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Revision of Use the continuity method from Sat, 09/05/2009 - 03:02

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Quick description

If you can show that the set of parameters obeying a property P is non-empty, open, and closed, and the parameter space is connected, then P must be obeyed by all choices of the parameter. Thus, for instance, if one wants to prove a property P(t) for all t in some interval I, it suffices to establish the following three facts:

  • A base case P(t_0) for some t_0 \in I;

  • (Openness) If P(t) is true for some t \in I, then P(t') is true for all t' \in I sufficiently close to t;

  • (Closedness) If P(t_n) is true for some sequence t_n\in I converging to a limit t \in I, then P(t) is also true.

Prerequisites

Point-set topology; partial differential equations

Example 1

(Solving an ODE in a potential well)