appearance of continuity, but in fact it consists of a discrete sequence of frames.
This is the analogue of what happens when one numerically solves the linear test problem y’=y, <0 and finds conditions (A-stability) to retain, inside the discrete solution, the qualitative asymptotic behavior of the continuous one. Indeed, the analytic solution of the continuous Verhulst equation clearly shows the global asymptotic stability nature of its equilibrium , no matter how the initial condition p0 or the growth parameter r>0 are chosen; we would like to reproduce this behavior after the application of a suitable method.
Hence, to characterize the asymptotic behavior of trajectories (or appropriate solutions), it is necessary to take a space with a large stock of functions and with a metric other than the metric of uniform convergence. One possibility is to use the Hausdorff metric for the graphs of functions, as in [19, 20], or the equivalent metric where wε(x)=w(Vε(x)), Vε(x) is the ε-neighborhood of a point x, and is the Euclidean distance. If we complete the space of C1-functions [0, 1]→R via this metric, we obtain a compact space, denoted further by C∆([0, 1], R).